Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914261567012864 |
|---|---|
| author | Pavlyukevich, Ilya Pilipenko, Andrey |
| author_facet | Pavlyukevich, Ilya Pilipenko, Andrey |
| contents | In this paper we study Markov chains with the state space given by the coordinate axes of $\mathbb R^m$, $m \geq 2$, whose step sizes on each positive half-axis are distributed according to a centered probability distribution with variance $v_i^2 \in (0, \infty)$, $i = 1,\ldots, m$. Under very mild assumptions on the jumps sizes on the negative half-axes, we show that the Donsker scaling limit of such Markov chains is a Walsh Brownian motion whose weights are determined explicitly in terms of stationary distributions of certain embedded Markov chains. This convergence result is applied to integer-valued random walks perturbed on a finite subset of $\mathbb Z$ called a membrane. We show that their Donsker scaling limit is an oscillating skew Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks Pavlyukevich, Ilya Pilipenko, Andrey Probability 60F17, 60G42, 60G50, 60H10, 60J10, 60J55, 60K37 In this paper we study Markov chains with the state space given by the coordinate axes of $\mathbb R^m$, $m \geq 2$, whose step sizes on each positive half-axis are distributed according to a centered probability distribution with variance $v_i^2 \in (0, \infty)$, $i = 1,\ldots, m$. Under very mild assumptions on the jumps sizes on the negative half-axes, we show that the Donsker scaling limit of such Markov chains is a Walsh Brownian motion whose weights are determined explicitly in terms of stationary distributions of certain embedded Markov chains. This convergence result is applied to integer-valued random walks perturbed on a finite subset of $\mathbb Z$ called a membrane. We show that their Donsker scaling limit is an oscillating skew Brownian motion. |
| title | Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks |
| topic | Probability 60F17, 60G42, 60G50, 60H10, 60J10, 60J55, 60K37 |
| url | https://arxiv.org/abs/2310.10809 |