Appendix A: Estimation of b(L)
We use the following procedure to estimate b(L) given by Eq. (11) from the actual data:
-
1.
We fix L.
-
2.
We calculate \(E[F_j^{(L)}]\) and \(V[\delta F_j^{(L)}]\) for all words \((j=1,2,\ldots ,W)\).
-
3.
We minimize \(Q^{(L)}\) with respect to b(L) under the condition \(b(L)>0\).
Here, \(Q^{(L)}\) is defined by
$$\begin{aligned}&Q^{(L)} \equiv s \cdot \left\{ \sum _{j \in \{j|Q_j^{(L)}>0,\; C_j \ge 100\}}\frac{{Q_j^{(L)}}^2}{N_p}\right\} \nonumber \\&\quad +(1-s) \cdot \left\{ \sum _{j \in \{j|Q_j^{(L)}<0,\; C_j \ge 100\}} \frac{{Q_j^{(L)}}^2}{N_m} \right\} , \end{aligned}$$
(A1)
where
$$\begin{aligned} Q_j^{(L)}=\log {(V[\delta F_j^{(L)}(\tau ^{L})]^{1/2})}-\log {((\frac{2}{L} \cdot C_j+b(L) \cdot C_j^2)^{1/2})} , \nonumber \\ \end{aligned}$$
(A2)
\(C_j=E[F_j^{(L)}]\), \(N_p=\sum _{j \in \{j|Q_j^{(L)}>0, C_j \ge 100\}} 1\), and \(N_m=\sum _{j \in \{j|Q_j^{(L)}<0,\; C_j \ge 100\}}1\). Note that the minimization of the first term of Eq. (A2) corresponds to a reduction of the data beneath the theoretical lower bound in Eq. (11). However, when we use only the first term, the estimation of b(L) is strongly affected by outliers. Therefore, we use the second term to accept the data beneath the theoretical curve, and s is the parameter used to control the ratio of acceptance. Here, we use \(s=0.9\) in our analysis. In addition, the reason why we only use \(\check{c}_j \ge 100\) is that we neglect words with a small \(c_j\), as these do not affect the estimation b(L) (see Eq. (11) and Fig. 2).
Appendix B: \(V[\delta F^{(L)}_j]\) for given \(\{r_j(t)\}\)
We calculate \(V[ \delta F_j^{(L)}]\) for given \(\{r_j(t)\}\). Using Eq. (30), we can decompose \(V[ \delta F_j^{(L)}]\) as
$$\begin{aligned}&V[ \delta F_j^{(L)} (\tau ^{(L)})]=\check{C_j}^2 V[\delta R_j^{(L)}]+ V[\delta W_j^{(L)}] \nonumber \\&\quad + 2 \cdot \check{C}_j E[ \delta R_j^{(L)} \delta W_j^{(L)}]. \end{aligned}$$
(B1)
First, we calculate the second term in Eq. (B1). The second term is written as \(V[\delta W_j^{(L)}]=E[\delta {W_j^{(L)}}^2]-E[\delta {W_j^{(L)}}]^2\), where \(E[\delta {W_j^{(L)}}^2]\) is given by
$$\begin{aligned} E[\delta {W_j^{(L)}}^2]=&\frac{1}{T^{(L)}-1} \sum ^{T^{(L)}-1}_{t^{(L)}=1} \{W_j^{(L)}(t^{(L)}+1) \nonumber \\&-W^{(L)}(t^{(L)})\}^2 \end{aligned}$$
(B2)
$$\begin{aligned} \approx&\frac{1}{L} \left\{ \sum ^{T}_{t=L+1}\frac{w_j(t)^2}{T-L} + \sum ^{T-L}_{t=1} \frac{w_j(t)^2}{T-L} \right\} , \nonumber \\ \end{aligned}$$
(B3)
where we use the assumption that \(T>>1\), \(<w_j(t)>=0\), and \(\{w_j(t)\}\) are independently distributed random variables. Approximating the sums in Eq. (B3) by Eq. (26) (using the assumption \(T>>1\) and \(L \le T/2\)), we write
$$\begin{aligned} E[\delta {W_j^{(L)}}^2] \approx \frac{2}{L}\{ E[1/m] \cdot \check{c}_j + {\check{\delta }_j}^2 (1+V[r_j]) \check{c}_j^2 \}. \end{aligned}$$
(B4)
In the calculation, we also use the approximations
$$\begin{aligned}&\sum ^{T}_{t=L+1} \frac{r_j(t)}{T-L} \approx 1, \end{aligned}$$
(B5)
$$\begin{aligned}&\sum ^{T-L}_{t=1} \frac{r_j(t)}{T-L} \approx 1, \end{aligned}$$
(B6)
and
$$\begin{aligned}&\frac{1}{2}\Bigg \{\sum ^{T}_{t=L+!}\frac{\{r_j(t)-\sum ^{T}_{t=L+1}\frac{r_j(t)}{T-L}\}^2}{T-L} \nonumber \\&\qquad + \sum ^{T-L}_{t=1}\frac{\{r_j(t)-\sum ^{T-L}_{t=1}\frac{r_j(t)}{T-L}\}^2}{T-L}\Bigg \} \nonumber \\&\quad \approx V[r_j]. \end{aligned}$$
(B7)
These approximations are based on the assumption that \(T>>L\) and \(\{r_j(t)\}\) do not have a particular trend.
Next, we calculate \(E[\delta {W_j^{(L)}}]^2\). Using Eq. (29), we can estimate \(E[\delta {W_j^{(L)}}]^2\) as follows:
$$\begin{aligned} E[\delta {W_j^{(L)}}]^2= & {} \left\{ \frac{W_j^{(L)}(T^{(L)})-W_j^{(L)}(1)}{T^{(L)}-1} \right\} ^2 \nonumber \\\approx & {} \left\{ \frac{O(1/\sqrt{L})}{T^{(L)}-1} \right\} ^2 \approx O(\frac{L}{(T-L)^2}) \end{aligned}$$
(B8)
$$\begin{aligned}\approx & {} {\left\{ \begin{array}{ll} O(1/T^2) &{}(L<<T), \\ O(1/T) &{} (L \approx T). \end{array}\right. } \end{aligned}$$
(B9)
Therefore, we can neglect this term for \(T>>1\).
Consequently, inserting Eqs. (B4) and (B9) into Eq. (B1), and using \(E[\delta C_j \delta W_j] \approx O(\sqrt{1/T}) \approx 0\) \((T>>1)\), we can obtain
$$\begin{aligned}&V[\delta F_j^{(L)}] \approx a(L) \check{C}_j +b(L) \check{C}_j^2 , \end{aligned}$$
(B10)
where
$$\begin{aligned} a(L)=\frac{2}{L} a_0 \end{aligned}$$
(B11)
and
$$\begin{aligned} b(L)= V[\delta R_j^{(L)}]+\frac{2b_0(1+V[r_j])}{L}. \end{aligned}$$
(B12)
Here, \(a_0=E[1/m]\) and \(b_0=\check{\delta }_j^2\).
Appendix C: \(V[\delta R^{(L)}_j]\) for a random walk
We calculate \(V[\delta R^{(L)}_j]\) for the following random walk with dissipation \(\kappa \ge 0\) and external force u(t):
$$\begin{aligned} r(t+1)=\kappa \cdot r(t)+u(t)+\eta (t), \end{aligned}$$
(C1)
where \(<\eta (t)>_{\eta }=0\), \(<(\eta (t)-<\eta (t)>_{\eta })^2>_{\eta }=\check{\eta }^2<<1\), \(u(t)>0\), and \(\kappa \ge 0\) and we omit the subscript j. Using Eq. (C1), R(I), defined by 28, is written as
$$\begin{aligned} R(I)= & {} \frac{1}{L} \sum ^{(LI-1)}_{t'=(I-1)L}\Bigg [\sum ^{t'-1}_{k=0} \kappa ^k u(1+t'-k) \nonumber \\&+ \kappa ^{t'} r(1) + \sum ^{t'-1}_{k=0}\kappa ^k \eta (1+t'-k)\Bigg ] \nonumber \\= & {} P_1(I)+P_2(I)+P_3(I). \end{aligned}$$
(C2)
Here, we define \(P_1(I)\), \(P_2(I)\), and \(P_3(I)\) as follows:
$$\begin{aligned}&P_1(I) \equiv \frac{1}{L} \sum ^{(LI-1)}_{t'=(I-1)L} \sum ^{t-1}_{k=0} \kappa ^k \cdot u(1+t'-k), \end{aligned}$$
(C3)
$$\begin{aligned}&P_2(I) \equiv \frac{1}{L} \sum ^{(LI-1)}_{t'=(I-1)L} \kappa ^{t'} \cdot r(1), \end{aligned}$$
(C4)
$$\begin{aligned}&P_3(I) \equiv \frac{1}{L} \sum ^{(LI-1)}_{t'=(I-1)L} \sum ^{t-1}_{k=0} \kappa ^k \cdot \eta (1+t'-k). \end{aligned}$$
(C5)
Because \(V[\delta R^{(L)}]\) can be decomposed as
$$\begin{aligned} V[\delta R^{(L)}_j] =V[\delta R]=E[\delta R^2]-E[\delta R]^2, \end{aligned}$$
(C6)
we calculate \(E[\delta R^2]\) and \(E[\delta R]^2\), respectively.
Calculation of \(E[\delta R^2]\). Here, we calculate the first term of Eq. (C6), \(E[\delta R^2]\). \(\delta R(I)^2\) is denoted by
$$\begin{aligned}&\delta R_j(I)^2 \nonumber \\&= (\delta P_1(I)+ \delta P_2(I)+ \delta P_3(I))^2 \nonumber \\&\approx \delta P_1(I)^2+ \delta P_2(I)^2+\delta P_3(I)^2+ 2 \cdot \delta P_1(I) \cdot \delta P_2(I). \nonumber \\ \end{aligned}$$
(C7)
We estimate the effects of the first term in Eq. (C7), \(\delta P_1(I)^2\). \(\delta P_1(I)\) can be written as
$$\begin{aligned} \delta P_1(I)=\frac{1}{L^2} \cdot \sum ^{L(I+1)}_{t=2}g_0(t,I) u(t), \end{aligned}$$
(C8)
where
$$\begin{aligned}&g_0(t,T) = \nonumber \\&{\left\{ \begin{array}{ll} \frac{ (\kappa ^L-1)^2 \cdot \kappa ^{-t+1} \cdot \kappa ^{(I-1)L} }{\kappa -1}, &{} 2 \le t \le (I-1)L+1,\\ \frac{(\kappa ^L-2) \cdot \kappa ^{(-t+1)} \cdot \kappa ^{I \cdot L}+1}{\kappa -1}, &{} (I-1)L+2 \le t \le LI+1, \\ \frac{\kappa ^{(-t+1)} \cdot \kappa ^{(I+1)L}-1}{\kappa -1}, &{} IL+2 \le t \le L(I+1). \\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(C9)
In addition, using the variables
$$\begin{aligned} u_0=\sum ^{T}_{t=1}u(t)/T \end{aligned}$$
(C10)
and
$$\begin{aligned} \delta u'(t)=u(t)-u_0, \end{aligned}$$
(C11)
we can write
$$\begin{aligned}&\delta P_1(I)^2 \nonumber \\&\quad = \frac{1}{L^2} \left\{ \sum ^{L(I+1)}_{t=2}g_0(t,I) u(t) \right\} ^2 \end{aligned}$$
(C12)
$$\begin{aligned}&=\frac{1}{L^2} \left\{ u_0^2 \cdot \left[ \sum ^{L(I+1)}_{t=2}g_0(t,I)\right] ^2+ \left[ \sum ^{L(I+1)}_{t=2}g_0(t,I) u'(t)\right] ^2 \right\} ^2 \nonumber \\&\approx \frac{1}{L^2} u_0^2 \cdot \left\{ \sum ^{L(I+1)}_{t=2}g_0(t,I) \right\} ^2 \end{aligned}$$
(C13)
$$\begin{aligned}&= \frac{1}{L^2} u_0^2 \cdot \left[ \frac{\kappa ^{(I-1)L} \cdot (\kappa ^L-1)^2}{(\kappa -1)^2}\right] ^2. \end{aligned}$$
(C14)
Hence, the temporal average of \(\delta P_1(I)^2\) is obtained by
$$\begin{aligned}&\bar{P}_1 \equiv \sum ^{T^{(L)}-1}_{I=1} \frac{\delta P_1(I)^2}{T^{(L)}-1} \end{aligned}$$
(C15)
$$\begin{aligned}&\approx \frac{1}{T^{(L)}-1} \cdot \sum ^{T^{(L)}-1}_{I=1} \frac{1}{L^2} u_0^2 \cdot [\frac{\kappa ^{(I-1)L} \cdot (\kappa ^L-1)^2}{(\kappa -1)^2}]^2 \nonumber \\&\quad = \frac{u_0^2}{L^2 \cdot (T^{(L)}-1)} \frac{(\kappa ^L-1)^3 (\kappa ^{2L(T^{(L)}-1)}-1)}{(\kappa -1)^4 \cdot (\kappa ^L+1)}.\nonumber \\ \end{aligned}$$
(C16)
Similarly, we estimate the effects of \(\delta P_2(I)^2\) as
$$\begin{aligned} \delta P_2(I)^2= & {} \frac{r(1)}{L^2} \cdot \frac{\kappa ^{2IL}(\kappa ^L+\kappa ^{-L}-2)^2}{(\kappa -1)^2}. \end{aligned}$$
(C17)
Therefore, the temporal average of \(\delta P_2(I)^2\) is obtained by
$$\begin{aligned}&\bar{P}_2 \equiv \sum ^{T^{(L)}-1}_{I=1}\frac{\delta P_2(I)^2}{T^{(L)}-1} \end{aligned}$$
(C18)
$$\begin{aligned}&\quad = \frac{r(1)^2}{L^2 \cdot (T^{(L)}-1)} \frac{(\kappa ^L-1)^3 (\kappa ^{2L(T^{(L)}-1)}-1)}{(\kappa -1)^2 \cdot (\kappa ^L+1)}.\nonumber \\ \end{aligned}$$
(C19)
Lastly, we investigate the effects of \(\delta P_3(I)^2\), i.e.,
$$\begin{aligned} \delta P_3(I)^2= & {} \frac{1}{L^2} \left\{ \sum ^{L(I+1)}_{t=2}g_0(t,I) \eta (t) \right\} ^2 \end{aligned}$$
(C20)
$$\begin{aligned}\approx & {} \frac{1}{L^2} \eta _0^2 \cdot \sum ^{L(I+1)}_{t=2}g_0(t,I)^2, \end{aligned}$$
(C21)
where, from the definition,
$$\begin{aligned} \eta _0^2 \approx \sum ^{T}_{t=1}\eta (t)^2/T. \end{aligned}$$
(C22)
We can calculate the sum of g(t, I) with respect to t as
$$\begin{aligned} \sum ^{L(I+1)}_{t=2}g_0(t,I)^2=Q_1(I)+Q_2+Q_3, \end{aligned}$$
(C23)
where
$$\begin{aligned}&Q_1(I)\equiv \sum ^{(I-1)L+1}_{t=2}g_0(t,I)^2 =\frac{(\kappa ^L-1)^4 \cdot (\kappa ^{2(I-1)L}-1)}{(\kappa -1)^3 \cdot (\kappa +1)}, \nonumber \\ \end{aligned}$$
(C24)
$$\begin{aligned}&Q_2 \equiv \sum ^{LI+1}_{(I-1)L+2}g_0(t,I)^2 \nonumber \\&\quad =\frac{L(\kappa ^2-1)+(\kappa ^{2L}-3\kappa ^L+2)(\kappa ^{2L}-\kappa ^{L}+2 \kappa )}{(\kappa -1)^3 \cdot (\kappa +1)},\nonumber \\ \end{aligned}$$
(C25)
$$\begin{aligned}&Q_3 \equiv \sum ^{(I+1)L}_{LI+2}g_0(t,I)^2 \nonumber \\&\quad =\frac{(\kappa ^L-1)(\kappa ^L-2\kappa -1)+L(\kappa ^2-1)}{(\kappa -1)^3 \cdot (\kappa +1)}. \end{aligned}$$
(C26)
From these results, we can obtain the temporal average of \(\delta P_3(I)^2\) as
$$\begin{aligned}&\bar{P}_3 \equiv \sum ^{T^{(L)}-1}_{I=1}\frac{\delta P_3(I)^2}{T^{(L)}-1} \end{aligned}$$
(C27)
$$\begin{aligned}&\quad =\frac{\check{\eta }^2}{L^2(T^{(L)}-1)}\cdot (\bar{P}_{3a}+\bar{P}_{3b}+\bar{P}_{3c}). \end{aligned}$$
(C28)
Here,
$$\begin{aligned}&\bar{P}_{3a}\nonumber \\&\quad = \frac{(\kappa ^L-1)^3 \cdot (\kappa ^{2L(T^{(L)}-1)}-(T^{(L)}-1)\kappa ^{2L}+T^{(L)}-2)}{(\kappa -1)^3 (\kappa +1) (\kappa ^L+1)}, \nonumber \\ \end{aligned}$$
(C29)
$$\begin{aligned}&\bar{P}_{3b}=(T^{(L)}-1) \nonumber \\&\quad \times \frac{L \cdot (\kappa ^2-1) +(\kappa ^{2L}-3\kappa ^L+2)(\kappa ^{2L}-\kappa ^L+2\kappa )}{(\kappa -1)^3 (\kappa +1)}, \end{aligned}$$
(C30)
$$\begin{aligned}&\bar{P}_{3c}=(T^{(L)}-1) \cdot \frac{L \cdot (\kappa ^2-1) + (\kappa ^L-1)(\kappa ^L-2\kappa -1)}{(\kappa -1)^3 (\kappa +1)}. \nonumber \\ \end{aligned}$$
(C31)
Calculation of \(E[\delta R]^2\). Next, we calculate \(E[\delta R]^2\). \(E[\delta R]^2\) can be decomposed as follows:
$$\begin{aligned}&E[\delta R]^2=(E[\delta P_1]+E[\delta P_2]+E[\delta P_3])^2 \nonumber \\&\quad = E[\delta P_1]^2+E[\delta P_2]^2+2 \cdot E[\delta P_1] \cdot E[\delta P_2]\nonumber \\ \end{aligned}$$
(C32)
$$\begin{aligned}&+ 2E[\delta P_3](E[\delta P_1]+E[\delta P_2]+E[\delta P_3]) \nonumber \\&\quad \approx E[\delta P_1]^2+E[\delta P_2]^2+2 \cdot E[\delta P_1] \cdot E[\delta P_2],\nonumber \\ \end{aligned}$$
(C33)
where we use \(E[\delta P_3] \approx 0\).
