Turning a Coin over Instead of Tossing It

Engländer, János; Volkov, Stanislav

doi:10.1007/s10959-016-0725-1

Cited by 15 publications

(57 citation statements)

References 9 publications

(7 reference statements)

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“…holds with a, b ∈ R, a ≤ b and n ∈ N, and some probability measure Q such that Q({0}) = 0. These scaling limits we did establish in many cases in [7].…”

Section: 4supporting

confidence: 56%

“…The relationship with [7] is explained in 4.2 in [2]. The paper builds on the authors' previous results with M. Benaïm in [1], and they point out in [2] that "In particular, the results we use provide functional convergence of the rescaled interpolating processes to the auxiliary Markov processes..." at which point the authors refer to [1] and another article.…”

Section: 2mentioning

confidence: 73%

“…3.1. Some results from [7]. Some of the basic results of [7] are summarized in the following theorem.…”

Section: Review Of Relevant Literaturementioning

confidence: 99%

“…Recent results by Benaïm, Bouguet and Cloez. In a recent follow up paper to [7] by Bouguet and Cloez [2], the setting has been generalized in such a way that instead of two states (heads and tails or ±1), one considers D ≥ 2 states, and with probability p n in the nth step the state changes according to a given irreducible Markov chain. 2 (They also allow a small error term.)…”

Section: 2mentioning

confidence: 99%

“…Then lim inf n→∞ a n > 0, where the a n are defined by (7). Moreover, if lim n→∞ q n+1 q n = 1, then lim n→∞ a n = 1/2.…”

Section: 33mentioning

confidence: 99%

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The coin-turning walk and its scaling limit

Engländer¹,

Volkov²,

Wang³

2020

Electron. J. Probab.

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Let S be the random walk obtained from "coin turning" with some sequence {p n } n≥2 , as introduced in [7]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics.We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const•n −1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold.In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed.We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.

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“…holds with a, b ∈ R, a ≤ b and n ∈ N, and some probability measure Q such that Q({0}) = 0. These scaling limits we did establish in many cases in [7].…”

Section: 4supporting

confidence: 56%

Section: 2mentioning

confidence: 73%

“…3.1. Some results from [7]. Some of the basic results of [7] are summarized in the following theorem.…”

Section: Review Of Relevant Literaturementioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…Then lim inf n→∞ a n > 0, where the a n are defined by (7). Moreover, if lim n→∞ q n+1 q n = 1, then lim n→∞ a n = 1/2.…”

Section: 33mentioning

confidence: 99%

See 3 more Smart Citations

The coin-turning walk and its scaling limit

Engländer¹,

Volkov²,

Wang³

2020

Electron. J. Probab.

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Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

Prigent

Roberts

2020

Probab. Theory Relat. Fields

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We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $$\varepsilon _n$$ ε n such that $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ . This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

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Impatient Random Walk

Engländer

Volkov

2019

J Theor Probab

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We introduce a new type of random walk where the definition of edge repellence/reinforcement is very different from the one in the "traditional" reinforced random walk models, and investigate its basic properties, such as null vs. positive recurrence, transience, as well as the speed. The two basic cases will be dubbed "impatient" and"ageing" random walks.Fix a vertex v 0 ∈ G which we call "origin", and assume that the walk starts at this point, X 0 = v 0 ; for G = Z d , the default will be v 0 := 0.Definition 1 (Passage times). A sequence s 0 , s 1 , s 2 , . . . of nonnegative real numbers will be called a sequence of passage times if s 0 = 1.Definition 2 (Walk modified by passage times). We will modify the walk in such a way that if it has crossed an edge e exactly k times before, then it takes s k units of time (as opposed to 1) to cross this edge again, in either direction; in particular, it takes one unit of time to cross the edge for the first time.The two basic cases are as follows:with the convention that for t < 1, the value of the first sum is zero and then T (t) = ts Z((X 0 ,X 1 ),0) = ts 0 = t.Let the random function U : [0, ∞) → [0, ∞) denote the right continuous generalized inverse of T , that is, U(t) := sup{s : T (s) ≤ t}. With the exception of one case, we will work with s k > 0 for all k ≥ 0, and then T is strictly increasing in t and U = T −1 .Definition 5 (Definition of X imp and X age via time change). The impatient random walk X imp (ageing random walk X age ) is defined via X imp (T (t)) = X ⌊t⌋ , t ≥ 0 (X age (T (t)) = X ⌊t⌋ , t ≥ 0), or, equivalently, by X imp (t) := X ⌊U (t)⌋ , t ≥ 0 (X age (t) := X ⌊U (t)⌋ , t ≥ 0), 1 The intuitive meaning is clear: the more the walker crosses the same edge, the faster it happens.

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Turning a Coin over Instead of Tossing It

Cited by 15 publications

References 9 publications

The coin-turning walk and its scaling limit

The coin-turning walk and its scaling limit

Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

Impatient Random Walk

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