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Distributional Behavior of Time Averages of Non-\(L^1\) Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

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Abstract

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

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Acknowledgments

This work was inspired by the conference of “Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics.” We are indebted to T. Miyaguchi and H. Takahashi for his helpful comments. This work was partially supported by Grant-in-Aid for Young Scientists (B) (Grant No. 26800204 to TA) and by the MEXT, Japan (Platform for Dynamic Approaches to Living System; KAKENHI 23115007 to SS).

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Correspondence to Takuma Akimoto.

Appendices

Appendix 1: First Moment

The Laplace transform of \(\langle X_t \rangle \) is given by

$$\begin{aligned} \langle X_s \rangle&= \left. -\frac{\partial {\hat{P}}(k,s)}{\partial k}\right| _{k=0} \nonumber \\&= -\frac{\hat{\psi }'(0,s) \int _0^\infty W(\tau ) e^{-s\tau } d\tau }{ [1 - {\hat{\psi }}(0,s)]^2} + \frac{\int _0^\infty w(\tau ) [\int _0^\tau I(t,\tau )e^{-st}dt] d\tau }{ 1 - {\hat{\psi }}(0,s)} \nonumber \\&=-\frac{\hat{\psi }'(0,s)}{ s[1 - \hat{w}(s)]} +\frac{\int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st}}{ 1 - {\hat{w}}(s)}. \end{aligned}$$
(48)

Because the asymptotic behavior of \(w(\tau ) I(t,\tau )\) is given by \(w(\tau ) I(t,\tau )\sim AB\gamma t\tau ^{-2+\gamma -\alpha }/|\Gamma (-\alpha )|\) for \(t/\tau \ll 1\), the integral in the second term can be calculated as follows:

$$\begin{aligned} \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st}&\sim \frac{AB\gamma }{|\Gamma (-\alpha )|} \int _0^\infty t\left[ \int _t^\infty \tau ^{-2-\alpha +\gamma }d\tau \right] e^{-st} dt \nonumber \\&= \frac{AB\gamma }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \int _0^\infty t^{\gamma -\alpha }e^{-st}dt \nonumber \\&\sim \frac{AB\gamma \Gamma (\gamma -\alpha +1) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \frac{1}{s^{\gamma -\alpha +1}}. \end{aligned}$$
(49)

Using the asymptotic behavior of \(\hat{\psi }'(0,s)\),

$$\begin{aligned} \hat{\psi }'(0,s) \sim - \frac{AB \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{1}{s^{\gamma -\alpha }}\quad (\gamma >\alpha ), \end{aligned}$$
(50)

we have

$$\begin{aligned} \langle X_s \rangle&= \frac{B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|} \frac{1}{s^{1+\gamma }} +\frac{B\gamma (\gamma -\alpha ) \Gamma (\gamma -\alpha ) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )} \frac{1}{s^{1+\gamma }} \nonumber \\&= \frac{B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|} \left[ 1+\frac{\gamma (\gamma -\alpha )}{1+\alpha -\gamma }\right] \frac{1}{s^{1+\gamma }}. \end{aligned}$$
(51)

Appendix 2: Second Moment

The Laplace transform for the second moment of \(X_t\) is given by

$$\begin{aligned} \langle X^{2}_s \rangle&= \left. \frac{\partial ^{2} {\hat{P}}(k,s)}{\partial k^{2}} \right| _{k=0} \nonumber \\&= \frac{\int _0^\infty w(\tau ) \int _0^\tau I(t,\tau )^2 e^{-st} dt d\tau }{1-\hat{w}(s)} - \frac{2\hat{\psi }'(0,s) \int _0^\infty w(\tau ) \int _0^\tau I(t,\tau ) e^{-st}dt d\tau }{[1-\hat{w}(s)]^2}\nonumber \\&\quad + \frac{2 \{\hat{\psi }'(0,s)\}^2}{s[1-\hat{w}(s)]^2} + \frac{\hat{\psi }''(0,s)}{s[1-\hat{w}(s)]} \nonumber \\&= \frac{\int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )^2\right] e^{-st}}{1-\hat{w}(s)} - \frac{2\hat{\psi }'(0,s) \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )\right] e^{-st} }{[1-\hat{w}(s)]^2}\nonumber \\&\quad + \frac{2 \{\hat{\psi }'(0,s)\}^2}{s[1-\hat{w}(s)]^2} + \frac{\hat{\psi }''(0,s)}{s[1-\hat{w}(s)]}. \end{aligned}$$
(52)