\(E[\delta P_1]\) and \(E[\delta P_2]\) are obtained as
$$\begin{aligned}&E[\delta P_{1}]=P_{1}[T^{(L)}]-P_{1}[1] \end{aligned}$$
(C34)
$$\begin{aligned}&\,\, =\frac{u_0}{T^{(L)}-1} \cdot \frac{1}{L}\left[ \frac{-\kappa ^{(T^{(L)}-1)L}+\kappa ^{T^{(L)}L}-\kappa ^L+1}{(\kappa -1)^2}\right] , \nonumber \\ \end{aligned}$$
(C35)
$$\begin{aligned}&E[\delta P_{2}]=P_{2}[T^{(L)}]-P_{2}[1] \end{aligned}$$
(C36)
$$\begin{aligned}&\,\, = \frac{r(1)}{T^{(L)}-1} \cdot \frac{1}{L}\left[ \frac{-\kappa ^{(T^{(L)}-1)L}+\kappa ^{T^{(L)}L}-\kappa ^L+1}{\kappa -1}\right] . \nonumber \\ \end{aligned}$$
(C37)
Calculation of \(V[R^{(L)}]\). Lastly, we calculate \(V[R^{(L)}]\). Substituting Eqs. (C7) and (C37) for Eq. (C6), we can obtain
$$\begin{aligned}&V[R^{(L)}] \nonumber \\&\,\, \approx R^{(1)} \cdot u_0^2 +R^{(2)} \cdot \check{\eta }^2 +R^{(3)} \cdot r(1)^2+R^{(4)} \cdot u_0 \cdot r(1),\nonumber \\ \end{aligned}$$
(C38)
Where, from Eqs. (C16) and (C35C36),
$$\begin{aligned}&R^{(1)} =R_1^{(1)}+R_2^{(1)}, \end{aligned}$$
(C39)
$$\begin{aligned}&R_1^{(1)}= \frac{1}{L^2 \dot{(}T^{(L)}-1)}\left[ \frac{(\kappa ^L-1)^3 \cdot (\kappa ^{2L(T^{(L)}-1)}-1)}{(\kappa -1)^4 \cdot (\kappa ^L+1)}\right] ,\nonumber \\ \end{aligned}$$
(C40)
$$\begin{aligned}&R_2^{(1)}=-\frac{1}{L^2 \cdot (T^{(L)}-1)^2} \cdot W_1^2, \end{aligned}$$
(C41)
$$\begin{aligned}&W_1=\frac{\kappa ^{T^{(L)}L}-\kappa ^{(T^{(L)}-1)L}+1-\kappa ^L}{(\kappa -1)^2}; \end{aligned}$$
(C42)
from Eq. (C28),
$$\begin{aligned}&R^{(2)} =R_1^{(2)}+R_2^{(2)}+R_3^{(2)}, \end{aligned}$$
(C43)
$$\begin{aligned}&R_{1}^{(2)}= \frac{1}{L^2 \cdot (T^{(L)}-1)} \nonumber \\&\quad \times \frac{(\kappa ^L-1)^3 \cdot (\kappa ^{2L(T^{(L)}-1)}-(T^{(L)}-1)\kappa ^{2L}+T^{(L)}-2)}{(\kappa -1)^3 (\kappa +1) (\kappa ^L+1)}, \nonumber \\ \end{aligned}$$
(C44)
$$\begin{aligned}&R_{2}^{(2)}=\frac{1}{L^2}\nonumber \\&\quad \cdot \frac{L \cdot (\kappa ^2-1) +(\kappa ^{2L}-3\kappa ^L+2)(\kappa ^{2L}-\kappa ^L+2\kappa )}{(\kappa -1)^3 (\kappa +1)}, \end{aligned}$$
(C45)
$$\begin{aligned}&{R}_{3}^{(2)}=\frac{1}{L^2} \cdot \frac{L \cdot (\kappa ^2-1) + (\kappa ^L-1)(\kappa ^L-2\kappa -1)}{(\kappa -1)^3 (\kappa +1)}; \end{aligned}$$
(C46)
from Eqs. (C19) and (C37),
$$\begin{aligned}&R^{(3)} =R_1^{(3)}+R_2^{(3)}, \end{aligned}$$
(C47)
$$\begin{aligned}&R_{1}^{(3)} = \frac{1}{L^2 \cdot (T^{(L)}-1)} \frac{(\kappa ^L-1)^3 \cdot (\kappa ^{2L(T^{(L)}-1)}-1)}{(\kappa -1)^2 \cdot (\kappa ^L+1)},\nonumber \\ \end{aligned}$$
(C48)
$$\begin{aligned}&R_{2}^{(3)}=-\frac{1}{L^2 \cdot (T^{(L)}-1)^2} \cdot W_2^2, \end{aligned}$$
(C49)
$$\begin{aligned}&W_2=\frac{\kappa ^{T^{(L)}L}-\kappa ^{(T^{(L)}-1)L}+1-\kappa ^L}{\kappa -1}; \end{aligned}$$
(C50)
and, from Eqs. C35 and C37,
$$\begin{aligned} R^{(4)}=-\frac{2}{L^2 \cdot (T^{(L)}-1)^2} \cdot W_1 \cdot W_2. \end{aligned}$$
(C51)
1.1 Appendix C.1: Calculation of \(V[{R^{(L)}}^2]\) for large L
We calculate \(V[{R^{(L)}}^2]\) for \(L>>1\). Here, we assume \(L/2 \ge T\).
Case of \(\kappa <1\):
Using \(\kappa ^L \approx 0\) and \(\kappa ^T \approx 0\), we can obtain
$$\begin{aligned}&R_1^{(1)} \approx \frac{1}{L(T-L)} \frac{1}{(\kappa -1)^4}, \end{aligned}$$
(C52)
$$\begin{aligned}&R_2^{(1)} \approx \frac{1}{(T-L)^2} \frac{1}{(\kappa -1)^4}, \end{aligned}$$
(C53)
$$\begin{aligned}&R_1^{(2)} \approx \frac{2L-T}{L^2 (T-L)} \cdot \frac{1}{(\kappa -1)^3 \cdot (\kappa +1)}, \end{aligned}$$
(C54)
$$\begin{aligned}&R_2^{(2)} \approx \frac{1}{L} \cdot \frac{1}{(\kappa -1)^2 }+\frac{1}{L^2} \cdot \frac{4k}{(\kappa -1)^3 \cdot (\kappa +1)},\nonumber \\ \end{aligned}$$
(C55)
$$\begin{aligned}&R_3^{(2)} \approx \frac{1}{L} \cdot \frac{1}{(\kappa -1)^2 }+\frac{1}{L^2} \cdot \frac{2\kappa +1}{(\kappa -1)^3 \cdot (\kappa +1)},\nonumber \\ \end{aligned}$$
(C56)
$$\begin{aligned}&R_1^{(3)} \approx \frac{1}{L(T-L)} \frac{1}{(\kappa -1)^2}, \end{aligned}$$
(C57)
$$\begin{aligned}&R_2^{(3)} \approx \frac{1}{(T-L)^2} \frac{1}{(\kappa -1)^4}, \end{aligned}$$
(C58)
$$\begin{aligned}&R^{(4)} \approx \frac{1}{(T-L)^2} \frac{1}{(\kappa -1)^3}. \end{aligned}$$
(C59)
Considering only dominant terms, we can obtain
$$\begin{aligned} V[\delta R^{(L)}]\approx \frac{2}{L} \frac{ \check{\eta }^2 }{(\kappa -1)^2 } + \frac{1}{L(T-L)}( \frac{u_0^2}{(\kappa -1)^4}+ \frac{r(1)^2}{(\kappa -1)^2}). \nonumber \\ \end{aligned}$$
In the case of \(T-L>>1\), we can obtain the simpler form
$$\begin{aligned} V[\delta R^{(L)}] \approx \check{\eta }^2 \frac{2}{L} \cdot \frac{1}{(\kappa -1)^2 } \propto \frac{1}{L}. \end{aligned}$$
(C60)
Case of \(\kappa =1\): Next, we calculate the case of \(\kappa =1\). Taking the limit of \(\kappa \rightarrow 1\), we get
$$\begin{aligned}&R_1^{(1)} \approx L^2, \end{aligned}$$
(C61)
$$\begin{aligned}&R_2^{(1)} \approx -L^2, \end{aligned}$$
(C62)
$$\begin{aligned}&R_1^{(2)} \approx 0, \end{aligned}$$
(C63)
$$\begin{aligned}&R_2^{(2)} \approx \frac{2L^3+3L^2+L}{6} \cdot \frac{1}{L^2}, \end{aligned}$$
(C64)
$$\begin{aligned}&R_3^{(2)} \approx \frac{2L^3-3L^2+L}{6} \cdot \frac{1}{L^2}, \end{aligned}$$
(C65)
$$\begin{aligned}&R_1^{(3)} \approx 0, \end{aligned}$$
(C66)
$$\begin{aligned}&R_2^{(3)} \approx 0, \end{aligned}$$
(C67)
$$\begin{aligned}&R^{(4)} \approx 0. \end{aligned}$$
(C68)
From these results, we can obtain
$$\begin{aligned} V[R^{(L)}] \approx \frac{1}{3} \cdot \check{\eta }^2 \cdot (2L+1/L) \propto L. \end{aligned}$$
(C69)
Case of \( \kappa >1\): Lastly, we calculate the case of \( \kappa >1\) for \(L>>1\).
Calculation of \(R^{(1)}\). For \(L>>1\), in the case of \( \kappa >1\), a dominant term of \(R_1^{(1)}\) is given by
$$\begin{aligned} R_1^{(1)} \approx \frac{1}{L(T-L)} \frac{k^{2T}}{(k-1)^4}. \end{aligned}$$
(C70)
In a similar way,
$$\begin{aligned} R_2^{(1)} \approx \frac{-1}{(T-L)^2}\frac{k^{2T}}{(k-1)^4}. \end{aligned}$$
(C71)
Calculation of \(R^{(2)}\).
