Big jump principle for heavy-tailed random walks with correlated increments

Abstract

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into physical sciences where systems usually exhibit correlations. Here, we study the principle for a random walk with correlated increments. Examples of the increments are the autoregressive model of first order and the discretised Ornstein–Uhlenbeck process both with heavy-tailed noise. The correlation leads to the dependence of large values of the sum not only on the big jump but also on the following increments. We describe this behaviour by two big jump principles, namely unconditioned and conditioned on the step number when the big jump occurs. The unconditional big jump principle is described by a correlation-dependent shift between the sum and maximum distribution tails. For the conditional big jump principle, the shift depends also on the step number of the big jump.

Appendices

Correction term for the BJPs

1.1 IID random variables

Fig. 13

Histograms of the IID maximum $\delta _\text {max}$ (blue) and the IID sum $x_N$ (red) compared with the IID BJP $N\alpha z^{-1-\alpha }$ (dotted line) and the correction $N\alpha (z-C)^{-1-\alpha }$ with $C=(N-1)\langle \delta _i \rangle$, see Eqs. (48) and (52). The IID random variables follow the Pareto PDF with $\alpha =1.5$. We used $N=10^2$ and $10^5$ realisations

We assume that the IID random variables $\delta _i$ follow the Pareto PDF with $\alpha >1$, i.e. the mean $\langle \delta _i \rangle$ exists. Clearly for the Pareto PDF all random variables are positive and hence $x_N>\delta _\text {max}$. Here, we find a correction to the IID BJP $x_N {\mathop {\sim }\limits ^{d}} \delta _\text {max}$ with the following idea.

The random walk is defined by the sum $x_N=\sum _{i=1}^N \delta _i$. Now, we assume that the maximum happened at b, i.e. $\delta _\text {max}=\delta _b$, is very large. Then, we replace all remaining random variables $\delta _i$, $i \ne b$, by the mean $\langle \delta _i \rangle$. This replacement neglects the fluctuations of the remaining variables due to the dominating large value of the maximum. We get

$$\begin{aligned} x_N {\mathop {\sim }\limits ^{d}} \delta _\text {max} + (N-1) \langle \delta _i \rangle , \end{aligned}$$

(48)

or similarly $x_N {\mathop {\sim }\limits ^{d}} \delta _\text {max} + \langle x_{N-1} \rangle$, see Fig. 13.

We can derive this relationship also from the expansion of the PDFs. The sum PDF is the convolution $f_{x_N}(z)=(f*\ldots *f)^{(N)}(z)$ which is in Laplace space the product ${\hat{f}}_{x_N}(s)=[{\hat{f}}_{\delta _i}(s)]^N$. The Laplace transform of the random variable PDF is ${\hat{f}}_{\delta _i}(s)=\alpha \varGamma (-\alpha )s^\alpha +1-\langle \delta _i \rangle s + {\mathcal {O}}(s^2)$. To get the tail of $f_{x_N}(z)$ we only need the lowest order non-integer exponents of s in the Laplace transform. Hence, we use the small s expansion ${\hat{f}}_{x_N}(s) \sim N \alpha \varGamma (-\alpha ) s^\alpha -N(N-1)\alpha \langle \delta _i \rangle \varGamma (-\alpha )s^{1+\alpha }$. Inverse Laplace transform gives

$$\begin{aligned} f_{x_N}(z)\sim N\alpha z^{-1-\alpha }+N(N-1)\alpha (1+\alpha )\langle \delta _i \rangle z^{-2-\alpha }. \end{aligned}$$

(49)

Now, we need the maximum PDF which is $f_{\delta _\text {max}}(z)=Nf_{\delta _i}(z)[F_{\delta _i}(z)]^{N-1}\sim Nf_{\delta _i}(z) = N\alpha z^{-1-\alpha }$. To get the correction of Eq. (48), we make the ansatz for the variable transform $\delta _\text {max}+C$. The PDF is $f_{\delta _\text {max}+C}(z)=f_{\delta _\text {max}}(z-C)$. The large argument behaviour $N\alpha (z-C)^{-1-\alpha }$ with large $z-C$ is

$$\begin{aligned} f_{\delta _\text {max}+C}(z) \sim N\alpha z^{-1-\alpha }+N\alpha (1+\alpha )C z^{-2-\alpha }. \end{aligned}$$

