On the central limit theorem for the elephant random walk with gradually increasing memory and random step size

1.   Introduction

The elephant random walk (ERW) represents a unique form of one-dimensional random walk along integers, notable for its retention of complete memory regarding its entire past trajectory. Schütz and Trimper ^[13] introduced the ERW as a means to investigate the memory effects inherent in a non-Markovian random walk. This model is inspired by the notion that elephants possess an exceptional memory, embodying the popular belief that elephants never forget their paths. It is well known that the asymptotic properties of the ERW depend on its memory parameter $p\in[0, 1]$. More precisely, the ERW exhibits three regimes: the diffusive regime for $0\leqslant p < 3/4$, the critical regime for $p = 3/4$, and the superdiffusive regime of $3/4 < p\leqslant 1$. Many interesting limit theorems are already known for the ERW. In particular, $n^{-1}S_n\to0$ a.s. for any $p\in]0, 1[$ and $n^{-1/2}S_n\to\mathcal N(0, (3-4p)^{-1})$ in distribution in the diffusive regime. In the critical regime, we have $(n\log n)^{-1/2}S_n\to\mathcal N(0, 1)$ in distribution and $n^{1-2p}S_n\to L$ almost surely and in $\mathbb{L}^4$ in the superdiffusive regime. The reader can refer to Baur and Bertoin ^[2], Bercu ^[3,4], Bercu and Laulin ^[5], Kubota and Takei ^[10], Coletti, Gava, and Schütz ^[6], Laulin ^[11] and the references therein and to Ma et al. ^[12] and Dedecker et al. ^[7] for contributions to the rate of convergence in the central limit theorem of the ERW in the diffusive and critical regimes. Since the seminal paper by Schütz and Trimper ^[13], many variants of the ERW have been introduced in this literature. Aguech et al. ^[1] extended the results of Bercu ^[3] and Kubota and Takei ^[10] in the case where the memory of the elephant is increasing (see Gut and Stadtmuller ^[8]): At any step $n$, the elephant remembers only the steps at times $1, \ldots, m_n$, where $m_n$ is a non-decreasing integer sequence such that $m_n < n$ and $\lim_n m_n/n = \theta\in [0, \, 1]$ and $\lim_n m_n = +\infty$. In this paper, we combine the results in ^[7] and ^[1], and we assume that the size of the step of the elephant is random and with a non-decreasing memory $m_n$. More precisely, suppose we have an ascending sequence of integers denoted as $m_n$, where each $m_n$ is less than or equal to $n$. Let $(U_n)_{n\geqslant 1}$ be a sequence of uniformly distributed random variables on the set $\{1, \ldots, m_n\}$, and let $(V_n)_{n\geqslant 1}$ denote a sequence of random variables with values either -1 or 1 such that $\mathbb{P}(V_n = 1) = p$ for some fixed $p\in]0, 1[$. In ^[7], the authors assume that the size of the step of the elephant is a sequence $(Z_n)_{n\geqslant 1}$ of independent identically distributed random variables with finite variance $\sigma^2 = \mathbb V(Z_1)$, and they assume that the sequences of random variables $(U_n)_{n\geqslant 1}$, $(V_n)_{n\geqslant 1}$, and $(Z_n)_{n\geqslant 1}$ are independent. Denote by $X_1$ the first step of the elephant and assume that $X_1$ is a random variable such that $\mathbb{P}(X_1 = 1) = 1- \mathbb{P}(X_1 = -1) = r$ for some fixed $r\in[0, 1]$. For any integer $n\geqslant 0$, we denote by $S_n$ the position of the elephant at time $n$. Following ^[7], we have $S_0 = 0$, and for any integer $n\geqslant 1$,

where, for all $n\geq 1$,

and recall that $U_n$ is uniformly distributed on $\{1, \ldots, m_n\}$ for any $n\geqslant 1$. Without loss of generality, one can assume that $\mathbb E(Z_n) = 1$ for any $n\geqslant 1$, and consequently, the following decomposition will be usefull: For any $n\geqslant 1$,

with

Denote $a_1 = 1$, and for any $n \geq 2$,

where $\Gamma(s) = \int_0^{\infty} t^{s-1} e^{-t} d t, s > 0$, is the Gamma function. For $m_n = n$, Dedecker et al. ^[7] proved the following result:

Theorem 1.1. [7, Theorems 2.1 and 3.1] Assume that $p\in(0, \, 1]$ and ${\mathbb E}\!\left[{Z^2}\right] < +\infty$.

(1) If $p\in(0, \, 3/4]$, then

(2) If $p\in (3/4, 1]$, then

where $L$ is a non-denerate and non-Gaussian random variable.