For \(L>>1\),
$$\begin{aligned} R_1^{(2)}\approx & {} \frac{\kappa ^{2T}}{L(T-L)} \frac{1}{(\kappa -1)^3 \cdot (\kappa +1)}. \end{aligned}$$
(C72)
Similarly,
$$\begin{aligned}&R_2^{(2)} \approx \frac{\kappa ^{4L}}{L^2} \frac{1}{(\kappa -1)^3 \cdot (\kappa +1)}, \end{aligned}$$
(C73)
$$\begin{aligned}&R_3^{(2)} \approx \frac{\kappa ^{2L}}{L^2} \frac{1}{(\kappa -1)^3 \cdot (\kappa +1)}. \end{aligned}$$
(C74)
Therefore,
$$\begin{aligned} R^{(2)} \approx \frac{\kappa ^{2T}}{L(T-L)} \frac{1}{(\kappa -1)^3 \cdot (\kappa +1)}. \end{aligned}$$
(C75)
Calculation of \(R^{(3)}\) and \(R^{(4)}\). For a large L,
$$\begin{aligned}&R_1^{(3)} \approx \frac{\kappa ^{2T}}{L(T-L)} \frac{1}{(\kappa -1)^2}, \end{aligned}$$
(C76)
$$\begin{aligned}&R_2^{(3)} \approx -\frac{\kappa ^{2T}}{(T-L)^2} \frac{1}{(\kappa -1)^2}. \end{aligned}$$
(C77)
Therefore,
$$\begin{aligned} R^{(3)} \approx \frac{\kappa ^{2T}}{(T-L) \cdot (\kappa -1)^2} [\frac{1}{L}-\frac{1}{T-L}]. \end{aligned}$$
(C78)
\(R^{(4)}\) is also approximated as
$$\begin{aligned} R^{(4)} \approx -\frac{2 \cdot \kappa ^{2T}}{(T-L)^2} [\frac{1}{(\kappa -1)^3}]. \end{aligned}$$
(C79)
Consequently, for \(T>>L>>1\), we can obtain
$$\begin{aligned} V[\delta R^{(L)}] \propto \frac{\kappa ^{2T}}{L(T-L)}. \end{aligned}$$
(C80)
Appendix D: \(V[\delta R_j^{(L)}]\) for the power-law forgetting
We calculate \(V[\delta R_j^{(L)}]\) for the power-law forgetting process given by Eq. (13). Here, we consider \(L \ge 2\) and omit the suffix j for simplification. R(I) is defined by
$$\begin{aligned} R(I)=\sum _{t=LI+1}^{L(I+1)}\frac{r(t)}{L}. \end{aligned}$$
(D1)
From the definition, r(t) is written as
$$\begin{aligned} r(t)=\sum _{s=0}^{\infty } \theta (s) \cdot \eta (t-s), \end{aligned}$$
(D2)
where
$$\begin{aligned} \theta (s) \equiv \frac{1}{Z(\beta )}(s+ d_\beta )^{-\beta }. \end{aligned}$$
(D3)
Then, we can calculate
$$\begin{aligned} R(I)=\frac{1}{L}\sum ^{LI+1}_{t=-\infty }\theta _1'(t) \eta (t) + \sum ^{L(I+1)}_{t=LI+2}\theta _2'(t)\eta (t), \end{aligned}$$
(D4)
where
$$\begin{aligned} \theta _1'(t)= & {} \sum ^{L-1}_{k=0} \theta (k+LI+1-t), \end{aligned}$$
(D5)
$$\begin{aligned} \theta _2'(t)= & {} \sum ^{L-(t-LI)}_{k=0} \theta (k). \end{aligned}$$
(D6)
In a similar way, we can also calculate \(R(I+1)\) as follows:
$$\begin{aligned} R(I+1)=\frac{1}{L}\sum ^{L(I+1)+1}_{t=-\infty }\theta _3'(t) \eta (t) + \sum ^{L(I+2)}_{t=L(I+1)+2}\theta _4'(t)\eta (t),\nonumber \\ \end{aligned}$$
(D7)
where
$$\begin{aligned} \theta _3'(t)= & {} \sum ^{L-1}_{k=0} \theta (k+L(I+1)+1-t), \end{aligned}$$
(D8)
$$\begin{aligned} \theta _4'(t)= & {} \sum ^{L-(t-L(I+1))}_{k=0} \theta (k). \end{aligned}$$
(D9)
From these results, \(\delta R(I)=R(I+1)-R(I)\) can be calculated as
$$\begin{aligned}&\delta R(I)= R(I+1)-R(I) \end{aligned}$$
(D10)
$$\begin{aligned}&=\frac{1}{L} \sum ^{LI+1}_{t=-\infty }(\theta _3'(t)-\theta _1'(t)) \eta (t) +\sum ^{L(I+1)}_{t=LI+2}(\theta _3'(t)-\theta _2'(t)) \eta (t) \nonumber \\&\quad +\sum ^{L(I+2)}_{t=L(I+1)+1}\theta _4'(t) \eta (t). \end{aligned}$$
(D11)
Taking the average of \(\delta R(I)\) with respect to \(\eta \) gives
$$\begin{aligned}&V_{\eta }[\delta R] \approx E_{\eta }[(R(I+1)-R(I))^2] \nonumber \\&\quad = \frac{\check{\eta }^2}{L^2} \Bigg \{\sum ^{LI+1}_{t=-\infty }(\theta _3'(t)-\theta _1'(t))^2 \nonumber \\&\qquad +\sum ^{L(I+1)}_{t=LI+2}(\theta _3'(t)-\theta _2'(t))^2 +\sum ^{L(I+2)}_{t=L(I+1)+1}\theta _4'(t)^2 \Bigg \} \nonumber \\&\quad =\frac{\check{\eta }^2}{L^2 Z(\beta )^2} (U_1(\beta ,L)+U_2(\beta ,L)+U_3(\beta ,L)),\nonumber \\ \end{aligned}$$
(D12)
where we use \(E[\delta R] \approx 0\) because \(<\eta (t)>=0\).
Here, \(U_1(\beta ,L)\), \(U_2(\beta ,L)\), and \(U_3(\beta ,L)\) are defined as
$$\begin{aligned}&U_1(\beta ,L)=\sum ^{LI+1}_{t=-\infty }\frac{ Z(\beta )^2}{\check{\eta }^2} (\theta _3'(t)-\theta _1'(t))^2, \end{aligned}$$
(D13)
$$\begin{aligned}&U_2(\beta ,L)=\sum ^{L(I+1)}_{t=LI+2} \frac{ Z(\beta )^2}{\check{\eta }^2}(\theta _3'(t)-\theta _2'(t))^2, \end{aligned}$$
(D14)
and
$$\begin{aligned} U_3(\beta ,L)=\sum ^{L(I+2)}_{t=L(I+1)+1} \frac{ Z(\beta )^2}{\check{\eta }^2} (\theta _4'(t))^2. \end{aligned}$$
(D15)
We factor out \(Z(\beta )^2\) for later calculations. \(U_1(\beta ,L)\) is given by Eq. (D95), \(U_2(\beta ,L)\) is given by Eq. (D127), and \(U_3(\beta ,L)\) is given by Eq. (D145). The details of the derivations of these equations are given in the following section and beyond.
1.1 Appendix D.1: Calculation of \(U_1(\beta ,L)\)
Here, we calculate \(U_1(\beta ,L)\) defined by Eq. (D13):
$$\begin{aligned}&U_1(\beta ,L)\nonumber \\&\quad = \frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{LI+1}_{t=-\infty }(\theta _3'(t)-\theta _1'(t))^2 \end{aligned}$$
(D16)
$$\begin{aligned}&=\frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{LI+1}_{t=-\infty }\Bigg (\sum ^{L}_{i=1}\theta (k+L(I+1)+1-t) \nonumber \\&\quad -\sum ^{L}_{i=1}\theta (k+LI+1-t)\Bigg )^2. \end{aligned}$$
(D17)
By replacing the index t with a new index \(t'=t+LI+1\), \(U_1(\beta ,L)\) can be written as
$$\begin{aligned} U_1(\beta ,L)= \sum ^{\infty }_{t'=0}\frac{ Z(\beta )^2}{\check{\eta }^2}\left( \sum ^{L-1}_{k=0}\theta (t'+k+L)-\sum ^{L-1}_{k=0}\theta (t'+k)\right) ^2.\nonumber \\ \end{aligned}$$
(D18)
Using the Euler–Maclaurin formula [42], we can obtain
$$\begin{aligned}&\sum ^{b}_{k=a}g(k) \approx \int ^{b}_{a}g(x)\mathrm{d}x \nonumber \\&\quad +\frac{1}{2}(g(b)+g(a))+\frac{1}{12}(\frac{\mathrm{d}}{\mathrm{d}x}g(x)|_{b}-\frac{\mathrm{d}}{\mathrm{d}x}g(x)|_{a}), \nonumber \\ \end{aligned}$$
(D19)
and then we make the approximation
$$\begin{aligned}&U_1 \approx \frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{\infty }_{t'=0} \left\{ \theta (t'+L)+\frac{\theta (t'+2L-1)+\theta (t'+L+1)}{2} \right. \nonumber \\&\qquad +\frac{\theta '(t'+2L-1)-\theta '(t'+L+1)}{12} \nonumber \\&\qquad +\int ^{L-1}_{1}\theta (t'+k+L)\mathrm{d}k \nonumber \\&\qquad - \, \theta (t')-\frac{\theta (t'+L-1)+\theta (t'+1)}{2} \nonumber \\&\qquad -\left. \frac{\theta '(t'+L-1)-\theta '(t'+1)}{12}-\int ^{L-1}_{1} \theta (t'+k) \mathrm{d}k \right\} ^2. \nonumber \\ \end{aligned}$$
(D20)
Substituting Eq. (D3) into \(\theta (t')\) and performing some calculations, for \(\beta >0\) and \(\beta \ne 1\), we obtain
$$\begin{aligned}&U_1 \approx \sum ^{\infty }_{t'=0}\left\{ \sum ^{3}_{k=1}J^{(-1)}_{A,k} (t+A^{(-1)}_k)^{-(\beta -1)} \right. \nonumber \\&\quad +\left. \sum ^{6}_{k=1}J^{(0)}_{A,k} ((t+A^{(0)}_k)^{-\beta }+ \sum ^{3}_{k=1}J^{(+1)}_{A,k}(t+A^{(+1)}_k)^{-(\beta +1)}\right\} ^2.\nonumber \\ \end{aligned}$$
(D21)
For \(\beta =1\),
$$\begin{aligned}&U_1 \approx \sum ^{\infty }_{t'=0}\left\{ \sum ^{3}_{k=1}J^{(-1)}_{A,k} \log (t+A^{(-1)}_k) \right. \nonumber \\&\quad \left. + \sum ^{6}_{k=1}J^{(0)}_{A,k} ((t+A^{(0)}_k)^{-\beta }+ \sum ^{3}_{k=1}J^{+1}_{A,k}(t+A^{(+1)}_k)^{-(\beta +1)}\right\} ^2. \nonumber \\ \end{aligned}$$
(D22)
We combine the two equations into
$$\begin{aligned}&U_1 \approx \sum ^{\infty }_{t'=0}\left\{ \sum ^{+1}_{m=-1} \sum _{k=1}^{p_m}J^{(m)}_{A,k} f_A(t,\beta +m,A^{(m)}_k) \right\} ^2,\nonumber \\ \end{aligned}$$
(D23)
where
$$\begin{aligned} f_A(x,\alpha ,U)= & {} {\left\{ \begin{array}{ll} (x+U)^{-\alpha } &{} (\alpha \ne 0), \\ \log (x+U) &{} (\alpha =0), \end{array}\right. } \end{aligned}$$
(D24)
$$\begin{aligned} A_1^{(-1)}= & {} =A_1^{(+1)}=2L-1-d_{\beta }, \end{aligned}$$
(D25)
$$\begin{aligned} A_2^{(-1)}= & {} A_2^{(+1)}=L+1+d_{\beta },\end{aligned}$$
(D26)
$$\begin{aligned} A_3^{(-1)}= & {} A_3^{(+1)}=L-1+d_{\beta }, \end{aligned}$$
(D27)
$$\begin{aligned} A_4^{(-1)}= & {} A_4^{(+1)}=1+d_{\beta },\end{aligned}$$
(D28)
$$\begin{aligned} A_1^{(0)}= & {} 2L-1+d_{\beta }, \end{aligned}$$
(D29)
$$\begin{aligned} A_2^{(0)}= & {} 1+L+d_{\beta }, \end{aligned}$$
(D30)
$$\begin{aligned} A_3^{(0)}= & {} L+d_{\beta }, \end{aligned}$$
(D31)
$$\begin{aligned} A_4^{(0)}= & {} L-1+d_{\beta }, \end{aligned}$$
(D32)
$$\begin{aligned} A_5^{(0)}= & {} 1+d_{\beta }, \end{aligned}$$
(D33)
$$\begin{aligned} A_6^{(0)}= & {} d_{\beta }, \end{aligned}$$
(D34)
\(p_{-1}=p_{1}=4\), \(p_{0}=6\),
$$\begin{aligned} J^{(-1)}_{A,1}= & {} J^{(-1)}_{A,4}= {\left\{ \begin{array}{ll} 1/(1-\beta ) &{} {(\beta \ne 1)}, \\ 1 &{} {(\beta = 1)}, \\ \end{array}\right. } \end{aligned}$$
(D35)
$$\begin{aligned} J^{(-1)}_{A,2}= & {} J^{(-1)}_{A,3}= {\left\{ \begin{array}{ll} -1/(1-\beta ) &{} {(\beta \ne 1)},\\ -1 &{} {(\beta = 1)},\\ \end{array}\right. } \end{aligned}$$
(D36)
$$\begin{aligned} J^{(0)}_{A,1}= & {} J^{(0)}_{A,2}=1/2, \end{aligned}$$
(D37)
$$\begin{aligned} J^{(0)}_{A,3}= & {} 1, \end{aligned}$$
(D38)
$$\begin{aligned} J^{(0)}_{A,4}= & {} J^{(0)}_{A,5}=-1/2, \end{aligned}$$
(D39)
$$\begin{aligned} J^{(0)}_{A,6}= & {} -1, \end{aligned}$$
(D40)
and
$$\begin{aligned} J^{(+1)}_{A,1}= & {} J^{(+1)}_{A,3}=1/12 \cdot \beta ,\end{aligned}$$
(D41)
$$\begin{aligned} J^{(+1)}_{A,2}= & {} J^{(+1)}_{A,4}=-1/12 \cdot \beta . \end{aligned}$$
(D42)
We expand \(U_1\) as
$$\begin{aligned}&U_1 \approx \sum ^{\infty }_{t'=0}\left\{ \sum ^{+1}_{m=-1} \sum _{k=1}^{p_m}J^{(m)}_{A,k} f_A(t,\alpha +m,A^{(m)}_k) \right\} ^2 \nonumber \\&\quad =\sum ^{+1}_{m_1=-1}U_1^{(2)}(t;\beta ,m_1,m_1)+2 \sum _{m_1>m_2}U_1^{(2)}(t;\beta ,m_1,m_2),\nonumber \\ \end{aligned}$$
(D43)
where
$$\begin{aligned}&U^{(2)}_1(t;\beta ,m_1,m_2) =\sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} \nonumber \\&\quad \lim _{Q \rightarrow \infty } \sum ^{Q}_{t=0}f_A(t,\beta +m_i,A^{(m)}_i)f_A(t,\beta +m_j,A^{(m)}_j).\nonumber \\ \end{aligned}$$
(D44)
We calculate \(U^{(2)}_1(t;\beta ,m_1,m_2)\). Using the Euler–Maclaurin formula in Eq. (D19), we can obtain
$$\begin{aligned}&\sum ^{Q}_{t=0}f_A(t,\alpha _1,V_1)f_A(t,\alpha _2,V_2) \nonumber \\&\quad \approx G_2^{(R)}(1,Q;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\qquad +\int ^Q_1 (t+V_1)^{-\alpha _1}(t+V_2)^{-\alpha _2} \mathrm{d}t \nonumber \\&\quad \approx G_2^{(R)}(1,Q;\alpha _1,\alpha _2,V_1,V_2)+G_2^{(I)}(Q;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\qquad -G_2^{(I)}(1;\alpha _1,\alpha _2,V_1,V_2). \end{aligned}$$
(D45)
(i) For \(\alpha _1 \ne 1\), \(\alpha _2 \ne 1\) \(x_1=0\), and \(x_2=0\), \(G_2^{(R)}(x_1,x_2;\alpha _1,\alpha _2,V_1,V_2)\) is given by
$$\begin{aligned}&G_2^{(R)}(x_1,x_2;\alpha _1,\alpha _2,V_1,V_2)=(V_1)^{-\alpha _1}(V_2)^{-\alpha _2} \nonumber \\&\quad +\frac{1}{2}\{(V_1+x_1)^{-\alpha _1}(V_2+x_1)^{-\alpha _2} \nonumber \\&\quad +(V_1+x_2)^{-\alpha _1}(V_2+x_2)^{-\alpha _2}\} \nonumber \\&\quad -\frac{1}{12}\{(\alpha _2(V_2+x_2)+\alpha _1(V_1+x_2)) \nonumber \\&\quad \times (V_1+x_2)^{-\alpha _1-1}(V_2+x_2)^{-\alpha _2-1} \nonumber \\&\quad -(\alpha _2(V_1+x_1)+\alpha _1(V_2+x_1)) \nonumber \\&\quad \times (V_1+x_1)^{-\alpha _1-1}(V_2+x_1)^{-\alpha _2-1}\}. \end{aligned}$$
(D46)
Here, because Q approaches zero for \(Q>>1\), \(G_2^{(R)}(x,Q;0,\alpha _2,V_1,V_2)\) is not dependent on Q. Accordingly, we can denote \(G_2^{(R)}(x,Q;0,\alpha _2,V_1,V_2)\) as \(G_2^{(R \rightarrow Q)}(x;\alpha _1,\alpha _2,V_1,V_2)\).