(50)

Comparison between Eqs. (49) and (50) yields

$$\begin{aligned} C=(N-1)\langle \delta _i \rangle , \end{aligned}$$

(51)

and therefore, shows Eq. (48). In terms of the PDFs, the correction yields

$$\begin{aligned} f_{x_N}(z) \sim f_{\delta _\text {max}}(z-C). \end{aligned}$$

(52)

Summarised, we learn from this IID case that Eq. (48) can be found quite easily by the theme: neglect the fluctuations of the remaining variables (which are not the maximum) and replace their values by the mean.

1.2 Correlated random walk

Fig. 14

a Histogram of ${\tilde{\delta }}_\text {max}$ (blue) conditioned on $b=4$ compared with the large z behaviour $\alpha z^{-1-\alpha }/\varPhi (b)$ (dotted line) and the correction of Eq. (54) (solid line). The $\delta _i$ follow the Pareto PDF with $\alpha =1.5$. The memory kernel is algebraic with $\beta =0.3$. We used $N=5$ and $10^6$ realisations. b Histogram of ${\tilde{\delta }}_\text {max}$ (blue) compared with the large z behaviour $N\alpha z^{-1-\alpha }$ (dotted line) and the correction of Eq. (55) (solid line). The IID random variables follow the Pareto PDF with $\alpha =1.5$. The memory kernel is algebraic with $\beta =0.3$. We used $N=10^2$ and $10^5$ realisations

Fig. 15

a Histogram of ${\tilde{x}}_N$ (red) conditioned on $b=4$ compared with the large z behaviour $(W_{N-b})^\alpha \alpha z^{-1-\alpha }/\varPhi (b)$ (dotted line) and the correction of Eq. (56) (solid line). The $\delta _i$ follow the Pareto PDF with $\alpha =1.5$. The memory kernel is algebraic with $\beta =0.3$. We used $N=5$ and $10^6$ realisations. b Histogram of ${\tilde{x}}_N$ (red) compared with the large z behaviour ${\tilde{\gamma }}_N\alpha z^{-1-\alpha }$ (dotted line) and the correction of Eq. (57) (solid line). The IID random variables follow the Pareto PDF with $\alpha =1.5$. The memory kernel is algebraic with $\beta =0.3$. We used $N=10^2$ and $10^5$ realisations

We consider the correlated random walk model and start with the maximum ${\tilde{\delta }}_\text {max}$. We found the relationship ${\tilde{\delta }}_\text {max} {\mathop {\sim }\limits ^{d}} \delta _\text {max}$, see Eq. (32). In addition, here we can find a correction similar to the IID case of Eq. (48). Per definition the correlated increment is the weighted sum ${\tilde{\delta }}_i=\sum _{j=1}^i M_{i-j} \delta _j$, see Eq. (4). We condition the appearance of the maximum ${\tilde{\delta }}_\text {max}$ at the step number b. Still per definition, it is ${\tilde{\delta }}_b=\delta _b+\sum _{j=1}^{b-1} M_{b-j} \delta _j$. When the maximum is very large, we replace the remaining variables by their mean (remember we assume $\alpha >1$ and Pareto). We get

$$\begin{aligned} {\tilde{\delta }}_b {\mathop {\sim }\limits ^{d}}\delta _b+ \langle \delta _i \rangle \sum _{j=1}^{b-1} M_{b-j} . \end{aligned}$$

(53)

The IID maximum $\delta _b$ is conditioned on the occurrence of ${\tilde{\delta }}_\text {max}$. Therefore, $\delta _b$ follows the conditional PDF $f_{\delta _\text {max}|b}(z|b)\sim \alpha z^{-1-\alpha }/\varPhi (b)$ where $\varPhi (b)$ is the probability that ${\tilde{\delta }}_\text {max}$ happens at b, see Eq. (38). We get the corrected conditional maximum PDF

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}|b}(z|b)\sim \frac{1}{\varPhi (b)} \alpha \left[ z- \langle \delta _i \rangle \sum _{j=1}^{b-1} M_{b-j}\right] ^{-1-\alpha }, \end{aligned}$$

(54)

see Fig. 14. The unconditional maximum PDF is obtained by summing over all b with the weight $\varPhi (b)$, see Eq. (38). We get

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z)\sim N \alpha \left[ z- \langle \delta _i \rangle \sum _{b=1}^N \left( \varPhi (b)\sum _{j=1}^{b-1} M_{b-j}\right) \right] ^{-1-\alpha }, \end{aligned}$$

(55)

see Fig. 14.

