Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks
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_ | 1866908728787206144 |
|---|---|
| author | Garbit, Rodolphe Raschel, Kilian |
| author_facet | Garbit, Rodolphe Raschel, Kilian |
| contents | In this paper we study the number of returns to the coordinate hyperplanes for multidimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks Garbit, Rodolphe Raschel, Kilian Probability Combinatorics In this paper we study the number of returns to the coordinate hyperplanes for multidimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations. |
| title | Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2512.19822 |