\(G_2^{(R \rightarrow Q)}(x;\alpha _1,\alpha _2,V_1,V_2)\) is written as
$$\begin{aligned}&G_2^{(R \rightarrow Q)}(x;\alpha _1,\alpha _2,V_1,V_2)=(V_1)^{-\alpha _1}(V_2)^{-\alpha _2} \nonumber \\&\quad + \frac{1}{2}\{(V_1+x_1)^{-\alpha _1}(V_2+x_1)^{-\alpha _2}\} \nonumber \\&\quad -\frac{1}{12}\{ -(\alpha _2(V_1+x_1)+\alpha _1(V_2+x_1)) \nonumber \\&\quad \times (V_1+x_1)^{-\alpha _1-1}(V_2+x_1)^{-\alpha _2-1}\}. \end{aligned}$$
(D47)
As a special point, for \(x_1=0\) and \(x_2=0\), we determine
$$\begin{aligned} G_2^{(R)}(0,0;\alpha _1,\alpha _2,V_1,V_2)&=G_2^{(R \rightarrow Q)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&=(V_1)^{-\alpha _1}(V_2)^{-\alpha _2}. \end{aligned}$$
(D48)
(ii) In the case of \(\alpha _1 = 1\) and \(\alpha _2 = 1\), we can also calculate
$$\begin{aligned}&G_2^{(R)}(x_1,x_2;0,0,V_1,V_2)=\log (V_1) \log (V_2) \nonumber \\&\quad + \frac{1}{2}\{\log (V_1+x_1)\log (V_2+x_1) \nonumber \\&\quad +\log (V_1+x_2)\log (V_2+x_2)\} \nonumber \\&\quad + \frac{1}{12}\{\frac{\log (V_1+x_2)}{V_2+x_2}+\frac{\log (V_2+x_2)}{V_1+x_2} \nonumber \\&\quad - (\frac{\log (V_1+x_1)}{V_2+x_1}+\frac{\log (V_2+x_1)}{V_1+x_1})\}, \end{aligned}$$
(D49)
and, for \(Q>>1\), we can obtain
$$\begin{aligned}&G_2^{(R)}(x_1,x_2;0,\alpha _2,V_1,V_2)=\log (V_1) V_2^{-\alpha _2}+ \nonumber \\&\quad \frac{1}{2}\Bigg \{\log (V_1+x_1)V_2^{-\alpha _2}+\log (V_1+x_2)(V_2+x_2)^{-\alpha _2}\Bigg \} \nonumber \\&\quad +\frac{1}{12} \times \Bigg \{ (V_2+x_2)^{-\alpha _2-1} \times \nonumber \\&\quad \frac{(-\beta (V_1+x_2)\log (V_1+x_2)+V_2+x_2)}{V_1+x_2} \nonumber \\&\quad - (V_2+x_1)^{-\alpha _2-1} \times \nonumber \\&\quad \frac{\left( -\beta (V_1+x_1)\log (V_1+x_1)+V_1+x_1\right) }{V_1+x_1}\Bigg \}. \end{aligned}$$
(D50)
As a special point, for \(x_1=0\) and \(x_2=0\), we determine
$$\begin{aligned}&G_2^{(R)}(0,0;0,0,V_1,V_2)=G_2^{(R \rightarrow Q)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\quad =\log (V_1) \log (V_2). \end{aligned}$$
(D51)
(iii) In the case of \(\alpha _1 = 1\) and \(\alpha _2 \ne 0\), we can also obtain
$$\begin{aligned}&G_2^{(R)}(x_1,x_2;0,\alpha _2,V_1,V_2)=\log (V_1) V_2^{-\alpha _2} \nonumber \\&\quad +\frac{1}{2}\Bigg \{\log (V_1+x_1)V_2^{-\alpha _2}+\log (V_1+x_2)(V_2+x_2)^{-\alpha _2}\Bigg \} \nonumber \\&\quad +\frac{1}{12} \times \nonumber \\&\quad \Bigg \{\frac{(V_2+x_2)^{-\alpha _2-1}(-\beta (V_1+x_2)\log (V_1+x_2)+V_2+x_2)}{V_1+x_2} \nonumber \\&\quad -(V_2+x_1)^{-\alpha _2-1} \times \nonumber \\&\quad \frac{(-\beta (V_1+x_1)\log (V_1+x_1)+V_1+x_1)}{V_1+x_1}\Bigg \}. \end{aligned}$$
(D52)
As a special point, for \(x_1=0\) and \(x_2=0\), we determine
$$\begin{aligned} G_2^{(R)}(0,0;0,\alpha _2,V_1,V_2)= \log (V_1) V_2^{-\alpha _2}. \end{aligned}$$
(D53)
Next, we calculate the integration term of Eq. (D45), \(G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2)\). \(G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2)\) is defined by
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\quad = {\left\{ \begin{array}{ll} \int (x+V_1)^{-\alpha _1}(x+V_2)^{-\alpha _2} \mathrm{d}x &{} (\alpha _1,\alpha _2>0), \\ \int \log (x+V_1) \log (x+V_2) \mathrm{d}x &{} (\alpha _1,\alpha _2=0), \\ \int \log (x+V_1) (x+V_2)^{-\alpha _2} \mathrm{d}x &{} (\alpha _1=0, \alpha _2>0), \\ \end{array}\right. }, \end{aligned}$$
(D54)
where we neglect an integral constant.
We calculate \(G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2)\).
(i) In the case of \(V_1=V_2\), \(\alpha _1 \ne 0\), and \(\alpha _2 \ne 0\), we can write
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2) =\frac{1}{1-\alpha _1-\alpha _2}(x+V_1)^{-\alpha _1-\alpha _2+1}. \nonumber \\ \end{aligned}$$
(D55)
(ii) When \(\alpha _1\) and \(\alpha _2\) are nonintegers, and \(V_1 \ne V_1\), we obtain
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2)= (V_j+x)^{1-\alpha _j}(V_i-V_j)^{-\alpha _i} \times \nonumber \\&\frac{{}_2F_1(\alpha _i,1-\alpha _j,2-\alpha _j,\frac{-(V_j+x)}{(V_i-V_j)})}{(1-\alpha _j)}, \end{aligned}$$
(D56)
where \(i={\arg \max }_{k \in \{1,2\}}\{V_k\}\) and \(j={\arg \min }_{k \in \{1,2\}}\{V_k\}\) (under this condition, \(G_2^{(I)}\) takes a real number).
(iii) For \(\alpha _1=1,2,3,\ldots \), \(\alpha _2=1,2,3,\ldots \) and \(V_1 \ne V_2\),
using a partial fraction decomposition,
$$\begin{aligned} \frac{1}{(z+q_1)^{n_1}(z+q_2)^{n_2}}=\sum _{k=1}^{n_1}\frac{h^{(1)}_k}{(z+q_1)^k}+\sum _{k=1}^{n_2}\frac{h^{(2)}_k}{(z+q_2)^k}, \nonumber \\ \end{aligned}$$
(D57)
with
$$\begin{aligned}&h^{(1)}_k=\frac{1}{(n_1-k)!}(-1)^{n_2}\frac{(n_1+n_2-k-1)!}{(n_2-1)!} \nonumber \\&\quad \times (-q_2+q_1)^{-n_1-n_2+k}, \nonumber \\&\quad h^{(2)}_k=\frac{1}{(n_2-k)!}(-1)^{n_1}\frac{(n_1+n_2-k-1)!}{(n_1-1)!}\nonumber \\&\quad \times (-q_1+q_2)^{-n_1-n_2+k}, \end{aligned}$$
(D58)
we can write
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\quad =h_1^{(1)}\log (x+V_1)+h_1^{(2)}\log (x+V_2) \nonumber \\&\qquad +\sum _{k=2}^{\alpha _1}\frac{1}{1-k}h^{(1)}_k (x+V_1)^{1-k}\nonumber \\&\qquad +\sum _{k=2}^{\alpha _2} \frac{1}{1-k} h^{(2)}_k (x+V_2)^{1-k}. \end{aligned}$$
(D59)
(iv) For \(\alpha _1=0\), \(\alpha _2=0\), and \(V_1 \ne V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,0,V_1,V_2) \nonumber \\&\quad =Li_{2}(\frac{V_1+x}{V_1-V_2}) (V_2-V_1) \nonumber \\&\qquad +\log (V_1+x)\{(V_1+x)\log (V_2+x) \nonumber \\&\qquad +(V_2-V_1)\log (-\frac{V_2+x}{V_1-V_2})-V_1-x\} \nonumber \\&\qquad +V_1-V_2\log (V_2+x)-x\log (V_2+x)+2x, \nonumber \\ \end{aligned}$$
(D60)
where \(Li_{2}(x)\) is the polylogarithm.
(v) For \(\alpha _1=0\), \(\alpha _2=0\), and \(V_1=V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,0,V_1,V_2) \nonumber \\&\quad =\log (V_1+x)^2 (V_1+x)+2\log (V_1+x)(-V_1-x) \nonumber \\&\qquad +2x+V_1. \end{aligned}$$
(D61)
(vi) For \(\alpha _1=0\), \(\alpha _2=1\), and \(V_1 \ne V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,1,V_1,V_2) \nonumber \\&\quad =Li_{2}(\frac{V_1+x}{V_1-V_2}) + \log (V_1+x)\{\log (-\frac{V_2+x}{V_1-V_2})\}. \nonumber \\ \end{aligned}$$
(D62)
(vii) For \(\alpha _1=0\), \(\alpha _2=1\), and \(V_1=V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,1,V_1,V_2)=\frac{1}{2}\log ^2(V_1+x). \end{aligned}$$
(D63)
(viii) For \(\alpha _1=0\), \(\alpha _2=2\), and \(V_1 \ne V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,2,V_1,V_2) \nonumber \\&\quad =\frac{(V_2+x)\log (V_2+x)-(V_1+x)\log (V_1+x)}{(V_1-V_2)(V_2+x)}.\nonumber \\ \end{aligned}$$
(D64)
(ix) For \(\alpha _1=0\), \(\alpha _2=2\), and \(V_1= V_2\), we can write
$$\begin{aligned}&G_2^{(I)}(x;0,2,V_1,V_2)=-\frac{\log (V_1+x)+1}{V_1+x}. \end{aligned}$$
(D65)
Consequently, the summary of \(G_2^{(I)}\) is given by
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\quad ={\left\{ \begin{array}{ll} {(\alpha _1 \ne 0, \alpha _2 \ne 0, V_1=V_2): }\\ \frac{1}{1-\alpha _1-\alpha _2}(x+V_1)^{-\alpha _1-\alpha _2+1} \\ {(\alpha _2 \mathrm{and} \alpha _2 {\text {are noninteger}}, V_1 \ne V_2): }\\ \frac{(V_j+x)^{(1-\alpha _j)}(V_i-V_j)^{(-\alpha _i)}{}_2F_1(\alpha _i,1-\alpha _j,2-\alpha _j,\frac{-(V_j+x)}{(V_i-V_j)})}{(1-\alpha _j)} \\ {(\alpha _1, \alpha _2 \ge 2, \alpha _1 \mathrm{and} \alpha _2 {\text {are integers}}, V_1 \ne V_2): }\\ h_1^{(1)}\log (x+V_1)+h_1^{(2)}\log (x+V_2) \\ +\sum _{k=2}^{\alpha _1}\frac{1}{1-k}h^{(1)}_k (x+V_1)^{1-k} \\ +\sum _{k=2}^{\alpha _2} \frac{1}{1-k} h^{(2)}_k (x+V_2)^{1-k} \\ {(\alpha _1=0, \alpha _2=0, V_1= V_2): }\\ \log (V_1+x)^2 (V_1+x)+2\log (V_1+x)(-V_1-x) \\ +2x+V_1 \\ {(\alpha _1=0, \alpha _2=0, V_1 \ne V_2): }\\ Li_{2}(\frac{V_1+x}{V_1-V_2}) (V_2-V_1) \\ +\log (V_1+x)\{(V_1+x)\log (V_2+x) \\ +(V_2-V_1)\log (-\frac{V_2+x}{V_1-V_2})-V_1-x\} \\ +V_1-V_2\log (V_2+x)-x\log (V_2+x)+2x \\ {(\alpha _1=0, \alpha _2=1, V_1 = V_2): }\\ \frac{1}{2}\log ^2(V_1+x) \\ {(\alpha _1=0, \alpha _2=1, V_1 \ne V_2): }\\ Li_{2}(\frac{V_1+x}{V_1-V_2}) + \log (V_1+x)\{\log (-\frac{V_2+x}{V_1-V_2})\} \\ {(\alpha _1=0, \alpha _2=2, V_1=V_2): }\\ -\frac{\log (V_1+x)+1}{V_1+x} \\ {(\alpha _1=0, \alpha _2=2, V_1 \ne V_2): }\\ \frac{(V_2+x)\log (V_2+x)-(V_1+x)\log (V_1+x)}{(V_1-V_2)(V_2+x)}. \\ \end{array}\right. } \end{aligned}$$
(D66)
1.2 Appendix D.2: Calculation of \(G_2^{(Q)}(\beta ,m_1,m_2)\)
\(U_1(t;\beta ,m_1,m_2)\) is decomposed into
$$\begin{aligned}&U_1(t;\beta ,m_1,m_2) \nonumber \\&\quad =\sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} G_2^{(R)}(1,Q;\beta +m_1,\beta +m_2,A^{(m_1)}_i,A^{(m_2)}_j) \nonumber \\&\qquad - \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} G_2^{(I)}(0;\beta +m_1,\beta +m_2,A^{(m_1)}_i,A^{(m_2)}_j) \nonumber \\&\qquad + \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} G_2^{(I)}(Q;\beta +m_1,\beta +m_2,A^{(m_1)}_i,A^{(m_2)}_j).\nonumber \\ \end{aligned}$$
(D67)
We have already calculated \(G_2^{(R)}(1,Q;\alpha _1,\alpha _2,A^{(m_1)}_i,A^{(m_2)}_j)\) as \(G_2^{R \rightarrow Q}(x,\alpha _1,\alpha _2,V_1,V_2)\) in the previous sections. Here, we calculate
$$\begin{aligned}&G_2^{(Q)}(\beta ,m_1,m_2) \nonumber \\&\quad \equiv \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} G_2^{(I)}(Q;\beta +m_1,\beta +m_2,A^{(m_1)}_i,A^{(m_2)}_j) \nonumber \\ \end{aligned}$$
(D68)
for \(Q>>1\).
(i) When
\(\beta \)
is a noninteger
We study the asymptotic behavior of \(G^{(I)}(Q;\alpha _1,\alpha _2,V_1,V_2)\) in Eq. (D68) for \(Q>>1\) in the case of \(V_1 \ne V_2\). Here, we use the following formulas of the asymptotic behavior of the hypergeometric function [43]. When \(b-a\), \(c-a\), a, b, and c are nonintegers for a large x,
$$\begin{aligned}&{}_2F_1(a,b,c,x) \nonumber \\&\quad \approx \frac{\varGamma (b-a) \varGamma (c)}{ \varGamma (b) \varGamma (c-a)}\frac{1}{(-x)^{a}} \nonumber \\&\qquad +\frac{(-1)^a a(1+a-c) \varGamma (b-a) \varGamma (c)}{(1+a-b)\varGamma (b) \varGamma (c-a)}\frac{1}{x^{a+1}} \nonumber \\&\qquad +\frac{(-1)^a a(1+a)(1+a-c)(2+a-c) \varGamma (b-a) \varGamma (c)}{2(1-a-b)(2-a+b)\varGamma (b) \varGamma (c-a)}\frac{1}{x^{a+2}} \nonumber \\&\qquad +\frac{\varGamma (a-b)\varGamma (c)}{\varGamma (a)\varGamma (c-b)}\frac{1}{(-x)^b}. \end{aligned}$$
(D69)
When both \(b-a\) and \(c-a\) are integers and \(c-b>0\) for a large x,
$$\begin{aligned}&{}_2F_1(a,b,c,x) \nonumber \\&\quad \approx \frac{\varGamma (b-a) \varGamma (c)}{ \varGamma (b) \varGamma (c-a)}\frac{1}{(-x)^{a}} \nonumber \\&\qquad +\frac{(-1)^a a(1+a-c) \varGamma (b-a) \varGamma (c)}{(1+a-b)\varGamma (b) \varGamma (c-a)}\frac{1}{x^{a+1}} \nonumber \\&\qquad +\frac{(-1)^a a(1+a)(1+a-c)(2+a-c) \varGamma (b-a) \varGamma (c)}{2(1+a-b)(2+a-b)\varGamma (b) \varGamma (c-a)}\frac{1}{x^{a+2}} \nonumber \\&\qquad +(-1)^{b-a}\varGamma (c) \nonumber \\&\qquad \frac{(\log (-x)+\psi (b-a+1)-\psi (c-b)-\psi (b)-\gamma )}{\varGamma (a)\varGamma (b-a+1)\varGamma (c-b)}\frac{1}{(-x)^b}. \nonumber \\ \end{aligned}$$
(D70)
When \(b=a\) and \(c-a\) are integers for a large x,
$$\begin{aligned}&{}_2F_1(a,b,c,x) \nonumber \\&\quad \approx \frac{ \varGamma (c)}{ \varGamma (a) \varGamma (c-a)}\frac{(\log (-z)-\psi (c-a)-\psi (a)-2\gamma )}{(-x)^{a}} \nonumber \\&\qquad +\frac{\varGamma (c)^2(-x)^{-c}}{\varGamma (a)^2(\varGamma (c-a+1))^2}. \end{aligned}$$
(D71)
Here, \(\gamma =0.5772\) is the Euler constant.
