Generalized Excited Random Walks under Bernoulli excitations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917534601576448 |
|---|---|
| author | Alves, Rodrigo B. Iacobelli, Giulio Valle, Glauco Zuaznábar, Leonel |
| author_facet | Alves, Rodrigo B. Iacobelli, Giulio Valle, Glauco Zuaznábar, Leonel |
| contents | We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ramírez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as $p_n$-GERW.
Assuming bounded jumps and $p_n \approx n^{-β}$, we show a series of results for the $p_n$-\Name{} depending on the value of $β$ and on the dimension $d$. Specifically, for every $β\in(0,1]$ and $d=2$ or $d>h(β)$, with $h$ a decreasing function of $β$, we prove a SLLN for the range, while for $β<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling, and we obtain a Functional Central Limit Theorem for $β> 1/2$ and $d\geq 2$, or $β=1/2$ and $d=2$. Finally, for $β=1/2$ and $d \ge 11$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12228 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized Excited Random Walks under Bernoulli excitations Alves, Rodrigo B. Iacobelli, Giulio Valle, Glauco Zuaznábar, Leonel Probability We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ramírez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as $p_n$-GERW. Assuming bounded jumps and $p_n \approx n^{-β}$, we show a series of results for the $p_n$-\Name{} depending on the value of $β$ and on the dimension $d$. Specifically, for every $β\in(0,1]$ and $d=2$ or $d>h(β)$, with $h$ a decreasing function of $β$, we prove a SLLN for the range, while for $β<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling, and we obtain a Functional Central Limit Theorem for $β> 1/2$ and $d\geq 2$, or $β=1/2$ and $d=2$. Finally, for $β=1/2$ and $d \ge 11$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time. |
| title | Generalized Excited Random Walks under Bernoulli excitations |
| topic | Probability |
| url | https://arxiv.org/abs/2303.12228 |