Because the asymptotic behavior of \(w(\tau ) I(t,\tau )^2\) is given by \(w(\tau ) I(t,\tau )^2\sim A(B\gamma t)^2\tau ^{-3+2\gamma -\alpha }/|\Gamma (-\alpha )|\) for \(t/\tau \ll 1\), the integral in the first term can be calculated as follows:

$$\begin{aligned} \int _0^\infty dt \left[ \int _t^\infty d\tau w(\tau ) I(t,\tau )^2 \right] e^{-st}&\sim \frac{A(B\gamma )^2}{|\Gamma (-\alpha )|} \int _0^\infty t^2\left[ \int _t^\infty \tau ^{-3-\alpha +2\gamma }d\tau \right] e^{-st} dt \nonumber \\&= \frac{A(B\gamma )^2 }{|\Gamma (-\alpha )| (2+\alpha -2\gamma )} \int _0^\infty t^{2\gamma -\alpha }e^{-st}dt \nonumber \\&\sim \frac{A(B\gamma )^2 \Gamma (2\gamma -\alpha +1) }{|\Gamma (-\alpha )| (2+\alpha - 2\gamma )} \frac{1}{s^{2\gamma -\alpha +1}}. \end{aligned}$$
(53)

Using Eq. (49) and

$$\begin{aligned} \hat{\psi }''(0,s) \sim \frac{AB^2 \Gamma (2\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{1}{s^{2\gamma -\alpha }}, \quad (\gamma >\alpha ) \end{aligned}$$
(54)

we have

$$\begin{aligned} \langle X^2_s \rangle&\sim \left[ \frac{(B\gamma )^2 \Gamma (2\gamma -\alpha +1) }{|\Gamma (-\alpha )| (2+\alpha - 2\gamma )} + \frac{2B \Gamma (\gamma -\alpha )}{|\Gamma (-\alpha )|}\frac{B\gamma \Gamma (\gamma -\alpha +1) }{|\Gamma (-\alpha )| (1+\alpha -\gamma )}\right. \nonumber \\&\left. \quad +\,\, \frac{2B^2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|^2} +\; \frac{B^2 \Gamma (2\gamma -\alpha )}{|\Gamma (-\alpha )|} \right] \frac{1}{s^{2\gamma +1}} \nonumber \\&= \frac{B^2 }{|\Gamma (-\alpha )|} \left[ \frac{\gamma ^2 (2\gamma -\alpha ) \Gamma (2\gamma -\alpha ) }{2+\alpha - 2\gamma } +\; \frac{2 \gamma (\gamma -\alpha ) \Gamma (\gamma -\alpha )^2 }{|\Gamma (-\alpha )| (1+\alpha -\gamma )}\right. \nonumber \\&\left. \quad +\,\, \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} + \Gamma (2\gamma -\alpha ) \right] \frac{1}{s^{2\gamma +1}} \nonumber \\&= \frac{B^2 }{|\Gamma (-\alpha )|} \left[ \left\{ 1+\frac{\gamma ^2 (2\gamma -\alpha ) }{2+\alpha - 2\gamma }\right\} \Gamma (2\gamma -\alpha )\right. \nonumber \\&\left. \quad +\,\, \left\{ 1 + \frac{\gamma (\gamma -\alpha ) }{1+\alpha -\gamma } \right\} \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} \right] \frac{1}{s^{2\gamma +1}}. \end{aligned}$$
(55)

The inverse Laplace transform reads

$$\begin{aligned} \langle X^2_t \rangle&\sim \frac{B^2 }{|\Gamma (-\alpha )|\Gamma (1+2\gamma )} \left[ \left\{ 1+\frac{\gamma ^2 (2\gamma -\alpha ) }{2+\alpha - 2\gamma }\right\} \Gamma (2\gamma -\alpha )\right. \nonumber \\&\left. \quad + \left\{ 1 + \frac{\gamma (\gamma -\alpha ) }{1+\alpha -\gamma } \right\} \frac{2 \Gamma (\gamma -\alpha )^2}{|\Gamma (-\alpha )|} \right] t^{2\gamma }. \end{aligned}$$
(56)