Using these formulas, the hypergeometric function in Eq. (D56) is written as
$$\begin{aligned}&{}_2F_1(\alpha _i,1-\alpha _j,2-\alpha _j,\frac{-(V_j+x)}{(V_i-V_j)})) \nonumber \\&\quad \approx P_1 (\frac{(V_i-V_j)}{Q+V_j})^{\alpha _i}+ P_2 (\frac{(V_i-V_j)}{Q+V_j})^{\alpha _i-1} \nonumber \\&\qquad + P_3 (\frac{(V_i-V_j)}{Q+V_j})^{\alpha _i-2} \nonumber \\&\qquad + P_4 (\frac{(V_i-V_j) }{Q+V_j})^{1-\alpha _j}\log (Q) + P_5(V_i,V_j) (\frac{(V_i-V_j)}{Q+V_j})^{1-\alpha _j},\nonumber \\ \end{aligned}$$
(D72)
where
$$\begin{aligned} P_1= & {} {\left\{ \begin{array}{ll} \frac{(1-\alpha _j)}{(1-\alpha _i-\alpha _j)} &{} {(\alpha _i+\alpha _j \ne 1)}, \\ 0 &{} {(\alpha _i+\alpha _j =1)}, \end{array}\right. } \end{aligned}$$
(D73)
$$\begin{aligned} P_2= & {} {\left\{ \begin{array}{ll} \frac{-\alpha _i(\alpha _i+\alpha _j-1)}{(\alpha _i+\alpha _j)}P_1 &{} {(\alpha _i+\alpha _j \ne 0)}, \\ 0 &{} {(\alpha _i+\alpha _j=0)}, \end{array}\right. } \end{aligned}$$
(D74)
$$\begin{aligned} P_3= & {} {\left\{ \begin{array}{ll} \frac{(1+\alpha _i)(\alpha _i+\alpha _j)}{2(1+\alpha _i+\alpha _j)}P_2 &{} {(\alpha _i+\alpha _j \ne -1)},\\ 0&{} {(\alpha _i+\alpha _j=-1)},\\ \end{array}\right. } \end{aligned}$$
(D75)
$$\begin{aligned} P_4= & {} {\left\{ \begin{array}{ll} 0 &{} (\alpha _i+\alpha _j\hbox { is a noninteger}), \\ \frac{\varGamma (2-\alpha _j)}{\varGamma (\alpha _i)} \frac{(-1)^{1-\alpha _i-\alpha _j}}{\varGamma (2-\alpha _i-\alpha _j)} &{} (\alpha _i+\alpha _j=0,-1,1), \end{array}\right. } \end{aligned}$$
(D76)
$$\begin{aligned} P_5(V_i,V_j)= & {} \frac{\varGamma (2-\alpha _j)}{\varGamma (\alpha _i)} \times \nonumber \\&{\left\{ \begin{array}{ll} (\alpha _i+\alpha _j\hbox { is a noninteger})\\ \varGamma (\alpha _i+\alpha _j-1) \\ {(\alpha _i+\alpha _j=-1,0,1)} \\ \frac{(-1)^{1-\alpha _i-\alpha _j}(-\log (V_i-V_j)+\psi (2-\alpha _i-\alpha _j)-\psi (1-\alpha _j))}{\varGamma (2-\alpha _i-\alpha _j)}.\\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(D77)
Substituting these results, we can obtain the following approximation:
$$\begin{aligned}&G_2^{(I)}(Q;\alpha _1,\alpha _2,V_1,V_2) \approx \frac{1}{1-\alpha _j} \nonumber \\&\quad \times \left\{ P_1 (Q+V_j)^{1-\alpha _i-\alpha _j} \right. \nonumber \\&\quad + P_2 (V_i-V_j)(Q+V_j)^{-\alpha _i-\alpha _j} \nonumber \\&\quad + P_3 (V_i-V_j)^2 (Q+V_j)^{-\alpha _i-\alpha _j-1} \nonumber \\&\quad + P_4 (V_i-V_j)^{1-\alpha _i-\alpha _j}\log (Q) \nonumber \\&\quad \left. + P_5(V_i,V_j)\times (V_i-V_j)^{1-\alpha _i-\alpha _j} \right\} , \end{aligned}$$
(D78)
where \(i={\arg \max }_{k \in \{1,2\}}\{V_k\}\) and \(j={\arg \min }_{k \in \{1,2\}}\{V_k\}\). (Under this condition, \(G_2^{(I)}\) takes a real number.)
In addition, using the asymptotic behavior of the power-law function,
$$\begin{aligned} (Q+V)^{\alpha } \approx Q^{\alpha }+(\alpha )VQ^{\alpha -1}+\frac{(\alpha )(\alpha -1)}{2}V^2 Q^{\alpha -2}, \nonumber \\ \end{aligned}$$
(D79)
we can obtain \(G_2^{(I)}\) as a sum of the power-law function of Q,
$$\begin{aligned}&G_2^{(I)}(Q;\alpha _1,\alpha _2,V_1,V_2) \approx \frac{1}{1-\alpha _j} \nonumber \\&\quad \times \left\{ P_1 Q^{1-\alpha _i-\alpha _j}+ (P_1(-\alpha _i-\alpha _j+1)V_j \right. \nonumber \\&\quad +P_2(V_i-V_j))Q^{-\alpha _i-\alpha _j} \nonumber \\&\quad + (\frac{P_1}{2}(-\alpha _i-\alpha _j+1)(-\alpha _i-\alpha _j)V_j^2 \nonumber \\&\quad +P_2(-\alpha _i-\alpha _j)(V_i-V_j)V_j \nonumber \\&\quad +P_3 (V_i-V_j)^2) Q^{-\alpha _i-\alpha _j-1} \nonumber \\&\quad + P_4 (V_i-V_j)^{1-\alpha _i-\alpha _j}\log (Q) \nonumber \\&\quad \left. + P_5(V_i,V_j)\times (V_i-V_j)^{1-\alpha _i-\alpha _j} \right\} . \end{aligned}$$
(D80)
For \(V_1=V_2\), using Eq. (D79), Eq. (D55) is approximated as
$$\begin{aligned}&G_2^{(I)}(x;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\quad \approx \frac{1}{1-\alpha _1-\alpha _2}(Q+V_2)^{-\alpha _1-\alpha _2+1} \nonumber \\&\quad \approx \frac{1}{1-\alpha _1-\alpha _2} (Q^{-\alpha _1-\alpha _2+1}+(-\alpha _1-\alpha _2+1)V_2Q^{-\alpha _1-\alpha _2} \nonumber \\&\qquad +\frac{(-\alpha _1-\alpha _2)(-\alpha _1-\alpha _2+1)}{2}V_2^2 Q^{-\alpha _1-\alpha _2-1}). \end{aligned}$$
(D81)
Hence, because \(Q^{a} \rightarrow 0\) (\(a<0\), \(Q>>1\)),
$$\begin{aligned}&G_2^{(Q)}(\beta ,m_1,m_2) \nonumber \\&\quad \approx \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} G^{(I)}(Q;\beta +m_1,\beta +m_2,A^{(m_1)}_i,A^{(m_2)}_j) \nonumber \\&\quad \approx \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)} \Bigg [ \nonumber \\&\quad {\left\{ \begin{array}{ll} \frac{ P_5(V_q,V_r) \times (V_q-V_r)^{1-2\beta -m_1-m_2}}{1-\beta -m_s} &{} {(V_q \ne V_r)}, \\ 0 &{} {(V_q = V_r)}, \\ \end{array}\right. } \nonumber \\&\qquad + {\left\{ \begin{array}{ll} 0 &{} {(2\beta +m_1+m_2 \ne 0)}, \\ V_r &{} {(2\beta +m_1+m_2 = 0)}, \\ \end{array}\right. } \Bigg ]\nonumber \\ \end{aligned}$$
(D82)
where \(V_q=\max \{A^{(m_1)}_i,A^{(m_2)}_j\}\), \(V_r=\min \{A^{(m_1)}_i,A^{(m_2)}_j\}\), \(s=\arg \min _{\{u=1,2\}}\{A_{q_u}^{(m_u)}\}\), and \((q_1,q_2)=(i,j)\).
Case of
\(\beta =1\)
We use the approximation formulas for \(x>>1\),
$$\begin{aligned}&Li_2(x) \approx -\frac{1}{2}\log (-x)^2-\frac{\pi ^2}{6}, \end{aligned}$$
(D83)
$$\begin{aligned}&\log (x+A) \approx \log (x)+\frac{A}{x} . \end{aligned}$$
(D84)
From Eqs. (D60) and (D61), we can obtain
$$\begin{aligned}&G_2^{(Q)}(1,-1,-1) \nonumber \\&\quad \approx \sum _{i=1}^{p_{-1}} \sum _{j=1}^{p_{-1}} J_{Ai}^{(-1)} J_{Aj}^{(-1)} G^{(I)}(Q;0,0,A^{(-1)}_i,A^{(-1)}_j) \nonumber \\&\quad \approx \sum _{i=1}^{p_{-1}} \sum _{j=1}^{p_{-1}} J_{Ai}^{(-1)} J_{Aj}^{(-1)} G_{(-1,-1)}^{(Q)}(A^{(-1)}_i,A^{(-1)}_j),\nonumber \\ \end{aligned}$$
(D85)
where
$$\begin{aligned}&G_{(-1,-1)}^{(Q)}(V_1,V_2) \nonumber \\&\quad ={\left\{ \begin{array}{ll} (V_2-V_1)\left( -\frac{1}{2}\log (V_2-V_1)^2-\frac{\pi }{6}\right) +V_1 &{} (V_1 \ne V_2), \\ V_1 &{} (V_1=V_2).\\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(D86)
Similarly, from Eqs. (D62) and (D63),
$$\begin{aligned}&G_2^{(Q)}(1,-1,0) \nonumber \\&\quad \approx \sum _{i=1}^{p_{-1}} \sum _{j=1}^{p_{0}} J_{Ai}^{(-1)} J_{Aj}^{(0)} G^{(I)}(Q;0,0,A^{(-1)}_i,A^{(0)}_j) \nonumber \\&\quad \approx \sum _{i=1}^{p_{-1}} \sum _{j=1}^{p_{0}} J_{Ai}^{(-1)} J_{Aj}^{(0)} G_{-1,0}^{(Q)}(A_i^{(-1)},A^{(0)}_j), \end{aligned}$$
(D87)
where
$$\begin{aligned} G_{(-1,0)}^{(Q)}(V_1,V_2) = {\left\{ \begin{array}{ll} -\frac{1}{2}\log (V_2-V_1)^2 &{} (V_1 \ne V_2),\\ 0 &{} (V_1=V_2).\\ \end{array}\right. }.\nonumber \\ \end{aligned}$$
(D88)
Because the order of Eqs. (D64) and (D65) is \(\log (Q)/Q\),
$$\begin{aligned} G_2^{(Q)}(1,-1,2) \approx 0. \end{aligned}$$
(D89)
In addition, from Eq. (D59),
$$\begin{aligned} G_2^{(Q)}(1,m_1,m_2) \approx 0 \quad (m_1=0,1,m_2=0,1). \end{aligned}$$
(D90)
Note that in Eq. (D59), because \(h_1^{(1)}=-h_1^{(2)}\), the terms of \(\log (Q)\) approach zero for \(Q>>1\) by cancelling each other out.
When \(\beta \) is an integer and \(\beta \ne 1\) The terms of \(\log (Q)\) in Eq. (D59) approach zero for \(Q>>1\) by cancelling each other out because \(h_1^{(1)}=-h_1^{(2)}\). We use the following result:
$$\begin{aligned} G_2^{(I)}(Q;\alpha _1,\alpha _2,V_1,V_2) \approx 0. \end{aligned}$$
(D91)
Therefore,
$$\begin{aligned} G_2^{(Q)}(\beta ,m_1,m_2) \approx 0. \end{aligned}$$
(D92)
We summarize \(G_2^{(Q)}\) as follows:
$$\begin{aligned}&G_2^{(Q)}(\beta ,m_1,m_2) \nonumber \\&\quad \approx \sum _{i=1}^{p_{m_1}} \sum _{j=1}^{p_{m_2}} J_{Ai}^{(m_1)} J_{Aj}^{(m_2)}G_2^{(I;Q)}(\beta ,m_1,m_2,A^{(m_1)}_i,A^{(m_2)}_j),\nonumber \\ \end{aligned}$$
(D93)
where
$$\begin{aligned}&G_2^{(I;Q)}(\beta ,m_1,m_2,A^{(m_1)}_i,A^{(m_2)}_j) \nonumber \\&\quad = {\left\{ \begin{array}{ll} {(2\beta +m_1+m_2 \ne 1, 0,-1, A^{(m_1)}_i \ne A^{(m_2)}_j)} \\ \frac{\varGamma (2\beta +m_1+m_2-1)\varGamma (2-\beta -m_{s})}{(1-\beta -m_{s}) \varGamma (\beta +m_{k})} (A_{q}-A_{r})^{1-2\beta -m_1-m_2} \\ {(\beta \hbox { is noninteger }2\beta +m_1+m_2=-1,1)} \\ \frac{(-1)^{1-2\beta -m_1-m_2} \varGamma (2-\beta -m_s)}{(1-\beta -m_s)\varGamma (\beta +m_k)} (A_{q}-A_{r})^{1-2\beta -m_1-m_2} \times \\ \frac{(-\log (A_{q}-A_{r})+\psi (2-2\beta -m_1-m_2)-\psi (1-\beta -m_s))}{\varGamma (2-2\beta -m_1-m_2)} \\ {(\beta \hbox { is noninteger, }2\beta +m_1+m_2=0)} \\ \frac{(-1)^{1-2\beta -m_1-m_2} \varGamma (2-\beta -m_s)}{(1-\beta -m_s)\varGamma (\beta +m_k)} (A_{q}-A_{r})^{1-2\beta -m_1-m_2} \times \\ \frac{(-\log (A_{q}-A_{r})+\psi (2-2\beta -m_1-m_2)-\psi (1-\beta -m_s))}{\varGamma (2-2\beta -m_1-m_2)} \\ +A_{r} \\ {(\beta =1, m_1=-1,m_2=-1, A^{(m_1)}_i \ne A^{(m_2)}_j)} \\ (A^{(m_2)}_j-A^{(m_1)}_i)(-\frac{1}{2}\log (A^{(m_2)}_j-A^{(m_1)}_i)^2-\frac{\pi }{6}) \\ +A^{(m_1)}_i \\ {(\beta =1, m_1=-1,m_2=-1, A^{(m_1)}_i = A^{(m_2)}_j)} \\ A^{(m_1)}_i \\ {(\beta =1, (m_1,m_2) \in \{(-1,0),(0,-1)\}, A^{(m_1)}_i \ne A^{(m_2)}_j)} \\ -\frac{1}{2}\log (A^{(m_2)}_j-A^{(m_1)}_i)^2 \\ {(\beta =1\hbox { and }[m_1=1\hbox { or }m_2=1])} \\ 0 \\ {(\beta =2,3,4, \ldots )} \\ 0\\ {(A^{(m_1)}_i = A^{(m_2)}_j, 2\beta +m_1+m_2 \ne 0)}\\ 0 \\ {(A^{(m_1)}_i = A^{(m_2)}_j, 2\beta +m_1+m_2=0, \beta \ne 1)}\\ A_{r}. \\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(D94)
Here, \(A_q=\max \{A^{(m_1)}_i,A^{(m_2)}_j\}\), \(A_r=\min \{A^{(m_1)}_i,A^{(m_2)}_j\}\), \(t=\arg \max _{\{u=1,2\}}\{A_{q_u}^{(m_u)}\}\), \(s=\arg \min _{\{u=1,2\}}\{A_{q_u}^{(m_u)}\}\), and \((q_1,q_2)=(i,j)\). The reason why we change the suffix is to avoid a constant of the integrations from becoming a complex number.
