Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913769448275968 |
|---|---|
| author | Roy, Rahul Takei, Masato Tanemura, Hideki |
| author_facet | Roy, Rahul Takei, Masato Tanemura, Hideki |
| contents | We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $β\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $β\in (-1, 0] $. Also, for the case $β\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $β\in (-1, p/(1-p))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates Roy, Rahul Takei, Masato Tanemura, Hideki Probability We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $β\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $β\in (-1, 0] $. Also, for the case $β\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $β\in (-1, p/(1-p))$. |
| title | Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates |
| topic | Probability |
| url | https://arxiv.org/abs/2504.00566 |