Now, we describe the corrections for the correlated sum PDF. We repeat the just presented ansatz. Per definition, it is ${\tilde{x}}_N = \sum _{k=1}^N W_{N-k}\delta _k$, see Eq. (8). We take out the b-th IID increment ${\tilde{x}}_N = W_{N-b} \delta _b + \sum _{k=1,k \ne b}^N W_{N-k} \delta _k$. We assume that the big jump ${\tilde{\delta }}_\text {max}|b$ happens at b and that it is also large so that ${\tilde{\delta }}_\text {max}|b {\mathop {\sim }\limits ^{d}} \delta _b$. Note that the IID big jump $\delta _b$ is conditioned on the step number of ${\tilde{\delta }}_\text {max}|b$. We replace the remaining increments by their mean and get

$$\begin{aligned} {\tilde{x}}_N|b {\mathop {\sim }\limits ^{d}} W_{N-b} \delta _b + \langle \delta _i \rangle \sum _{k=1,k \ne b}^N W_{N-k}. \end{aligned}$$

(56)

Since $\delta _b {\mathop {\sim }\limits ^{d}}\alpha z^{-1-\alpha }/\varPhi (b)$, we find the correction for the conditional BJP

$$\begin{aligned} f_{{\tilde{x}}_N|b}(z|b) \sim \frac{(W_{N-b})^\alpha }{\varPhi (b)} \alpha \left[ z- \langle \delta _i \rangle \sum _{k=1,k \ne b}^N W_{N-k}\right] ^{-1-\alpha }, \nonumber \\ \end{aligned}$$

(57)

see Fig. 15. The corrections for the unconditional BJP are obtained by summing over all b with the weight $\varPhi (b)$, see Eq. (38). We get

$$\begin{aligned} f_{{\tilde{x}}_N}(z) \sim {\tilde{\gamma }}_N \alpha \left[ z- \langle \delta _i \rangle \sum _{b=1}^N\left( \varPhi (b) \sum _{k=1,k \ne b}^N W_{N-k}\right) \right] ^{-1-\alpha }, \nonumber \\ \end{aligned}$$

(58)

see Fig. 15.

Distribution of ${{\tilde{x}}_N}$ in the large N limit

We calculate the large N limit of the correlated sum PDF $f_{{\tilde{x}}_N}(z)$ where the uncorrelated increments follow the Pareto PDF $f_{\delta _i}(z) = \alpha z^{-1-\alpha }\varTheta (z-1)$ with the Heaviside step function $\varTheta (z-1)=1$ for $z \ge 1$ and $\varTheta (z-1)=1$ for $z < 0$.

We use the characteristic function which is the product

$$\begin{aligned} \varphi _{{\tilde{x}}_N}(k) = \prod _{i=1}^N\varphi _{\delta _i}(W_{N-i}k) \end{aligned}$$

(59)

First, we calculate the characteristic function of the uncorrelated increments

$$\begin{aligned} \varphi _{\delta _i}(k)&= \int \limits _{-\infty }^\infty e^{ikz}f_{\delta _i}(z)\mathrm {d}z \nonumber \\&=\int \limits _{-\infty }^\infty \text {cos}(kz)f_{\delta _i}(z)\mathrm {d}z + i \int \limits _{-\infty }^\infty \text {sin}(kz)f_{\delta _i}(z)\mathrm {d}z \nonumber \\&= \alpha \varGamma (-\alpha ) |k|^\alpha \text {cos}\left( \frac{\pi \alpha }{2}\right) \nonumber \\&\quad + {}_pF_q \left( \left\{ - \frac{\alpha }{2} \right\} , \left\{ \frac{1}{2},1-\frac{\alpha }{2} \right\} ,-\frac{k^2}{2} \right) \nonumber \\&\quad +i\Bigg [-\alpha \varGamma (-\alpha ) |k|^{-1+\alpha } \text {sin}\left( \frac{\pi \alpha }{2}\right) \nonumber \\&\quad +\frac{\alpha }{\alpha -1}k {}_pF_q \left( \left\{ \frac{1}{2}- \frac{\alpha }{2} \right\} , \left\{ \frac{3}{2},\frac{3}{2}-\frac{\alpha }{2} \right\} ,-\frac{k^2}{4} \right) \Bigg ]. \end{aligned}$$

