On the almost sure central limit theorem for the elephant random walk

Guevara, Víctor Hugo Vázquez

doi:10.1088/1751-8121/ab4b5f

Cited by 17 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By (12), we have a 2 n /v n ≍ 1/n as n → ∞, and exp ka 2 n vn ≤ 1 + C k n for all 1 ≤ k = o(n 2/3 ). Thus, it holds for all 1 ≤ k = o(n 2/3 ),…”

Section: Proof Of Theorem 23mentioning

confidence: 94%

“…Since the seminal work of Schütz and Trimper [11], the ERW has attracted a lot of attentions. A wide range of literature is available for the asymptotic behavior of the ERW and its extensions, see [1,5,2,12,3]. Baur and Bertoin [1] derived the functional limit theorems via a method of connection to Pólya-type urns.…”

Section: Introductionmentioning

confidence: 99%

“…They also showed that if p ∈ (3/4, 1], then the ERW converges to a non-degenerate random variable which is not normal. Vázquez Guevara [12] gave the almost sure central limit theorems. For the extensions of ERW, we refer to Bercu and Lucile [3] and Gut and Stadtmüller [10].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cramér moderate deviations for the elephant random walk

Fan¹,

Hu²,

Ma³

2020

Preprint

View full text Add to dashboard Cite

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter p which lies between 0 and 3/4.

show abstract

“…By (12), we have a 2 n /v n ≍ 1/n as n → ∞, and exp ka 2 n vn ≤ 1 + C k n for all 1 ≤ k = o(n 2/3 ). Thus, it holds for all 1 ≤ k = o(n 2/3 ),…”

Section: Proof Of Theorem 23mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cramér moderate deviations for the elephant random walk

Fan¹,

Hu²,

Ma³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Over the last decade, the ERW has received a growing attention in mathematics and statistical physics in the diffusive regime p < 3/4 and the critical regime p = 3/4, see [1,11,12], as well as in the superdiffusive regime p > 3/4, see [2,24]. We also refer the reader to the extension to the multi-dimensional ERW [4,5,6,17] as well as to the recent contributions [7,8,9,10,15,25,28].…”

Section: Introductionmentioning

confidence: 99%

On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution

Bercu¹

2022

Preprint

View full text Add to dashboard Cite

The aim of this paper is to investigate the asymptotic behavior of the so-called elephant random walk with stops (ERWS). In contrast with the standard elephant random walk, the elephant is allowed to be lazy by staying on his own position. We prove that the number of ones of the ERWS, properly normalized, converges almost surely to a Mittag-Leffler distribution. It allows us to carry out a sharp analysis on the asymptotic behavior of the ERWS. In the diffusive and critical regimes, we establish the almost sure convergence of the ERWS. We also show that it is necessary to self-normalized the position of the ERWS by the random number of ones in order to prove the asymptotic normality. In the superdiffusive regime, we establish the almost sure convergence of the ERWS, properly normalized, to a nondegenerate random variable. Moreover, we also show that the fluctuation of the ERWS around its limiting random variable is still Gaussian.

show abstract

“…A wide literature is now available on the ERW in dimesion d = 1 where p d = 3/4. A strong law of large numbers and a central limit theorem for the position S n , properly normalized, were established in the diffusive regime p < 3/4 and the critical regime p = 3/4, see [1], [8], [9], [17] and the recent contributions [3], [5], [7], [20]. The superdiffusive regime p > 3/4 is much harder to handle.…”

Section: Introductionmentioning

confidence: 99%

On the center of mass of the elephant random walk

Bercu¹,

Laulin²

2019

Preprint

View full text Add to dashboard Cite

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratric strong law for the center of mass of the elephant random walk. The asymptotic normality of the center of mass, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.Curiously, no other references are availabe on the asymptotic behavior of the center of mass. The proofs of many results on the convex hull C n as well as on the center of mass

show abstract

On the almost sure central limit theorem for the elephant random walk

Abstract: In this paper we present versions of the almost sure central limit theorem for both the scalar and multi-dimensional elephant random walk based in the almost sure central limit for martingales. In addition, convergence to even moments of Gaussian distribution also will be discussed.

Cited by 17 publications

References 31 publications

Cramér moderate deviations for the elephant random walk

Cramér moderate deviations for the elephant random walk

On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution

On the center of mass of the elephant random walk

Contact Info

Product

Resources

About