Remark 1.1. An important observation regarding the model investigated by Dedecker et al. ^[7] is its strong connection with the urn model. Specifically, the model operates as follows: Initially, an urn contains one white ball and one black ball. At each step $n$, a ball is randomly drawn from the urn. Subsequently, the drawn ball is replaced, with the addition of $Y_nZ_n$ balls of the same color and $(1-Y_n)Z_n$ balls of the opposite color, where $Y_n$ has a Bernoulli distribution with parameter $p$ and is independent of $Z_n$. The addition matrix for this urn is random and is given by

Some of the results of Dédecker et al. ^[7] and some extra results, can be obtained using this connection from ^[9], where, if we denote by $W_n$ and $B_n$, respectively, the number of white and black balls at time $n$, the position of the elephant at step $n$ can be given, due to this connection, by $S_n = W_n-B_n$. This connection with the Pólya urn model is not possible if we consider gradually increasing memory random walks: Practically, it is not possible at step $n$ to draw a ball only from the first $m_n$ added balls.

It is important to note that Theorem 1.1 holds in the particular case of $m_n = n$. Our aim in this work is to extend Theorem 1.1 to the case where $m_n\leqslant n$ such that $m_n/n\to\theta$ for some $\theta\in[0, 1]$. Additionally, for $p > 3/4$ (superdiffusive case), we are going to give the asymptotic distribution for the fluctuation of the elephant random walk around the random variable $\tau L$, where $L$ is defined in Theorem 1.1 and $\tau$ in Theorem 2.1.

2.   Results

First case: Diffusive regime ($0 < p < 3/4$)

Our first result concerns the case where $p\in(0, \, 3/4)$ and we give the asymptotic distribution of the elephant random walk.

Theorem 2.1. Let $\theta\in[0, 1]$ such that $\lim_{n\to\infty}m_n = +\infty$, and $m_n/n\to\theta$ as $n$ goes to infinity, and suppose that $p\in(0, \, 3/4)$. Assume that $(Z_n)_{n\geqslant 1}$ is iid positive with mean 1 and variance $\sigma^2$, and denote

Then,

Remark 2.1. If the step size is deterministic, then $\sigma = 0$, and we obtain the result in ^[1]. If $\theta = 1$, we obtain the result in ^[7].

Proof of Theorem 2.1. Let

By ^[1], we know that for $0 < p < 3/4$, we have

with

For all $t\in \mathbb{R}$, denote $\varphi_n(t) = {\mathbb E}\!\left[{e^{it\bar{S}_n}}\right]$ and consider the $\sigma$-algebra ${\mathcal{F}} = \sigma\Big(X_i, \, U_i, \, V_i, i\in\mathbb{N}\Big)$. So, for any $t$ in $\mathbb{R}$, we have

Since the random variables $\left(Z_i-1\right)_i$ are iid centered with finite variance $\sigma^2$ and that $X_i^2 = 1$ a.s., we get

In fact, if we denote

then,

where in the last three equations, the $o-$ term depends on the moment of order two of $Z_1$, which is finite by assumption.

Using (2.2), we obtain

As a first conclusion, we have

where $\varphi$ is the characteristic function of the standard normal law. On the other hand, for any $n\geqslant 1$ and any $t\in \mathbb{R}$,

Using (2.1) and noting that

we derive ^[1]

The proof of Theorem 2.1 is complete.□

Second case: Critical regime ($p = 3/4$)

Assume that $p = 3/4$ and consider the decompositon

Theorem 2.2. Let $\theta\in[0, 1]$ such that $m_n/n\to\theta$ as $n$ goes to infinity, and suppose that $p = 3/4$. If $(Z_n)_{n\geqslant 1}$ are iid positive with mean 1 and finite variance $\sigma^2$, then

Remark 2.2. Notice that in this case, the step size does not influence the asymptotic behavior of the normalized position of the elephant. This is due to the fact that the normalized remainder term $D_{m_n, \, n}$ converges to $0$ in probability.