(60)

The last step can be checked with Mathematica using $f_{\delta _i}(z) = \alpha z^{-1-\alpha }\varTheta (z-1)$. The small k expansion is

$$\begin{aligned} \varphi _{\delta _i}(k)&\sim \alpha \varGamma (-\alpha )\text {cos}\left( \frac{\pi \alpha }{2}\right) |k|^\alpha \left[ 1-i\text {sign}(k) \text {tan}\left( \frac{\pi \alpha }{2}\right) \right] \nonumber \\&\quad + 1+i\frac{\alpha }{\alpha -1}k+ {\mathcal {O}}(k^2) \end{aligned}$$

(61)

Therefore, for large N, we get

$$\begin{aligned} \varphi _{{\tilde{x}}_N}(k) \sim \text {exp}\left[ ik\mu - c |k|^\alpha \left( 1-i\beta \text {sign}(k)\text {tan}\left( \frac{\pi \alpha }{2}\right) \right) \right] \end{aligned}$$

(62)

with

$$\begin{aligned} \beta&=1,\nonumber \\ c&=-\alpha \varGamma (-\alpha ) \text {cos}\left( \frac{\pi \alpha }{2}\right) \sum _{i=1}^N (W_{N-i})^\alpha ,\nonumber \\ \mu&= {\left\{ \begin{array}{ll} 0 &{} \text { for } \alpha \in (0,1),\\ \frac{\alpha }{\alpha -1}\sum _{i=1}^N W_{N-i} &{} \text { for } \alpha \in (1,2). \end{array}\right. } \end{aligned}$$

(63)

Finally, the characteristic function of the shifted and rescaled correlated sum is

$$\begin{aligned} \varphi _{A_N({\tilde{x}}_N-B_N)}(k) = e^{-iA_NB_Nk}\prod _{i=1}^N\varphi _{\delta _i}(A_NW_{N-i}k). \end{aligned}$$

(64)

$A_N$ and $B_N$ can be chosen such that the limiting distribution is independent of N. One finds $A_N=({\tilde{\gamma }}_N)^{-1/\alpha }$ and $B_N=\mu$ so that the rescaled random variable converges for large N to $L_{\alpha ,\kappa ,c/{\tilde{\gamma }}_N,0}(z)$, see Fig. 16.

Fig. 16

Histogram for $({\tilde{\gamma }}_N)^{-1/\alpha }({\tilde{x}}_N-\mu )$ (red squares) compared with the limiting PDF $L_{\alpha ,\kappa ,c/{\tilde{\gamma }}_N,0}(z)$ (black line) where $N=10^4$, see the explanation for Eq. (64). The $\delta _i$ follow the Pareto PDF with $\alpha =3/2$. The memory kernel is exponential with the parameter $m=0.6$. The number of realisations is $10^4$

Scaling factor ${{\tilde{\gamma }}_N}$ in the large N limit

Here, we calculate the large N limit of the scaling factor defined in Eq. (13) as

$$\begin{aligned} {\tilde{\gamma }}_N = \sum _{k=1}^N (W_{N-k})^\alpha . \end{aligned}$$

(65)

First, we begin with the exponential memory kernel. From Eq. (10) we know that the large N behaviour of the weight is $W_{N-k} \sim 1/(1-m)$. Hence, we can conclude

$$\begin{aligned} {\tilde{\gamma }}_N \sim \frac{1}{(1-m)^\alpha } N. \end{aligned}$$

(66)

Second, for the algebraic memory kernel, we find the large N limit of the scaling factor

$$\begin{aligned} {\tilde{\gamma }}_N = \sum _{k=1}^N \left( \sum _{j=0}^{N-k}(j+1)^{-\beta }\right) ^\alpha , \end{aligned}$$

