Universality of Noise Reinforced Brownian Motions

Bertoin, Jean

doi:10.48550/arxiv.2002.09166

Cited by 3 publications

(5 citation statements)

References 19 publications

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“…We now immediately deduce from Theorem 1.2 and the continuous mapping theorem that its asymptotic behavior is given by: Although this argument only enables us to deal with real bounded random variables ξ, we stress that more generally, the assertions (i), (ii) and (iii) still hold when the generic step ξ is an arbitrary square integrable and centered variable in R d (for d ≥ 2, ς 2 is then of course the covariance matrix of ξ). Specifically, (i) follows from the invariance principle for step reinforced random walks (see Theorem 3.3 in [7]), whereas (iii) is Theorem 3.2 in the same work; see also [6]. In the critical case p = 1/2, (ii) can be deduced from the basic identity…”

Section: Relation To Step Reinforced Random Walksmentioning

confidence: 93%

“…Indeed, the almost sure convergence in (7) holds simultaneously for all dyadic rational numbers, and uniform convergence on [0, 1] then can be derived by a monotonicity argument à la Dini.…”

Section: A First Moment Calculationmentioning

confidence: 99%

“…Proof. It has been observed in [7] that, in the study of reinforcement induced by Simon's algorithm, it may be convenient to perform a time-substitution based on a Yule process. We shall use this idea here again, and introduce a standard Yule process Y = (Y t ) t≥0 , which we further assume to be independent of the preceding variables.…”

Section: Critical Regime P = 1/2mentioning

confidence: 99%

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How linear reinforcement affects Donsker's Theorem for empirical processes

Bertoin¹

2020

Preprint

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A reinforcement algorithm introduced by H.A. Simon [27] produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability p ∈ (0, 1), Ûn+1 is sampled uniformly from Û1, . . . , Ûn, and with complementary probability 1−p, Ûn+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, and that a further rescaling is needed when 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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Section: Relation To Step Reinforced Random Walksmentioning

confidence: 93%

Section: A First Moment Calculationmentioning

confidence: 99%

Section: Critical Regime P = 1/2mentioning

confidence: 99%

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How linear reinforcement affects Donsker's Theorem for empirical processes

Bertoin¹

2020

Preprint

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“…Then, Bercu [4] obtained some refinements on the asymptotic behaviour of the ERW. Indeed, most of the related work on limit theorems of random walks with memory can be subdivided into two categories: the study of limit theorems such as law of large numbers, central limit theorems and invariance principles, see for instance [2][3][4][5][6][7] and references therein; and hypergeometric identities arising from this kind of processes, see [8].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of the elephant random walk

Coletti¹,

Papageorgiou²

2021

J. Stat. Mech.

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In this work we study asymptotic properties of a long range memory random walk known as elephant random walk. First we prove recurrence and positive recurrence for the elephant random walk. Then, we establish the transience regime of the model. Finally, under the Poisson hypothesis, we study the replica mean field limit for this random walk and we obtain an upper bound for the expected distance of the walker from the origin.

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“…Several limit theorems are obtained in [1,2,9,10,19]. Variations of elephant random walks studied mainly from mathematical viewpoint are found in [3,4,5,7,12,13].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for the 'laziest' minimal random walk model of elephant type

Miyazaki,

Takei

2020

Preprint

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We consider a minimal model of one-dimensional discretetime random walk with step-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): The walker can move forward (never backward), or remain at rest. For each n = 1, 2, • • • , a random time Un between 1 and n is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time Un, then at time n + 1 it can move forward with probability p [resp. q], or with probability 1 − p [resp. 1 − q] it remains at its present position. For the case q > 0, several limit theorems are obtained by Coletti, Gava, and de Lima (2019). In this paper we prove limit theorems for the case q = 0, where the walker can exhibit all three forms of asymptotic behavior as p is varied. As a byproduct, we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees.

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Universality of Noise Reinforced Brownian Motions

Cited by 3 publications

References 19 publications

How linear reinforcement affects Donsker's Theorem for empirical processes

How linear reinforcement affects Donsker's Theorem for empirical processes

Asymptotic analysis of the elephant random walk

Limit theorems for the 'laziest' minimal random walk model of elephant type

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