Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Abstract

Elephant random walk is a kind of one-dimensional discretetime random walk with infinite memory: For each step, with probability α 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 phase transition from diffusive to superdiffusive behavior at the critical value αc = 1/2. For α ∈ (αc, 1), there is a scaling factor an of order n α such that the position Sn of the walker at time n scaled by an converges to a nondegenerate random variable W , whose distribution is not Gaussian. Our main result shows that the fluctuation of Sn around W · an is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

How linear reinforcement affects Donsker’s theorem for empirical processes

“…We mention that Kürsten [23] first pointed at the role of Bernoulli bond percolation on random recursive trees in this framework, see also [10]. It is moreover interesting to recall that, for the elephant random random walk, Kubota and Takei [22] have established that the fluctuations corresponding to (iii) are Gaussian. Whether or not the same holds for general step reinforced random walks is still open; this also suggests that for p > 1/2, Theorem 1.2 (iii) might be refined and yield a second order weak limit theorem involving again a Brownian bridge in the limit.…”

Scaling exponents of step-reinforced random walks

“…The almost sure convergence then follows from the observation that the procesŝ S(n)/π(n) is a martingale (in the setting of the elephant random walk, a similar property has been pointed at in [6,12,13,19]). Indeed, we see from Simon's algorithm and the assumption E(X ) = 0 that…”

“…• It would be interesting to complete the strong limit results (Theorems 1 and 2) and investigate the fluctuations n −1/αŜ (n) − V as n → ∞. In the setting of the elephant random walk, Kubota and Takei [19] have recently established that these fluctuations are Gaussian. • The case where the generic step X has the standard Cauchy distribution is remarkable, due to the feature that for any a, b > 0, a X 1 + bX 2 has the same distribution as (a + b)X , where X 1 and X 2 are two independent copies of X .…”

“…The cornerstone of our approach is provided by Lemma 2, where we observe that the process that counts the number of occurrences of a given item in Simon's algorithm can be turned into a square integrable martingale. The latter is a close relative to another martingale that occurs naturally in the setting of the elephant random walk; see [6,12,13,19], among others. The upshot of Lemma 2 is that this yield useful estimates for these numbers of occurrences and their asymptotic behaviors, which hold uniformly for all items.…”

“…Indeed, Kürsten [20] observed thatŜ is then a version of the so-called elephant random walk, a nearest neighbor process with memory which was introduced by Schütz and Trimper [23] and has then raised much interest. The description of the asymptotic behavior of the elephant random walk has motivated many works, see notably [3,5,6,12,13,19]. Further, when the typical step X has a symmetric stable distribution with index α ∈ (0, 2],Ŝ is the so-called shark random swim, which has been studied in depth by Businger [11].…”