(67)

by the following arguments. The inner sum behaves asymptotically as

$$\begin{aligned} \sum _{j=0}^{N-k}(j+1)^{-\beta } \sim \int _0^{n-k}(j+1)^{-\beta } \mathrm {d}j \sim \frac{(N-k+1)^{1-\beta }}{1-\beta }. \end{aligned}$$

(68)

The outer sum behaves asymptotically as the integral

$$\begin{aligned} \sum _{k=1}^N \left( \frac{(N-k+1)^{1-\beta }}{1-\beta }\right) ^\alpha \sim \int _1^N \left( \frac{(N-k+1)^{1-\beta }}{1-\beta }\right) ^\alpha \mathrm {d}k \end{aligned}$$

(69)

which can be calculated to

$$\begin{aligned} {\tilde{\gamma }}_N \sim \frac{1}{(1-\beta )^\alpha } \frac{N^{1+\alpha (1-\beta )}}{1+\alpha (1-\beta )} \end{aligned}$$

(70)

with the exponent range $1<1+\alpha (1-\beta )<3$.

Maximum CDF

The maximum PDF of the correlated increments follows the same tail as the maximum PDF of the uncorrelated increments

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z) \sim N A z^{-1-\alpha }, \end{aligned}$$

(71)

i.e it is independent of the correlations. We derive this result by rewriting the definition of the maximum PDF and finding a formula suitable for a proof by induction.

The maximum PDF is $f_{{\tilde{\delta }}_\text {max}}(z) = \mathrm {d}/\mathrm {d}z F_{{\tilde{\delta }}_\text {max}}(z)$ with the cumulative distribution function (CDF) $F_{{\tilde{\delta }}_\text {max}}(z)=\text {Prob}({\tilde{\delta }}_1 \le z ,\ldots , {\tilde{\delta }}_N \le z)$ which is

$$\begin{aligned} \begin{aligned} F_{{\tilde{\delta }}_\text {max}}(z) = \int \limits _{-\infty }^z \mathrm {d}z_1\ldots \int \limits _{-\infty }^z \mathrm {d}z_N f_{{\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_N}(z_1,\ldots ,z_N).\qquad \end{aligned}\end{aligned}$$

(72)

$f_{{\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_N}(z_1,\ldots ,z_N)$ is the joint PDF of the correlated increments $({\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_N)$. We rewrite this joint PDF using the chain rule of probability so that

$$\begin{aligned} f_{{\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_N}(z_1,\ldots ,z_N) = \prod _{i=1}^N f_{{\tilde{\delta }}_i|{\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_{i-1}}(z_i|z_1,\ldots ,z_{i-1}) \end{aligned}$$

(73)

Note that for $i=1$ the right hand side gives $f_{{\tilde{\delta }}_1}(z_1)$. We can simplify these conditional PDFs. When the first $i-1$ uncorrelated increments $\delta _1,\ldots ,\delta _{i-1}$ are given, then ith correlated increment is the sum of a constant and the i-th uncorrelated increment ${\tilde{\delta }}_i = \sum _{j=1}^{i-1}M_{i-j}\delta _j + \delta _i$, see Eq. (4). Therefore, the conditional PDFs of Eq. (73) are the single PDFs with shifted argument

$$\begin{aligned} f_{{\tilde{\delta }}_i|{\tilde{\delta }}_1,\ldots ,{\tilde{\delta }}_{i-1}}(z_i|z_1,\ldots ,z_{i-1}) = f_{\delta _i}\left( z_i-\sum _{j=1}^{i-1}M_{i-j}z_j \right) . \end{aligned}$$

(74)

Hence, we get the correlated maximum CDF in a suitable form

$$\begin{aligned} F_{{\tilde{\delta }}_\text {max}}(z) = \ \int \limits _{-\infty }^z\mathrm {d} z_1 \ldots \int \limits _{-\infty }^z \mathrm {d}z_N \prod _{i=1}^N f_{\delta _i}\left( z_i-\sum _{j=1}^{i-1}M_{i-j}z_j \right) . \end{aligned}$$

(75)

Now, we approximate this formula and find an expression predestinated for a proof by induction to show Eq. (71).

