Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Abstract

Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $\alpha$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits a phase transition from diffusive to superdiffusive behavior at the critical value $\alpha _c=1/2$. For $\alpha \in (\alpha _c, 1)$, there is a scaling factor $a_n$ of order $n^{\alpha }$ such that the position $S_n$ of the walker at time n scaled by $a_n$ converges to a nondegenerate random variable ${\widehat{W}}$, whose distribution is not Gaussian. Our main result shows that the fluctuation of $S_n$ around ${\widehat{W}} \cdot a_n$ is still Gaussian. We also give a description of a phase transition induced by bias decaying polynomially in time.

Appendix

1.1 Lemmas from Calculus

Lemma A1

(Schütz and Trimper [20], (12) and (13)) The general solution $\{x_n\}$ to the recursion

$$\begin{aligned} {\left\{ \begin{array}{ll} x_1=f_0, \\ x_{n+1} = f_n+g_n \cdot x_n &{}[n=1,2,\cdots ] \\ \end{array}\right. } \end{aligned}$$

is given by

$$\begin{aligned} x_n = \sum _{\ell =0}^{n-1} f_\ell \cdot \prod _{k=\ell +1}^{n-1} g_k. \end{aligned}$$

Lemma A2

(see e.g. Knopp [18], p.34) If a real sequence $\{a_n\}$ and a positive sequence $\{b_n\}$ satisfy

$$\begin{aligned} \lim _{n\rightarrow \infty } \dfrac{a_n}{b_n} = L \in {\mathbb {R}} \cup \{ \pm \infty \},\quad \text{ and }\quad \sum _{n=1}^{\infty } b_n = +\infty , \end{aligned}$$

then

$$\begin{aligned} \lim _{n\rightarrow \infty } \dfrac{\sum _{k=1}^n a_k}{\sum _{k=1}^n b_k} = L. \end{aligned}$$

Lemma A3

Consider a positive real sequence $\{a_n\}$ which monotonically diverges to $+\infty$, and another real sequence $\{b_n\}$.

• (i) (Kronecker’s lemma) If $\displaystyle \sum _{k=1}^{\infty } \dfrac{b_k}{a_k}$ converges, then $\displaystyle \lim _{n \rightarrow \infty } \dfrac{1}{a_n}\sum _{k=1}^n b_k =0$.

• (ii) (Heyde [14], Lemma 1 (ii)) If $\displaystyle \sum _{k=1}^{\infty } a_k b_k$ converges, then$\displaystyle \lim _{n \rightarrow \infty } a_n\sum _{k=n}^{\infty } b_k =0$.

1.2 Martingale Limit Theorems

Theorem A1

(Hall and Heyde [15], Theorem 2.15) Suppose that $\{M_n\}$ is a square-integrable martingale with mean 0. Let $d_k=M_k-M_{k-1}$ for $k=1,2,\cdots$, where $M_0=0$. On the event

$$\begin{aligned} \left\{ \sum _{k=1}^{\infty } E[(d_k)^2 \mid {\mathcal {F}}_{k-1}] <+\infty \right\} , \end{aligned}$$

$\{M_n\}$ converges a.s..

The following theorem is a special case of Heyde [14], Theorem 1 (b).

Theorem A2

Suppose that $\{M_n\}$ is a square-integrable martingale with mean 0. Let $d_k=M_k-M_{k-1}$ for $k=1,2,\cdots$, where $M_0=0$. If

$$\begin{aligned} \displaystyle \sum _{k=1}^{\infty } E[(d_k)^2] < +\infty \end{aligned}$$

holds in addition, then we have the following: Let $\displaystyle s_n^2 := \sum _{k=n}^{\infty } E[ (d_k)^2]$.

• (i) The limit $M_{\infty }:=\sum _{k=1}^{\infty } d_k$ exists a.s., and $M_n {\mathop {\rightarrow }\limits ^{L^2}} M_{\infty }$.

• (ii) Assume that

  • a) $\displaystyle \dfrac{1}{s_n^2} \sum _{k=n}^{\infty } (d_k)^2 \rightarrow 1$ as $n \rightarrow \infty$ in probability, and

  • b) $\displaystyle \lim _{n \rightarrow \infty } \dfrac{1}{s_n^2}E\left[ \sup _{k \ge n} (d_k)^2\right] =0$.

  Then we have

  $$\begin{aligned} \dfrac{M_{\infty } - M_n}{s_{n+1}} = \dfrac{ \sum _{k=n+1}^{\infty } d_k}{s_{n+1}}{\mathop {\rightarrow }\limits ^{\text {d}}} N(0,1). \end{aligned}$$

• (iii) Assume that the following three conditions hold:

  • a’) $\displaystyle \dfrac{1}{s_n^2} \sum _{k=n}^{\infty } (d_k)^2 \rightarrow 1$ as $n \rightarrow \infty$ a.s.,

  • c) $\displaystyle \sum _{k=1}^{\infty } \dfrac{1}{s_k} E[ |d_k| : |d_k| > \varepsilon s_k] < +\infty$ for any $\varepsilon > 0$, and

  • d) $\displaystyle \sum _{k=1}^{\infty } \dfrac{1}{s_k^4} E[ (d_k)^4 : |d_k| \le \delta s_k] < +\infty$ for some $\delta > 0$.

  Then $\displaystyle \limsup _{n \rightarrow \infty } \pm \dfrac{M_{\infty } - M_n}{{\widehat{\phi }}(s_{n+1}^2)} =1$ a.s., where ${\widehat{\phi }}(t):=\sqrt{2t\log |\log t|}$.

Remark A1

A sufficient condition for b) in Theorem A2 is that

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty }\dfrac{1}{s_n^2} \sum _{k=n}^{\infty } E[ (d_k)^2 : |d_k| > \varepsilon s_n] =0 \end{aligned}$$

for any $\varepsilon > 0$. (See the proof of Corollary 1 in Heyde [14]).