Consequently, \(U_1\) is obtained by
$$\begin{aligned}&U_1(\beta ,L) \approx \sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1} \sum _{i=1}^{p^{(A)}_{m_1}} \sum _{j=1}^{p^{(A)}_{m_2}} J_{A_i}^{(m_1)}J_{A_j}^{(m_2)} \nonumber \\&\quad \times \{ G_2^{(R;Q)}(1,\beta +m_1,\beta +m_2; A_i^{(m_1)},A_j^{(m_2)}) \nonumber \\&\quad +G_2^{(I;Q)}(\beta ,m_1,m_2,A_i^{(m_1)},A_j^{(m_2)}) \nonumber \\&\quad -G_2^{(I)}(1,\beta +m_1,\beta +m_2; A_i^{(m_1)},A_j^{(m_2)}) \}. \end{aligned}$$
(D95)
1.3 Appendix D.3: Calculation of \(U_2(\beta ,L)\)
\(U_2\) is defined by Eq. (D14) as
$$\begin{aligned} U_2(\beta ,L)=\sum ^{L(I+1)}_{t=LI+2}\frac{ Z(\beta )^2}{\check{\eta }^2} (\theta _3'(t)-\theta _2'(t))^2. \end{aligned}$$
(D96)
Substituting Eqs. (D6) and (D8) into Eq. (D96), we obtain
$$\begin{aligned}&U_2(\beta ,L) \nonumber \\&\quad =\sum ^{L(I+1)}_{t=LI+2}\frac{ Z(\beta )^2}{\check{\eta }^2} \Bigg (\sum ^{L-1}_{k=0}\theta (k+L(I+1)+1-t) \nonumber \\&\qquad -\sum ^{L(I+1)-t}_{k=0}\theta (k)\Bigg )^2. \end{aligned}$$
(D97)
Using a shifted index \(t'=t+LI+2\), \(U_2\) is written as
$$\begin{aligned}&U_2=\sum ^{L-2}_{t'=0}\frac{ Z(\beta )^2}{\check{\eta }^2} \Bigg (\sum ^{L-1}_{k=0}\theta (k+L-1-t') \nonumber \\&\quad -\sum ^{L-2-t'}_{k=0}\theta (k)\Bigg )^2. \end{aligned}$$
(D98)
Using the Euler–Maclaurin formula in Eq. (D19), we can obtain
$$\begin{aligned}&U_2 \approx \frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{L-3}_{t'=0} \left\{ \theta (L-1-t')+\frac{\theta (L-t')+\theta (2L-2-t')}{2} \right. \nonumber \\&\quad +\frac{\theta (2L-2-t')-\theta '(L-1-t')}{12} \nonumber \\&\quad +\int ^{L-1}_{1}\theta (k+L-1-t')\mathrm{d}k \nonumber \\&\quad - \theta (0)-\frac{\theta (1)+\theta (L-2-t)}{2} \nonumber \\&\quad - \left. \frac{\theta '(L-2-t)-\theta '(1)}{12}-\int ^{L-2-t}_{1} \theta (k) \mathrm{d}k \right\} ^2 \nonumber \\&\quad + u_2^{(0)}(\beta )^2, \end{aligned}$$
(D99)
where \(u_2^{(0)}(\beta )^2\) is given by
$$\begin{aligned}&u_2^{(0)}(\beta ) = (1+d_{\beta })^{-\beta }+\frac{1}{2}((2+d_{\beta })^{-\beta }+(L+d_{\beta })^{-\beta }) \nonumber \\&\quad +(-\beta )\frac{1}{12}((L+d_{\beta })^{-\beta -1}-(2+d_{\beta })^{-\beta -1}) \nonumber \\&\quad +{\left\{ \begin{array}{ll} \frac{1}{-\beta +1}((L+d_{\beta })^{-\beta +1}-(2+d_{\beta })^{-\beta +1} &{} {(\beta \ne 1)}, \\ \log (L+d_{\beta })-\log (2+d_{\beta }) &{} {(\beta =1)}.\\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(D100)
Here, \(u_2^{(0)}(\beta )\) corresponds to the term of \(t'=L-2\) in Eq. (D98). We separated and directly calculated this term to improve the accuracy.
Substituting Eq. (D3) into \(\theta (t')\) and performing some calculations, for \(\beta >0\) and \(\beta \ne 1\), we obtain
$$\begin{aligned} U_2\approx & {} \sum ^{L-3}_{t=0}\left\{ B_0+ \sum ^{p_B^{(-1)}}_{k=1}J^{(-1)}_{B,k} (-t+B^{(-1)}_k)^{-(\beta -1)} \right. \nonumber \\&+\sum ^{p_B^{(0)}}_{k=1}J^{(0)}_{B,k} ((-t+B^{(0)}_k)^{-\beta } \nonumber \\&+\left. \sum ^{p_B^{(+1)}}_{k=1}J^{(+1)}_{B,k}(-t+B^{(+1)}_k)^{-(\beta +1)}\right\} ^2 \nonumber \\&+u_2^{(0)}(\beta )^2, \end{aligned}$$
(D101)
and, for \(\beta =1\),
$$\begin{aligned} U_2\approx & {} \sum ^{L-3}_{t=0}\left\{ B_0+ \sum ^{p_B^{(-1)}}_{k=1}J^{(-1)}_{B,k} \log (-t+B^{(-1)}_k) \right. \nonumber \\&+ \sum ^{p_B^{(0)}}_{k=1}J^{(0)}_{B,k} ((-t+B^{(0)}_k)^{-\beta } \nonumber \\&\left. + \sum ^{p_B^{(+1)}}_{k=1}J^{+1}_{B,k}(-t+B^{(+1)}_k)^{-(\beta +1)}\right\} ^2 \nonumber \\&+u_2^{(0)}(\beta )^2. \end{aligned}$$
(D102)
By combining these results, \(U_2\) can be written as
$$\begin{aligned} U_2\approx & {} \sum ^{L-3}_{t=0}\left\{ B_0+\sum ^{+1}_{m=-1} \sum _{k=1}^{p^{(m)}_B}J^{(m)}_{B,k} f_A(-t,\beta +m,B^{(m)}_k) \right\} ^2 \nonumber \\&+u_2^{(0)}(\beta )^2, \end{aligned}$$
(D103)
where
$$\begin{aligned}&B_1^{(-1)}=B_1^{(0)}=B_1^{(+1)}=2L-2-d_{\beta }, \end{aligned}$$
(D104)
$$\begin{aligned}&B_2^{(-1)}=B_2^{(0)}=B_2^{(+1)}=L+d_{\beta }, \end{aligned}$$
(D105)
$$\begin{aligned}&B_3^{(-1)}=B_4^{(0)}=B_3^{(+1)}=L-2+d_{\beta }, \end{aligned}$$
(D106)
$$\begin{aligned}&B_3^{(0)}=L-1+d_{\beta }, \end{aligned}$$
(D107)
\(p^{(-1)}_B=3\), \(p^{(0)}_B=4\), \(p^{(+1)}_B=3\),
$$\begin{aligned}&B_0 \nonumber \\&\quad = {\left\{ \begin{array}{ll} -a^{-\beta }-\frac{1}{2}(1+d_{\beta })^{-\beta }-\frac{\beta }{12}(1+d_{\beta })^{-\beta -1}+\frac{(d_{\beta }+1)^{-\beta +1}}{-\beta +1},\\ -a^{-\beta }-\frac{1}{2}(1+d_{\beta })^{-\beta }-\frac{\beta }{12}(1+d_{\beta })^{-\beta -1}+\log (d_{\beta }+1), \\ \end{array}\right. } \end{aligned}$$
(D108)
$$\begin{aligned}&J_{B,1}^{(-1)}= {\left\{ \begin{array}{ll} 1/(-\beta +1)&{} (\beta \ne 1), \\ 1 &{} (\beta = 1), \end{array}\right. } \end{aligned}$$
(D109)
$$\begin{aligned}&J_{B,2}^{(-1)}=J_{B,3}^{(-1)}= {\left\{ \begin{array}{ll} -1/(-\beta +1)&{} (\beta \ne 1), \\ -1 &{} (\beta = 1), \end{array}\right. } \end{aligned}$$
(D110)
$$\begin{aligned}&J_{B,1}^{(0)}=J_{B,3}^{(0)}=1/2, \end{aligned}$$
(D111)
$$\begin{aligned}&J_{B,2}^{(0)}=1, \end{aligned}$$
(D112)
$$\begin{aligned}&J_{B,4}^{(0)}=-1/2, \end{aligned}$$
(D113)
and
$$\begin{aligned}&J_{B,1}^{(+1)}=-\beta /12,\end{aligned}$$
(D114)
$$\begin{aligned}&J_{B,2}^{(+1)}=J_{B,2}^{(+1)}=\beta /12. \end{aligned}$$
(D115)
Expanding the squared term gives
$$\begin{aligned}&U_2 \nonumber \\&\quad \approx \sum ^{L-3}_{t'=0}\left\{ B_0+\sum ^{+1}_{m=-1} \sum _{k=1}^{p_m}J^{(m)}_{B,k} f_A(t,\alpha +m,B^{(m)}_k) \right\} ^2 \nonumber \\&\quad =B_0^2(L-2)+2B_0\sum ^{+1}_{m=-1}U^{(1)}_2(-t;\beta ,m) \nonumber \\&\qquad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1}U^{(2)}_2(t;\beta ,m_1,m_2) \nonumber \\&\qquad +u_2^{(0)}(\beta )^2, \end{aligned}$$
(D116)
where
$$\begin{aligned}&U^{(1)}_2(t;\beta ,m_1)= \sum _{i=1}^{p^{(B)}_{m_1}} J_{Bi}^{(m_1)} \sum ^{L-2}_{t=0}f_A(-t,\beta +m_i,B^{(m)}_i)\nonumber \\ \end{aligned}$$
(D117)
and
$$\begin{aligned}&U^{(2)}_2(t;\beta ,m_1,m_2) \nonumber \\&\quad =\sum _{i=1}^{p^{(B)}_{m_1}} \sum _{j=1}^{p^{(B)}_{m_2}} J_{Bi}^{(m_1)} J_{Bj}^{(m_2)} \nonumber \\&\qquad \sum ^{L-2}_{t=0}f_A(-t,\beta +m_i,B^{(m)}_i)f_A(-t,\beta +m_j,B^{(m)}_j).\nonumber \\ \end{aligned}$$
(D118)
Using the Euler–Maclaurin formula in Eq. (D19), we can approximate the sum in \(U^{(1)}_2(t;\beta ,m_1,m_2))\) for \(L \ge 3\) as
$$\begin{aligned}&\sum ^{L-3}_{t=0}f_A(t,\alpha _1,V_1) \nonumber \\&\quad \approx G_1^{(R)}(3-L,\max \{-1,3-L\};\alpha _1,V_1) \nonumber \\&\qquad +\int ^{\max \{L-3,1\}}_1 (-t+V_1)^{-\alpha _1} \mathrm{d}t \nonumber \\&\quad \approx G_1^{(R)}(3-L,\max \{-1,3-L\},;\alpha _1,V_1) \nonumber \\&\qquad -G_1^{(I)}(\min \{-L+3,-1\};\alpha _1,V_1) \nonumber \\&\qquad +G_1^{(I)}(-1;\alpha _1,V_1), \end{aligned}$$
(D119)
where \(G_1^{(R)}(x_1,x_2;\alpha _1,V_1)\), for \(\alpha _1 \ne 1\) and \(\alpha _2 \ne 1\), is given by
$$\begin{aligned}&G_1^{(R)}(x_1,x_2;\alpha _1,V_1)=(V_1)^{-\alpha _1} \nonumber \\&\quad +\frac{1}{2}\{(V_1+x_1)^{-\alpha _1}+(V_1+x_2)^{-\alpha _1}\} \nonumber \\&\quad -\frac{1}{12}\{(-\alpha _1)(V_1+x_2)^{-\alpha _1-1} \nonumber \\&\quad -(-\alpha _1)(V_1+x_1)^{-\alpha _1-1}\}. \end{aligned}$$
(D120)
And, for \(x_1=0\) and \(x_2=0\), we define
$$\begin{aligned}&G_1^{(R)}(x_1,x_2;\alpha _1,V_1)=V_1^{-\alpha _1}. \end{aligned}$$
(D121)
For \(\alpha _1 = 0\), the corresponding term is also written as
$$\begin{aligned}&G_1^{(R)}(x_1,x_2;\alpha _1,V_1) \nonumber \\&\quad =\log (V_1)+ \frac{1}{2}\{\log (V_1+x_1)+\log (V_1+x_2)\} \nonumber \\&\qquad -\frac{1}{12}\{(V_1+x_2)^{-1} \nonumber \\&\qquad -(V_1+x_1)^{-1}\}. \end{aligned}$$
(D122)
And, for \(x_1=0\) and \(x_2=0\), we define
$$\begin{aligned}&G_1^{(R)}(x_1,x_2;\alpha _1,V_1)=\log (V_1). \end{aligned}$$
(D123)
In addition, \(G_1^{(I)}(x;\alpha ,V)\) is calculated as
$$\begin{aligned}&G_1^{(I)}(x;\alpha ,V)\nonumber \\&\quad = {\left\{ \begin{array}{ll} {(\alpha \ne 0, \alpha \ne 1)} \\ \int (x+V)^{-\alpha } \mathrm{d}x = \frac{1}{-\alpha +1} (x+V)^{-\alpha +1} \\ {(\alpha =1)} \\ \int (x+V)^{-1} \mathrm{d}x= \log (x+V) \\ {(\alpha =0)} \\ \int \log (x+V) \mathrm{d}x= (x+V)\log (x+V)-x. \\ \end{array}\right. } \end{aligned}$$
(D124)
Here, we omit a constant of integration.