We approximate the first inner integral over $z_N$, namely

$$\begin{aligned}&\int \limits _{-\infty }^z \mathrm {d}z_N f_{\delta _N} \left( z_N - \sum _{j=1}^{N-1}M_{N-1}z_j\right) \nonumber \\&\quad = F_{\delta _N} \left( z - \sum _{j=1}^{N-1}M_{N-1}z_j\right) \sim F_{\delta _N}(z). \end{aligned}$$

(76)

The approximation in the last step is due the binomial theorem for large values of z. The remaining $N-1$ integrals over $z_1,\ldots ,z_{N-1}$ in Eq. (75) are exactly the maximum CDF for $N-1$ variables $F_{{\tilde{\delta }}_\text {max}}(z;N-1)$. We added the number of variables $N-1$ into the notation and will continue with this notation for the following formulas. Therefore, the maximum CDF Eq. (75) behaves as

$$\begin{aligned} F_{{\tilde{\delta }}_\text {max}}(z;N) \sim F_{{\tilde{\delta }}_\text {max}}(z;N-1) F_{\delta _N}(z). \end{aligned}$$

(77)

It is important to mention that we implied the condition $M_{N-j}<1$ for the validity of this formula. We explain it for $N=2$. The CDF $F_{\delta _2}(z-M_1z_1)$ in Eq. (76) requires $z-M_1z_1>1$ which yields $z_1<(z-1)/M_1 \sim z/M_1$ for large z. Only for $M_1<1$ it is $z/M_1>z$ so that the integral over $z_1$ in Eq. (75) goes until z and not only until $z/M_1$. And this gives Eq. (77).

The maximum PDF is the derivative of the maximum CDF Eq. (77) and behaves as

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z;N)&\sim f_{{\tilde{\delta }}_\text {max}}(z;N-1) F_{\delta _N}(z)+F_{{\tilde{\delta }}_\text {max}}(z;N-1)\nonumber \\&\quad f_{\delta _N}(z). \end{aligned}$$

(78)

Furthermore, we can use here that both CDFs are about 1 for large z so that

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z;N) \sim f_{{\tilde{\delta }}_\text {max}}(z;N-1)+ f_{\delta _N}(z). \end{aligned}$$

(79)

We finally found the formula suitable for a proof by induction to show the tail behaviour Eq. (71). Starting with $N=2$, we can directly get $f_{{\tilde{\delta }}_\text {max}}(z;2) \sim 2 A z^{-1-\alpha }$. Furthermore, assuming the scaling $f_{{\tilde{\delta }}_\text {max}}(z;N) \sim NA z^{-1-\alpha }$ of Eq. (71) is correct, we see from Eq. (79) that $f_{{\tilde{\delta }}_\text {max}}(z;N+1) \sim (N+1)A z^{-1-\alpha }$. Thus, the scaling Eq. (71) is indeed correct.

1.1 Pareto IID random variables for $N=2$

We calculate the maximum PDF $f_{{\tilde{\delta }}_\text {max}}(z)$ for $N=2$ when the IID random variables follow the Pareto PDF $f_{\delta _i}(z)=\alpha z^{-1-\alpha }$ with $z>1$. We use $m=M_1$. From Eq. (75), we get

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z)&= \frac{\mathrm {d}}{\mathrm {d} z} F_{{\tilde{\delta }}_\text {max}}(z) = \int \limits _{-\infty }^z f_{\delta _i}(z) f_{\delta _i}(z_2-mz) \mathrm {d}z_2 \nonumber \\&\quad + \int \limits _{-\infty }^z f_{\delta _i}(z_1) f_{\delta _i}(z-mz_1) \mathrm {d}z_1. \end{aligned}$$

(80)

The first integral gives

$$\begin{aligned} \int \limits _{-\infty }^z f_{\delta _i}(z) f_{\delta _i}(z_2-mz) \mathrm {d}z_2 = f_{\delta _i}(z)F_{\delta _i}(z-mz). \end{aligned}$$

(81)

For the second integral, we derive the indefinite integral

$$\begin{aligned} \int f_{\delta _i}(z_1) f_{\delta _i}(z-mz_1) \mathrm {d}z_1 = I(z_1,z). \end{aligned}$$

(82)

