A strong invariance principle for the elephant random walk

Coletti, Cristian F.; Gava, Renato; Schütz, Gunter M.

doi:10.1088/1742-5468/aa9680

Cited by 49 publications

(46 citation statements)

References 14 publications

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“…The following theorems are generalizations of the results obtained in [1,2,7,8] for the original elephant random walk, although they are essentially proved in existing literatures ( [15,19] among others) for the correlated Bernoulli process. To make this paper reasonably self-contained, we indicate the main lines of proofs in section 3.…”

Section: Resultsmentioning

confidence: 79%

“…On the other hand, if α > 1/2, then {M n } is an L 2bounded martingale, and the martingale convergence theorem shows that S n /n α converges to a non-degenerate random variable with mean β Γ(α+1) , whose distribution turns out to be non-Gaussian (see [2,3] among others), a.s. and in L 2 . These and further strong limit theorems are obtained by [1,2,7,8]. Kürsten [17] relates the phase transition described above to the behavior of a spin system on random recursive trees.…”

Section: Introductionmentioning

confidence: 93%

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Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Kubota

Takei

2019

J Stat Phys

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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.

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Section: Resultsmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 93%

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Kubota

Takei

2019

J Stat Phys

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“…These results have also appeared more recently in the framework of the so-called elephant random walk, a random walk with memory which has been introduced by Schütz and Trimper [27]. See notably [2,3,11,12], and also [4,5,14,15] and references therein for some further developments. We mention that Kürsten [23] first pointed at the role of Bernoulli bond percolation on random recursive trees in this framework, see also [10].…”

Section: Relation To Step Reinforced Random Walksmentioning

confidence: 86%

How linear reinforcement affects Donsker’s theorem for empirical processes

Bertoin

2020

Probab. Theory Relat. Fields

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A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

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“…Initially, the distribution was thought to be normal (i.e. Gaussian) irrespective of the value of  ; subsequently it was found that it is actually non-Gaussian in the super-diffusion regime, although it is difficult to be specific and this is a topic of continuing research [13][14][15][16][17][18][19]. Minor modifications to the original formulation lead to sub-diffusion as well as super-diffusion [20,21], and various other models turn out to have a close connection to the ERW as well, see e.g.…”

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confidence: 99%

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

Kearney¹,

Martin²

2018

J. Stat. Mech.

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A random walk model is presented which exhibits a transition from standard to anomalous diffusion as a parameter is varied. The model is a variant on the elephant random walk and differs in respect of the treatment of the initial state, which in the present work consists of a given number N of fixed steps. This also links the elephant random walk to other types of history dependent random walk. As well as being amenable to direct analysis, the model is shown to be asymptotically equivalent to a non-linear urn process. This provides fresh insights into the limiting form of the distribution of the walker's position at large times. Although the distribution is intrinsically non-Gaussian in the anomalous diffusion regime, it gradually reverts to normal form when N is large under quite general conditions.

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A strong invariance principle for the elephant random walk

Cited by 49 publications

References 14 publications

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

How linear reinforcement affects Donsker’s theorem for empirical processes

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

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