Invariance principle for biased bootstrap random walks

Collevecchio, Andrea; Liu, Yunxuan

doi:10.1016/j.spa.2018.03.022

Cited by 4 publications

(6 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been called the coin-turning random walk by Engländer and Volkov who introduced more general versions in [9], and these were further studied by Engländer et al [10]. It has also been called the bootstrap random walk by Collevecchio, Hamza and Shi, who studied the pair (Y , Z ) in [8]; Collevecchio, Hamza and Liu gave a further generalisation in [7].…”

Section: Motivation and Existing Resultsmentioning

confidence: 99%

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

Prigent

Roberts

2020

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

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.

show abstract

Section: Motivation and Existing Resultsmentioning

confidence: 99%

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

Prigent

Roberts

2020

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

show abstract

“…great details in [3] (see also [4]). In mathematical finance, it was used to discuss the continuity of utility maximization under weak convergence (see [1]).…”

Section: Literature Reviewmentioning

confidence: 99%

“…We extend the model introduced in [3] and [4] to the case η n = ξ n k∈Mn ξ k , n ≥ 1, where M n is not necessarily the entire set {1, . .…”

Section: The Extended Bootstrap Random Walkmentioning

confidence: 99%

Limit theorems and ergodicity for general bootstrap random walks

Collevecchio

Hamza

Shi

et al. 2022

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

Given the increments of a simple symmetric random walk (Xn) n≥0 , we characterize all possible ways of recycling these increments into a simple symmetric random walk (Yn) n≥0 adapted to the filtration of (Xn) n≥0 . We study the long term behavior of a suitably normalized two-dimensional process ((Xn, Yn)) n≥0 . In particular, we provide necessary and sufficient conditions for the process to converge to a two-dimensional Brownian motion (possibly degenerate). We also discuss cases in which the limit is not Gaussian. Finally, we provide a simple necessary and sufficient condition for the ergodicity of the recycling transformation, thus generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and solving the discrete version of the open problem of the ergodicity of the general Lévy transformation (see Mansuy and Yor, 2006).

show abstract

“…From a different perspective, the bivariate flip-flop processes can be seen as a continuous-time analogue of the discrete-time bootstrapping random walk concept proposed in [3,2]. There, the authors start by constructing a discrete-time random walk from Bernoulli random variables, and then reuse (bootstrap) these random variables to create additional walks; together, the random walks converge weakly to twoor higher-dimensional Brownian motions.…”

Section: Introductionmentioning

confidence: 99%

“…For the values of λ a , λ b and P in that theorem, take λ a = λ 0 , λ b = λ n − λ 0 , and P as (3.3) 2. We use the notation J λn (• − 0) and J λn (• + 0) to denote the left and right limits, respectively.…”

mentioning

confidence: 99%

Strong convergence to two-dimensional alternating Brownian motion processes

Latouche¹,

Nguyen²,

Peralta³

2019

Preprint

View full text Add to dashboard Cite

Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional aspects of the univariate SBM and MMBM can be studied through the limiting behaviour of flip-flop processes. Here, we construct two classes of bivariate flip-flop processes whose marginals converge strongly to SBMs and are dependent on each other, which we refer to as alternating twodimensional Brownian motion processes. While the limiting bivariate processes are not Gaussian, they possess desirable qualities, such as being tractable and having a time-varying correlation coefficient function.

show abstract

Invariance principle for biased bootstrap random walks

Cited by 4 publications

References 6 publications

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

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

Limit theorems and ergodicity for general bootstrap random walks

Strong convergence to two-dimensional alternating Brownian motion processes

Contact Info

Product

Resources

About