Mathematica gives us

$$\begin{aligned} I(z_1,z)&= \frac{\alpha }{1-\alpha } z^{-2} {z_1}^{-\alpha }(z-mz_1)^{-\alpha }\left( 1-\frac{mz_1}{z}\right) ^\alpha \nonumber \\&\quad \times \Big [\alpha mz_1 \times {}_2F_1(1-\alpha ,1+\alpha ,2-\alpha ,mz_1/m)\nonumber \\&\quad +(\alpha -1)z \times {}_2F_1(-\alpha ,\alpha ,1-\alpha ,mz_1/z)\Big ] \end{aligned}$$

(83)

where ${}_2F_1$ is the hypergeometric function. The integrand of Eq. (82) yields 3 different regions for z. Thus, we get the second integral as

$$\begin{aligned}&\int \limits _{-\infty }^z f_{\delta _i}(z_1) f_{\delta _i}(z-mz_1) \mathrm {d}z_1\nonumber \\&\quad = {\left\{ \begin{array}{ll} 0, &{} z \in (-\infty , 1+m),\\ I\left( \frac{z-1}{m},z \right) -I(1,z), &{} z \in \left( 1+m,\frac{1}{1-m}\right) ,\\ I(z,z)-I(z,1), &{} z \in \left( \frac{1}{1-m},\infty \right) .\\ \end{array}\right. } \end{aligned}$$

(84)

Therefore, we find the maximum PDF exactly as the sum of the first integral Eq. (81) and the second integral Eq. (84).

The large z behaviour is the sum of the large z behaviours of Eqs. (81) and (84). The first integrals behaves as $f_{\delta _i}(z)=\alpha z^{-1-\alpha }$. The second integral gives also (after some calculations) the same behaviour $\alpha z^{-1-\alpha }$. So that we finally have

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z)\sim 2 \alpha z^{-1-\alpha }. \end{aligned}$$

(85)

This is the same large z behaviour as for the IID maximum $f_{\delta _\text {max}}(z)\sim 2\alpha z^{-1-\alpha }$. Therefore, it shows ${\tilde{\delta }}_\text {max} {\mathop {\sim }\limits ^{d}} \delta _\text {max}$ for $N=2$ where we get the large z behaviour from the exact expression of $f_{{\tilde{\delta }}_\text {max}}(z)$.

1.2 Uniform IID random variables for $N=2$

We calculate the maximum PDF $f_{{\tilde{\delta }}_\text {max}}(z)$ for $N=2$ when the IID random variables follow the uniform PDF on the interval [0, 1], i.e. $f_{\delta _i}(z)=\varTheta (1-z)\varTheta (z)$ where $\varTheta$ is the Heaviside step function. We use $m=M_1$. From Eq. (75), we get

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z)&= \frac{\mathrm {d}}{\mathrm {d} z} F_{{\tilde{\delta }}_\text {max}}(z)= \int \limits _{-\infty }^z f_{\delta _i}(z) f_{\delta _i}(z_2-mz) \mathrm {d}z_2 \nonumber \\&\quad + \int \limits _{-\infty }^z f_{\delta _i}(z_1) f_{\delta _i}(z-mz_1) \mathrm {d}z_1. \end{aligned}$$

(86)

The first integral gives

$$\begin{aligned} \int \limits _{-\infty }^z f_{\delta _i}(z) f_{\delta _i}(z_2-mz) \mathrm {d}z_2 = (1-m)z \varTheta (z) \varTheta (1-z). \end{aligned}$$

(87)

The second integral gives

$$\begin{aligned}&\int \limits _{-\infty }^z f_{\delta _i}(z_1) f_{\delta _i}(z-mz_1) \mathrm {d}z_1\nonumber \\&\quad = z \varTheta (z) \varTheta (1-z) + \frac{1+m-z}{m}\varTheta (z-1) \varTheta (1+m-z). \end{aligned}$$

(88)

Finally, the maximum PDF is

$$\begin{aligned} f_{{\tilde{\delta }}_\text {max}}(z) = {\left\{ \begin{array}{ll} (2-m)z &{} \text { for } 0<z<1,\\ (1+m-z)/m &{} \text { for } 1<z<1+m \end{array}\right. } \end{aligned}$$

(89)

which we observe in Fig. 2.