Proof of Theorem 2.2. From ^[7], we know that

Moreover, given $\mathcal{F}_{m_n}: = \sigma(X_i\, ; \, i\leqslant m_n)$, for any $k\geqslant m_n+1$, we have

Note also that, given $\mathcal{F}_{m_n}$, the random variables $\left(X_kZ_k\right)_{m_n+1\leq k\leq n}$ are i.i.d. In the other part, we have the following decomposition:

where

For any $t\in \mathbb{R}$ and any $n\geqslant 1$, we denote $\phi_n(t) = {\mathbb E}\!\left[{\exp\left(it\tilde{S}_n\right)}\right]$ and $\mathcal{G}_n: = \sigma\left(Z_1, \cdots, Z_n\right)$. Then, we have

Using the conditional distribution of $X_{m_n+1}$ given $\mathcal{F}_{m_n}$, we obtain

and recall that $m_n^{-1}T_{m_n}\xrightarrow[n \rightarrow +\infty]{{\rm{a.s.}}}0$ (see ^[3]) and $\vert m_n^{-1}T_{m_n}\vert\leqslant 1$ a.s. Then, for $n$ sufficiently large

Then, the characteristic function $\phi_n(t)$ has an asymptotic expression

Since

we deduce that

Finally, by ^[3], we conclude

The proof of Theorem 2.2 is complete. □

Third case: Superdiffusive regime ($p > 3/4$)

In this section, we consider the case, where $p\in(3/4, \, 1)$, called the superdifusive regime. First, we give the almost sure convergence of $S_n$.

Theorem 2.3. Let $\theta\in[0, 1]$ such that $m_n/n\to\theta$ as $n$ goes to infinity, and suppose that $p\in(3/4, \, 1)$. If $(Z_n)_{n\geqslant 1}$ are iid positive with mean 1 and finite variance $\sigma^2$, then

where $L$ is a non-Gaussian random variable.

Proof of Theorem 2.3. Recall that

and consequently,

From ^[7], we know that

On the other hand, according to $\mathcal{F}_{m_n}$, the sequence of random variables $\left(X_iZ_i\right)_{m_n+1\leq i\leq n}$ is i.i.d., and for all $i\geq m_n+1$,

Applying the strong law of large numbers condionally to $\mathcal{F}_m$, for $n$ large, we have

Then, almost surely, for $n$ large

□

The following result shows that the fluctuations of the elephant random walk are still gaussian around the random variable $L$ as given in Theorem 2.3.

Theorem 2.4. Let $\theta\in[0, 1]$ such that $m_n/n\to\theta$ as $n$ goes to infinity, and suppose that $p\in(3/4, \, 1)$. If $(Z_n)_{n\geqslant 1}$ are iid positive with mean 1 and finite variance $\sigma^2$, then

where

Remark 2.3. This result generalizes the result obtained in ^[7], it precises in addition the asymptotic distribution of the fluctuation, and coincides with the result of Aguech et al. ^[1], in the case where $\sigma = 0$.

Proof of Theorem 2.4. We start again with the following decomposition:

First, using ^[1], if $\sqrt{m_n^{4p-3}}|n^{-1}m_n-\theta|\to 0$, then

Moreover, we have

Let, for all $n$, $\mathcal{H}_n = \sigma\left(X_1, \cdots, X_n, \, L\right)$. For all real $t$, let $\psi_Z(t) = {\mathbb E}\!\left[{\exp\left(it\left[Z-1\right]\right)}\right]$ the characteristic function of $Z-1$ and $\Psi_n(t)$ is the characteristic function of $\sqrt{m_n^{4p-3}}\left(\frac{S_nm_n^{2(1-p)}}{n}-\tau L\right)$. So, we have

If $\psi_{Z|{\mathcal{H}_n}}$ denotes the characteristic function of $Z-1$ conditionally to $\mathcal{H}_n$, then

Consequently, if $n$ is large, then

The proof of Theorem 2.4 is complete. □

3.   Conclusions and discussion

In this work, we place emphasis on the fact that the asymptotic normality for the elephant random walk with gradually increasing memory and random step size still holds in the three regimes of the model. Our results extend previous ones established by Dedecker et al. ^[7] and Aguech et al. ^[1]. Additionally, we observed that the connection with Polya urns is not feasible for gradually increasing memory. Finally, we argue that our approach can be used for deriving many others limit theorems for the elephant random walk with gradually increasing memory and random step size (law of the iterated logarithm, rate of convergence in the central limit theorem, invariance principle).

What about the asymptotic normality of the ERW model, remembering only its last $m_n$ steps $n-m_n, \ldots, n-1$? This very interesting question will serve as the basis for a new research project. Another problem that we think is very interesting, and which will be one of our future projects, is to consider the same question but with an ERW model with a random step size. A more difficult problem is to establish the rate of convergence in the central limit theorems for the ERW model with restricted memory. In particular, it will be very interesting to understand the way that the restricted memory will influence the rate of convergence in the central limit theorem.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to extend his sincere appreciation to Deanship of Scientific Research at King Saud University: Researchers Supporting Program number (RSPD2024R987).

Conflict of interest

The author declares no conflict of interest.