Next, we calculate \(U^{(2)}_2(t;\beta ,m_1,m_2))\) in Eq. (D116). Using the Euler–Maclaurin formula in Eq. (D19), we can approximate the sum in \(U^{(2)}_2(t;\beta ,m_1,m_2))\) as
$$\begin{aligned}&\sum ^{L-3}_{t=0}f_A(-t,\alpha _1,V_1)f_A(-t,\alpha _2,V_2) \end{aligned}$$
(D125)
$$\begin{aligned}&\quad \approx G_2^{(R)}(3-L,\max \{-1,3-L\};\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\qquad +\int ^{\max \{L-3,1\}}_1 (-t+V_1)^{-\alpha _1}(-t+V_2)^{-\alpha _2} \mathrm{d}t \nonumber \\&\quad \approx G_2^{(R)}(3-L,\max \{-1,3-L\},;\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\qquad -G_2^{(I)}(\min \{3-L,1\};\alpha _1,\alpha _2,V_1,V_2) \nonumber \\&\qquad +G_2^{(I)}(-1;\alpha _1,\alpha _2,V_1,V_2). \end{aligned}$$
(D126)
Therefore, substituting Eqs. (D120), (D124), and D126 into Eq. (D116), for \(L>3\), we have
$$\begin{aligned}&U_2 \approx (L-2)B_0^2 \nonumber \\&\quad +2B_0\sum ^{+1}_{m=-1}\sum _{i=1}^{p^{(B)}_{m}} J_{Bi}^{(m)} \{ G_1^{(R)}(-1,3-L,\beta +m,B_i^{(m)}) \nonumber \\&\quad -G_1^{(I)}(-L+3,\beta +m,B_i^{(m)}) \nonumber \\&\quad +G_1^{(I)}(-1,\beta +m,B_i^{(m)}) \} \nonumber \\&\quad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1} \sum _{i=1}^{p^{(B)}_{m_1}} \sum _{j=1}^{p^{(B)}_{m_2}} J_{Bi}^{(m_1)}J_{Bj}^{(m_2)} \{ \nonumber \\&\quad G_2^{(R)}(3-L,-1,\beta +m_1,\beta +m_2; B_i^{(m_1)},B_j^{(m_2)}) \nonumber \\&\quad -G_2^{(I)}(3-L,\beta +m_1,\beta +m_2; B_i^{(m_1)},B_j^{(m_2)}) \nonumber \\&\quad +G_2^{(I)}(-1,\beta +m_1,\beta +m_2; B_i^{(m_1)},B_j^{(m_2)}) \} \nonumber \\&\quad + u^{(0)}_2(\beta )^2 \end{aligned}$$
(D127)
and, for \(L=3\), we have
$$\begin{aligned}&U_2 \approx (L-2)B_0^2 \nonumber \\&\quad +2B_0\sum ^{+1}_{m=-1}\sum _{i=1}^{p^{(B)}_{m}} J_{Bi}^{(m)} G_1^{(R)}(0,0,\beta +m,B_i^{(m)}) \nonumber \\&\quad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1} \sum _{i=1}^{p^{(B)}_{m_1}} \sum _{j=1}^{p^{(B)}_{m_2}} J_{Bi}^{(m_1)}J_{Bj}^{(m_2)} \{ \nonumber \\&\quad G_2^{(R)}(0,0,\beta +m_1,\beta +m_2; B_i^{(m_1)},B_j^{(m_2)}) \} \nonumber \\&\quad + u^{(0)}_2(\beta )^2. \end{aligned}$$
(D128)
For \(L=2\), from Eq. (D124), we have
$$\begin{aligned}&U_2 \approx u^{(0)}_2(\beta )^2. \end{aligned}$$
(D129)
Note that, because we cannot use the integral approximation method, we calculated directly from the sums for \(L=2\).
1.4 Appendix D.4: Calculation of \(U_3(\beta ,L)\)
\(U_3\) is defined by Eq. (D15) as
$$\begin{aligned} U_3= \frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{L(I+2)}_{t=L(I+1)+1}(\theta _4'(t))^2. \end{aligned}$$
(D130)
Substituting Eqs. (D6) and (D8) into Eq. (D130) gives
$$\begin{aligned} U_3= & {} \sum ^{L(I+2)}_{t=L(I+1)+1}\sum ^{L-(t-L(I+1))}_{k=0}(k+d_{\beta })^{-\beta }. \end{aligned}$$
(D131)
Using a shifted index \(t'=t-L(I+1)-1\), we can write
$$\begin{aligned} U_3= & {} \sum ^{L-1}_{t'=0}\left( \sum ^{L-(1+t')}_{k=0}(k+d_{\beta })^{-\beta }\right) ^2. \end{aligned}$$
(D132)
Using the Euler–Maclaurin formula in Eq. (D19), we can obtain
$$\begin{aligned}&U_3 \approx \frac{ Z(\beta )^2}{\check{\eta }^2} \sum ^{L-2}_{t'=0} \left\{ \theta (0)+\frac{\theta (1)+\theta (L-1-t))}{2} \right. \nonumber \\&\quad +\left. \frac{\theta '(L-1-t)-\theta '(1)}{12} +\int ^{L-1-t}_{1}\theta (k)\mathrm{d}k \right\} ^2 \nonumber \\&\quad +\theta (0)^2. \end{aligned}$$
(D133)
Here, because we cannot use the integral approximation method, we calculated directly from the sums for \(t'=L-1\).
Substituting Eq. (D3) into \(\theta (t')\), for \(\beta >0\) and \(\beta \ne 1\), we get
$$\begin{aligned} U_3\approx & {} \sum ^{L-2}_{t=0}\left\{ C_0+ \sum ^{p_C^{(-1)}}_{k=1}J^{(-1)}_{C,k} (-t+C^{(-1)}_k)^{-(\beta -1)} \right. \nonumber \\&+ \sum ^{p_C^{(0)}}_{k=1}J^{(0)}_{C,k} ((-t+C^{(0)}_k)^{-\beta } \nonumber \\&\left. + \sum ^{p_C^{(+1)}}_{k=1}J^{(+1)}_{C,k}(-t+C^{(+1)}_k)^{-(\beta +1)}\right\} ^2 \nonumber \\&+d_{\beta }^{-2\beta }, \end{aligned}$$
(D134)
and, for \(\beta =1\), we get
$$\begin{aligned} U_3\approx & {} \sum ^{L-2}_{t=0}\left\{ C_0+ \sum ^{p_C^{(-1)}}_{k=1}J^{(-1)}_{C,k} \log (-t+C^{(-1)}_k) \right. \nonumber \\&+ \sum ^{p_C^{(0)}}_{k=1}J^{(0)}_{C,k} ((-t+C^{(0)}_k)^{-\beta } \nonumber \\&\left. + \sum ^{p_C^{(+1)}}_{k=1}J^{+1}_{C,k}(-t+C^{(+1)}_k)^{-(\beta +1)}\right\} ^2 \nonumber \\&+d_{\beta }^{-2\beta }. \end{aligned}$$
(D135)
Combining these two equations gives
$$\begin{aligned}&U_3 \approx \sum ^{L-2}_{t=0}\left\{ C_0+\sum ^{+1}_{m=-1} \sum _{k=1}^{p^{(m)}_C}J^{(m)}_{C,k} f_A(-t,\beta +m,C^{(m)}_k) \right\} ^2 \nonumber \\&\quad +d_{\beta }^{-2\beta }, \end{aligned}$$
(D136)
where
$$\begin{aligned}&C_1^{(-1)}=C_1^{(0)}=C_1^{(-1)}=L-1+a, \end{aligned}$$
(D137)
\(p^{(-1)}_C=1\), \(p^{(0)}_C=1\), \(p^{(+1)}_C=1\),
$$\begin{aligned}&C_0 \nonumber \\&\quad = {\left\{ \begin{array}{ll} d_{\beta }^{-\beta }+\frac{1}{2}(1+d_{\beta })^{-\beta }+\frac{\beta }{12}(1+d_{\beta })^{-\beta -1}-\frac{(d_{\beta }+1)^{-\beta +1}}{-\beta +1},\\ d_{\beta }^{-\beta }+\frac{1}{2}(1+d_{\beta })^{-\beta }+\frac{\beta }{12}(1+d_{\beta })^{-\beta -1}-\log (d_{\beta }+1), \\ \end{array}\right. } \end{aligned}$$
(D138)
$$\begin{aligned}&J_{C,1}^{(-1)}= {\left\{ \begin{array}{ll} 1/(-\beta +1)&{} (\beta \ne 1), \\ 1 &{} (\beta = 1), \end{array}\right. } \end{aligned}$$
(D139)
$$\begin{aligned}&J_{C,1}^{(0)}=1/2, \end{aligned}$$
(D140)
and
$$\begin{aligned}&J_{C,1}^{(+1)}=-\beta /12. \end{aligned}$$
(D141)
Expanding the squared term gives
$$\begin{aligned}&U_3 \approx \sum ^{L-2}_{t'=0}\left\{ C_0+\sum ^{+1}_{m=-1} \sum _{k=1}^{p_m}J^{(m)}_{B,k} f_A(t,\alpha +m,B^{(m)}_k) \right\} ^2 \nonumber \\&\quad =C_0^2+2C_0\sum ^{+1}_{m=-1}U^{(1)}_3(-t;\beta ,m) \nonumber \\&\qquad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1}U^{(2)}_3(t;\beta ,m_1,m_2) \nonumber \\&\qquad +d_{\beta }^{-2\beta }, \end{aligned}$$
(D142)
where
$$\begin{aligned}&U^{(1)}_3(t;\beta ,m) \nonumber \\&\quad = \sum _{i=1}^{p^{(C)}_{m}} J_{Ci}^{(m)} \sum ^{L-2}_{t=0}f_A(-t,\beta +m,C^{(m)}_i) \end{aligned}$$
(D143)
and
$$\begin{aligned}&U^{(2)}_3(t;\beta ,m_1,m_2) \nonumber \\&\quad = \sum _{i=1}^{p^{(C)}_{m_1}} \sum _{j=1}^{p^{(C)}_{m_2}} J_{Ci}^{(m_1)} J_{Cj}^{(m_2)} \nonumber \\&\qquad \sum ^{L-2}_{t=0}f_A(-t,\beta +m_i,C^{(m)}_i)f_A(-t,\beta +m_j,C^{(m)}_j). \end{aligned}$$
(D144)
Consequently, as with \(U_2\), for \(L>2\), \(U_3\) is also calculated by
$$\begin{aligned}&U_3 \approx (L-1)C_0^2 \nonumber \\&\quad +2C_0\sum ^{+1}_{m=-1}\sum _{i=1}^{p^{(C)}_{m}} J_{C_i}^{(m)} \{ G_1^{(R)}(-1,2-L,\beta +m,C_i^{(m)}) \nonumber \\&\quad -G_1^{(I)}(-L+2,\beta +m,C_i^{(m)}) \nonumber \\&\quad +G_1^{(I)}(-1,\beta +m,C_i^{(m)}) \} \nonumber \\&\quad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1} \sum _{i=1}^{p^{(C)}_{m_1}} \sum _{j=1}^{p^{(C)}_{m_2}} J_{C_i}^{(m_1)}J_{C_j}^{(m_2)} \{ \nonumber \\&\quad G_2^{(R)}(2-L,-1,\beta +m_1,\beta +m_2; B_i^{(m_1)},B_j^{(m_2)}) \nonumber \\&\quad -G_2^{(I)}(2-L,\beta +m_1,\beta +m_2; C_i^{(m_1)},C_jj^{(m_2)}) \nonumber \\&\quad +G_2^{(I)}(-1,\beta +m_1,\beta +m_2; C_i^{(m_1)},C_j^{(m_2)}) \}\nonumber \\&\quad +d_{\beta }^{-2\beta }, \end{aligned}$$
(D145)
and, for \(L=2\),
$$\begin{aligned}&U_3 \approx (L-1)C_0^2 \nonumber \\&\quad +2C_0\sum ^{+1}_{m=-1}\sum _{i=1}^{p^{(C)}_{m}} J_{C_i}^{(m)} G_1^{(R)}(0,0,\beta +m,C_i^{(m)}) \nonumber \\&\quad +\sum ^{+1}_{m_1=-1}\sum ^{+1}_{m_2=-1} \sum _{i=1}^{p^{(C)}_{m_1}} \sum _{j=1}^{p^{(C)}_{m_2}} J_{C_i}^{(m_1)}J_{C_j}^{(m_2)} \{ \nonumber \\&\quad G_2^{(R)}(0,0,\beta +m_1,\beta +m_2; C_i^{(m_1)},C_j^{(m_2)}) \} \nonumber \\ \nonumber \\&\quad +d_{\beta }^{-2\beta }. \end{aligned}$$
(D146)
1.5 Appendix D.5: Calculation of \(V[R_j^{(L)}]\)
We can obtain \(V[R_j^{(L)}]\) by substituting \(U_1(\beta ,L)\) (Eq. (D95)), \(U_2(\beta ,L)\) (Eq. (D127)), and \(U_3(\beta ,L)\) (Eq. (D145)) into Eq. (D12) approximately.
Appendix E: Asymptotic behavior of \(V[R_j^{(L)}]\) in the case of the power-law forgetting process for \(L>>1\)
In this section, we calculate the asymptotic behavior of \(V[R_j^{(L)}]\) for \(L>>1\). Because \(V[R_j^{(L)}]\) is decomposed into \(U_1(\beta ,L)\), \(U_2(\beta ,L)\), and \(U_3(\beta ,L)\) (Eq. (D12)), we calculate the asymptotic behaviors of \(U_1(\beta ,L)\) (Eq. (D95)), \(U_2(\beta ,L)\) (Eq. (D127)), and \(U_3(\beta ,L)\) (Eq. (D145)), respectively.
1.1 Appendix E.1: Asymptotic behavior of \(U_1(\beta ,L)\) for \(L>>1\)
The dominant terms of Eq. (D95) are the cases of \(m_1=-1\) and \(m_2=-1\), namely,
$$\begin{aligned} \sum ^{p^{(A)}_{(-1)}}_{i=1} \sum ^{p^{(A)}_{(-1)}}_{j=1} J^{(-1)}_{A_i}J^{(-1)}_{A_j} G_2^{(I;Q)}(\beta ,-1,-1,A_i^{(-1)},A_j^{(-1)})\nonumber \\ \end{aligned}$$
(E1)
and
$$\begin{aligned} \sum ^{p^{(A)}_{(-1)}}_{i=1} \sum ^{p^{(A)}_{(-1)}}_{j=1} J^{(-1)}_{A_i}J^{(-1)}_{A_j} G_2^{(I)}(1,\beta -1,\beta -1,A_i^{(-1)},A_j^{(-1}).\nonumber \\ \end{aligned}$$
(E2)
Calculating these terms for \(L>>1\), we can write
$$\begin{aligned}&U_1(\beta ,L) \approx u_1^{(1)}(\beta )L^{3-2\beta }, \end{aligned}$$
(E3)
where, for \(\beta \ne 1\) and \(\beta >0\),
$$\begin{aligned}&u_1^{(1)}(\beta )=\frac{1}{(2-\beta )(1-\beta )^2} \nonumber \\&\quad \times \biggl \{ 4 {}_2F_1(\beta -1,2-\beta ,3-\beta ,-1) -\frac{(2-\beta )(4+2^{3-2\beta })}{3-2\beta } \nonumber \\&\quad +2 \frac{\varGamma (3-\beta )}{\varGamma (\beta -1)}(2^{3-2\beta }q_1(\beta ,L)-4q_2(\beta ,L)) \biggr \} \end{aligned}$$
(E4)
and, for \(\beta =1\),
$$\begin{aligned}&u_1^{(1)}(\beta ) \nonumber \\&\quad =-\frac{4\pi ^2}{6}-4\psi (2)-4\log (2)^2-2\log (-1)^2. \end{aligned}$$
(E5)
In addition,
$$\begin{aligned}&q_1(\beta ,L) \nonumber \\&\quad ={\left\{ \begin{array}{ll} \frac{-\log (2)+\psi (3-2\beta )+\psi (2-\beta )}{\varGamma (4-2\beta )} &{} {(\beta =0.5)}, \\ \varGamma (2\beta -3) &{} {(\beta \ne 0.5)},\\ \end{array}\right. } \end{aligned}$$
(E6)
and
$$\begin{aligned}&q_2(\beta ,L) \nonumber \\&\quad ={\left\{ \begin{array}{ll} \frac{(\psi (3-2 \beta )+\psi (2-\beta ))}{\varGamma (4-2\beta )} &{} {(\beta =0.5)}, \\ \varGamma (2 \beta -3) &{} {(\beta \ne 0.5)}. \\ \end{array}\right. } \end{aligned}$$
(E7)
Here, for the calculations, we use the following approximation formulas of the hypergeometric function [43]:
$$\begin{aligned} {}_2F_1(a,b,c;x) \rightarrow 1 \quad (x \rightarrow 0), \end{aligned}$$
(E8)
in Eqs. (D69), (D70), and (D71) and for the logarithmic function of Eq. (D84). In addition, we replace \(L+\mathrm{constant}\) with L.
