Laws of the iterated logarithm for iterated perturbed random walks
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915217850499072 |
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| author | Braganets, Oksana |
| author_facet | Braganets, Oksana |
| contents | Let $(ξ_k, η_k)_{k\geq 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$for $k\in\mathbb{N}$. We call the random sequence {T_k, k=1,2...} a (globally) perturbed random walk. Consider a general branching process generated by {T_k, k=1,2...} and let Y_j(t) denote the number of the jth generation individuals with birth times less or equal t. Assuming that Var ξ_1 is finite and allowing the distribution of η_1 to be arbitrary, we prove a law of the iterated logarithm (LIL) for Y_j(t). In particular, a LIL for the counting process of {T_k, k=1,2...} is obtained. The latter result was previously established in the article Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that Eη^a is finite for some positive a. In this paper, we show that the aforementioned additional assumption is not needed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Laws of the iterated logarithm for iterated perturbed random walks Braganets, Oksana Probability Let $(ξ_k, η_k)_{k\geq 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$for $k\in\mathbb{N}$. We call the random sequence {T_k, k=1,2...} a (globally) perturbed random walk. Consider a general branching process generated by {T_k, k=1,2...} and let Y_j(t) denote the number of the jth generation individuals with birth times less or equal t. Assuming that Var ξ_1 is finite and allowing the distribution of η_1 to be arbitrary, we prove a law of the iterated logarithm (LIL) for Y_j(t). In particular, a LIL for the counting process of {T_k, k=1,2...} is obtained. The latter result was previously established in the article Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that Eη^a is finite for some positive a. In this paper, we show that the aforementioned additional assumption is not needed. |
| title | Laws of the iterated logarithm for iterated perturbed random walks |
| topic | Probability |
| url | https://arxiv.org/abs/2503.20029 |