Appendix 3: \(n\)th (\(n>1\)) Moment and Its Coefficient \(M_n(\alpha ,\gamma )\)

The \(n\)-th (\(n>1\)) differentiation of \(\hat{P}(k,s)\) is given by the recursion relation:

$$\begin{aligned} \hat{P}^{(n)}(k,s)&= \frac{1}{1-\hat{\psi }(k,s)} \left[ \sum _{i=1}^{n-1} c_{n,i}\hat{P}^{(i)}(k,s) \hat{\psi }^{(n-i)}(k,s) + \hat{P}(k,s) \hat{\psi }^{(n)}(k,s)\right. \nonumber \\&\left. \quad + \int _0^\infty d\tau w(\tau ) \hat{\Psi }^{(n)}(k,s;\tau ) \right] , \end{aligned}$$
(57)

where \(c_{n,i}=c_{n-1,i}+c_{n-1,i-1}\) (\(i=2, \ldots , n-2\)) and \(c_{n,n-1}=c_{n,1}=n\).

Here,

$$\begin{aligned} \int _0^\infty d\tau w(\tau ) \hat{\Psi }^{(n)}(0,s;\tau )&= \int _0^\infty d\tau w(\tau ) \int _0^\tau I(t,\tau )^n e^{-st}dt \nonumber \\&= \int _0^\infty dt \int _t^\infty d\tau w(\tau ) I(t,\tau )^n e^{-st} \nonumber \\&\sim \frac{A(B\gamma )^n}{|\Gamma (-\alpha )|} \int _0^\infty dt t^n \int _t^\infty \tau ^{-(n+1) -\alpha +n \gamma } d\tau e^{-st} \nonumber \\&\sim \frac{AB^n \gamma ^n \Gamma (n\gamma -\alpha +1)}{|\Gamma (-\alpha )|(n+\alpha -n\gamma )}\frac{1}{s^{1-\alpha + n\gamma }}. \end{aligned}$$
(58)

We assume

$$\begin{aligned} \hat{P}^{(i)}(0,s) \sim (-1)^i\frac{B^i M_i(\alpha ,\gamma )}{|\Gamma (-\alpha )|} \frac{1}{s^{1+i\gamma }}, \end{aligned}$$
(59)

for \(i<n\). It follows that

$$\begin{aligned} \hat{P}^{(n)}(0,s)&= \left[ \sum _{i=1}^{n-1} c_{n,i} \frac{M_i(\alpha ,\gamma )}{|\Gamma (-\alpha )|} \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|} +\; \frac{\Gamma (n\gamma - \alpha )}{|\Gamma (-\alpha )|}\right. \nonumber \\&\left. \quad +\,\, \frac{\gamma ^n\Gamma (n\gamma - \alpha +1)}{|\Gamma (-\alpha )|(n+\alpha -n\gamma )} \right] \frac{(-B)^n}{s^{1 + n\gamma }} \nonumber \\&= \left[ \sum _{i=1}^{n-1} c_{n,i} M_i(\alpha ,\gamma ) \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|}\right. \nonumber \\&\left. \quad + \left\{ 1+ \frac{\gamma ^n(n\gamma -\alpha )}{n+\alpha -n\gamma }\right\} \Gamma (n\gamma - \alpha ) \right] \frac{(-B)^n}{|\Gamma (-\alpha )| s^{1 + n\gamma }}. \end{aligned}$$
(60)

Therefore,

$$\begin{aligned} M_n (\alpha ,\gamma ) = \sum _{i=1}^{n-1} c_{n,i} M_i(\alpha ,\gamma ) \frac{\Gamma ((n-i)\gamma - \alpha )}{|\Gamma (-\alpha )|} + \left\{ 1+ \frac{\gamma ^n(n\gamma -\alpha )}{n+\alpha -n\gamma }\right\} \Gamma (n\gamma - \alpha ). \end{aligned}$$
(61)

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Akimoto, T., Shinkai, S. & Aizawa, Y. Distributional Behavior of Time Averages of Non-\(L^1\) Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures. J Stat Phys 158, 476–493 (2015). https://doi.org/10.1007/s10955-014-1138-0

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