1.2 Appendix E.2: Asymptotic behavior of \(U_2(\beta ,L)\) for \(L>>1\)
As with \(U_1(\beta ,L)\), we can calculate the asymptotic behavior of \(U_2(\beta ,L)\). We focus on terms of higher order than O(L) in Eq. (D127). The highest order terms are
$$\begin{aligned} \sum ^{p^{(B)}_{(-1)}}_{i=1} \sum ^{p^{(B)}_{(-1)}}_{j=1} J^{(-1)}_{B_i}J^{(-1)}_{B_j} G_2^{(I;Q)}(3-L,\beta -1,\beta -1,B_i^{(-1)},B_j^{(-1)})\nonumber \\ \end{aligned}$$
(E9)
and
$$\begin{aligned} \sum ^{p^{(B)}_{(-1)}}_{i=1} \sum ^{p^{(B)}_{(-1)}}_{j=1} J^{(-1)}_{B_i}J^{(-1)}_{B_j} G_2^{(I)}(-1,\beta -1,\beta -1,B_i^{(-1)},B_j^{(-1)}).\nonumber \\ \end{aligned}$$
(E10)
The second highest order terms are
$$\begin{aligned} 2B_0 \sum ^{p^{(B)}_{(-1)}}_{i=1} J^{(-1)}_{B_i} G_1^{(I;Q)}(3-L,\beta -1,B_i^{(-1)}) \end{aligned}$$
(E11)
and
$$\begin{aligned} 2B_0 \sum ^{p^{(B)}_{(-1)}}_{i=1} J^{(-1)}_{B_i} G_1^{(I;Q)}(-1,\beta -1,B_i^{(-1)}). \end{aligned}$$
(E12)
The term of O(L) is
$$\begin{aligned} (L-1)B_0^2. \end{aligned}$$
(E13)
Calculating these terms, we can obtain
$$\begin{aligned}&U_2(\beta ,L) \nonumber \\&\quad \approx {\left\{ \begin{array}{ll} u^{(2)}_1(\beta )L^{3-2\beta } u^{(2)}_2(\beta )L^{2-\beta }+u^{(2)}_3(\beta )L &{} {(0<\beta <1)}, \\ u^{(2)}_a \log (L)^2 L + u^{(2)}_b \log (L)L+u^{(2)}_cL &{}{(\beta =1)}, \\ u^{(2)}_3(\beta )L &{} {(\beta >1)}, \end{array}\right. } \end{aligned}$$
(E14)
where
$$\begin{aligned}&u^{(2)}_1(\beta )=\frac{-4 {}_2F_1(\beta -1,2-\beta ,3-\beta ,-1)}{(2-\beta )(1-\beta )^2} \nonumber \\&\quad +\frac{(2^{3-2\beta }+3)}{(3-2\beta )(1-\beta )^2}, \end{aligned}$$
(E15)
$$\begin{aligned}&u^{(2)}_2(\beta )=2 B_0 \frac{2^{2-\beta }-3}{(2-\beta )(1-\beta )}, \end{aligned}$$
(E16)
$$\begin{aligned}&u^{(2)}_3(\beta )= B_0^2, \end{aligned}$$
(E17)
$$\begin{aligned}&u^{(2)}_a=1, \end{aligned}$$
(E18)
$$\begin{aligned}&u^{(2)}_b=-4\log (2)-2-2B_0, \end{aligned}$$
(E19)
$$\begin{aligned}&u^{(2)}_c= 2+4\psi (2)-4\psi (1) \nonumber \\&\quad +\log (2)(4+2\log (2)-4\log (-1)+4B_0).\nonumber \\ \end{aligned}$$
(E20)
Here, for the calculations, we use these approximation formulas of the hypergeometric functions in Eqs. (E8), (D69), (D70), and (D71) and in the logarithmic function of Eq. (D84). In addition, we replace \(L+ \mathrm{constant}\) with L.
1.3 Appendix E.3: Asymptotic behavior of \(U_3(\beta ,L)\) for \(L>>1\)
As with \(U_2(\beta ,L)\), we can calculate the asymptotic behavior of \(U_2(\beta ,L)\). We focus on terms of higher order than O(L) in Eq. (D145). The highest order terms are
$$\begin{aligned} \sum ^{p^{(C)}_{(-1)}}_{i=1} \sum ^{p^{(C)}_{(-1)}}_{j=1} J^{(-1)}_{C_i}J^{(-1)}_{C_j} G_2^{(I;Q)}(2-L,\beta -1,\beta -1,C_i^{(-1)},C_j^{(-1)})\nonumber \\ \end{aligned}$$
(E21)
and
$$\begin{aligned} \sum ^{p^{(C)}_{(-1)}}_{i=1} \sum ^{p^{(C)}_{(-1)}}_{j=1} J^{(-1)}_{C_i}J^{(-1)}_{C_j} G_2^{(I)}(-1,\beta -1,\beta -1,C_i^{(-1)},C_j^{(-1)}).\nonumber \\ \end{aligned}$$
(E22)
The second highest order terms are
$$\begin{aligned} 2C_0 \sum ^{p^{(C)}_{(-1)}}_{i=1} J^{(-1)}_{C_i} G_1^{(I;Q)}(2-L,\beta -1,C_i^{(-1)}) \end{aligned}$$
(E23)
and
$$\begin{aligned} 2C_0 \sum ^{p^{(C)}_{(-1)}}_{i=1} J^{(-1)}_{C_i} G_1^{(I;Q)}(-1,\beta -1,C_i^{(-1)}). \end{aligned}$$
(E24)
The term of O(L) is
$$\begin{aligned} (L-1)C_0^2. \end{aligned}$$
(E25)
Calculating these terms, we can obtain
$$\begin{aligned}&U_3(\beta ,L) \nonumber \\&\quad \approx {\left\{ \begin{array}{ll} u^{(3)}_1(\beta )L^{3-2\beta } u^{(3)}_2(\beta )L^{2-\beta }+u^{(3)}_3(\beta )L &{} {(0<\beta <1)}, \\ u^{(3)}_a \log (L)^2 L + u^{(3)}_b \log (L)L+u^{(3)}_cL &{} {(\beta =1)},\\ u^{(3)}_3(\beta )L &{} {(\beta >1)}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(E26)
where
$$\begin{aligned}&u^{(3)}_1(\beta )=\frac{1}{(1-\beta )^2 (3-2\beta )}, \end{aligned}$$
(E27)
$$\begin{aligned}&u^{(3)}_2(\beta )=2 C_0 \frac{2-\beta }{1-\beta }, \end{aligned}$$
(E28)
$$\begin{aligned}&u^{(3)}_3(\beta )= C_0^2, \end{aligned}$$
(E29)
$$\begin{aligned}&u^{(3)}_a=1, \end{aligned}$$
(E30)
$$\begin{aligned}&u^{(3)}_b=(-2+2C_0), \end{aligned}$$
(E31)
$$\begin{aligned}&u^{(3)}_c= -1+C_0^2+2C_0. \end{aligned}$$
(E32)
Here, for the calculations, we use the approximation formulas of the hypergeometric functions in Eqs. (E8), (D69), (D70), and (D71) and for the logarithmic function in Eq. (D84). In addition, we replace \(L+ \mathrm{constant}\) with L.
1.4 Appendix E.4: Asymptotic behavior of \(V[R_j^{(L)}]\) for \(L>>1\)
Substituting Eqs. (E3), (E14), and (E26) into Eq. (D12), we can obtain
$$\begin{aligned}&V[\delta R_j^{(L)}] \approx \frac{\check{\eta }^2}{Z(\beta )^2} \nonumber \\&\quad \times {\left\{ \begin{array}{ll} u_1(\beta )L^{1-2\beta }+u_2(\beta )L^{-\beta }+u_3(\beta )L^{-1} &{} {(0<\beta <1)}, \\ u_a \log (L)^2 L^{-1} + u_b \log (L) L^{-1}+u_c L^{-1} &{} {(\beta =1)}, \\ u_3(\beta )L^{-1} &{} {(\beta >1)}, \end{array}\right. } \end{aligned}$$
(E33)
$$\begin{aligned}&u_1(\beta )=u_1^{(1)}(\beta )+u_1^{(2)}(\beta )+u_1^{(3)}(\beta ), \end{aligned}$$
(E34)
$$\begin{aligned}&u_2(\beta )=u_2^{(2)}(\beta )+u_2^{(3)}(\beta ), \end{aligned}$$
(E35)
$$\begin{aligned}&u_3(\beta )= u_3^{(2)}(\beta )+u_3^{(3)}(\beta ), \end{aligned}$$
(E36)
$$\begin{aligned}&u_a=1, \end{aligned}$$
(E37)
$$\begin{aligned}&u_b=(-2+2C_0), \end{aligned}$$
(E38)
$$\begin{aligned}&u_c= -1+C_0^2+2C_0. \end{aligned}$$
(E39)
Consequently, the highest order term is obtained by
$$\begin{aligned} V[\delta R_j^{(L)}] \propto {\left\{ \begin{array}{ll} L^{1-2\beta } &{} {(0<\beta <1)}, \\ \log (L)^2 L^{-1} &{}{(\beta =1)}, \\ L^{-1} &{} {(\beta >1)}. \end{array}\right. } \end{aligned}$$
(E40)
Appendix F: Estimation of scaled total number of blogs, m(t), from the data
We estimate the scaled total number of blogs, m(t), using the moving median as follows:
-
1.
We create a set S consisting of indexes of words such that \(\check{c}_j\) takes a value larger than the threshold \(\check{c}_j \ge 100\), where \(\check{c}_j=\sum ^{T}_{t=1}g_j(t)/T\).
-
2.
We estimate m(t) as the median of \(\{g_j(t)/\check{c}_j:j \in S \}\) with respect to j.
-
3.
For \(t=1,2,\ldots ,T\), we calculate m(t) using step 2.
Here, we use only words with \(\check{c}_j \ge 100\) in step 1 because we neglect the discreteness. In step 2, we apply the median because of its robustness to outliers.
Appendix G: MSD of the power-law forgetting process for \(\beta =0.5\) and \(L>>1\)
By rough approximate calculations, we show that the logarithmic diffusion can be derived by the power-law forgetting process given by Eq. (13) or Eq. (D2) for \(\beta =0.5\) and \(L>>1\). More detailed and accurate derivation can be found in Ref. [15].
The MSD of the model given by Eq. (D2) can be calculated as
$$\begin{aligned}&\langle (r(t+L)-r(t))^2 \rangle \nonumber \\&\quad =\langle (\sum ^{\infty }_{s=0} \theta (s) \eta (t+L-s) - \sum ^{\infty }_{s'=0}\theta (s') \eta (t-s'))^2 \rangle \nonumber \\&\quad =\Bigg \langle \Bigg \{\sum ^{-1}_{s=-L} \theta (s+L) \eta (t-s) +\sum ^{\infty }_{s=0} (\theta (s+L) \nonumber \\&\qquad - \theta (s) ) \eta (t-s) \Bigg \}^2 \Bigg \rangle . \end{aligned}$$
(G1)
Using \(\sum ^{B}_{t=A} \eta (t)^2/(B-A) \approx \hat{\eta }^2\) for \(B-A>>1\) and \(L>>1\),
$$\begin{aligned}&\langle (r(t+L)-r(t))^2 \rangle \nonumber \\&\quad \approx \left\langle \hat{\eta }^2 \left( \sum ^{-1}_{s=-L} \theta (s+L)^2 +\sum ^{\infty }_{s=0} (\theta (s+L) - \theta (s) )^2\right) \right\rangle \nonumber \\&\quad =\hat{\eta }^2 (S_1+S_2), \end{aligned}$$
(G2)
where for the case of \(\beta =0.5\) and \(L>>1\),
$$\begin{aligned} S_1\equiv & {} \sum ^{-1}_{s=-L} \theta (s+L)^2 \approx \int ^{-1}_{-L} \theta (s+L)^2 \mathrm{d}s \nonumber \\= & {} \int ^{-1}_{-L} \frac{1}{Z(\beta )^2}(s+L+d_\beta )^{-1}\mathrm{d}s \nonumber \\= & {} \frac{1}{Z(\beta )^2}\left[ \log (s+L+d_\beta ) \right] _{-L}^{-1} \nonumber \\\propto & {} \log (L) \quad (L>>1) \end{aligned}$$
(G3)
and
$$\begin{aligned}&S_2 \equiv \sum ^{\infty }_{s=0} (\theta (s+L) - \theta (s) )^2) \approx \int ^{\infty }_{0} (\theta (s+L) - \theta (s) )^2 \mathrm{d}s \nonumber \\&\quad = \int ^{\infty }_{0} \frac{1}{Z(\beta )^2}((s+L+d_\beta )^{-0.5}-(s+d_\beta )^{-0.5})^2\mathrm{d}s \nonumber \\&\quad = \frac{1}{Z(\beta )^2} \left[ \log \left( \frac{(s+d_\beta +L)(s+d_\beta )}{(\sqrt{\frac{s+d_\beta }{L}}+\sqrt{\frac{s+d_\beta }{L}+1})^4}\right) \right] _{0}^{\infty } \nonumber \\&\quad \propto 2\log (L)-\log (L) = \log (L) \quad (L>>1). \end{aligned}$$
(G4)
Therefore, for \(\beta =0.5\) we can obtain
$$\begin{aligned} \langle (r(t+L)-r(t))^2 \rangle \propto \log (L) \quad (L>>1). \end{aligned}$$
(G5)
This approximation is simple but very rough. Hence, it is not a good approximation for the time scale of blog data (\(L<1000\)). More accurate approximations that hold for small L are discussed in Ref. [15].
Note that the ensemble MSD of the power-law forgetting process on a finite time scale can be roughly calculated as follows:
$$\begin{aligned}&\langle r'(L)_j^2 \rangle _{j} =\left\langle \sum _{s=0}^{(L-1)} \theta (s) \cdot \eta _j(L-s) \right\rangle _{j} \end{aligned}$$
(G6)
$$\begin{aligned}&\approx \check{\eta }^2 \sum _{s=0}^{L-1} \theta (s)^2=\check{\eta }^2 S_1, \end{aligned}$$
(G7)
where \(\langle A_j(t) \rangle _j= \sum ^{W}_{j}A_j(t)/W\) is the ensemble average for \(W>>1\) and we defined the power-law forgetting process for a finite time scale as
$$\begin{aligned} r_j'(t)=\sum _{s=0}^{t-1} \theta (s) \cdot \eta _j(t-s). \end{aligned}$$
(G8)
From these rough calculations, we may say that the difference between the time-averaged MSD given by Eq. (G2) and the ensemble MSD given by Eq. (G8) is S2 term in Eq. (G2).
Figure 5 shows the results of numerical calculations comparing the time-averaged MSD and the ensemble MSD. From these figures, it can be confirmed that the time-averaged MSD is different from the ensemble MSD. In addition, we can also confirm that the numerical simulations agree well with the theoretical curves. Since the current analysis is rough, more detailed analysis of the differences between the time-averaged MSD and the ensemble MSD, including analysis of real data, is needed in the future.
