On decoupled standard random walks
Gerold Alsmeyer, Alexander Iksanov, Zakhar Kabluchko
Abstract
Let $S_{n}=\sum_{k=1}^{n}ξ_{k}$, $n\in\mathbb{N}$, be a standard random walk with i.i.d. nonnegative increments $ξ_{1},ξ_{2},\ldots$ and associated renewal counting process $N(t)=\sum_{n\ge 1}1_{\{S_{n}\le t\}}$, $t\ge 0$. A decoupling of $(S_{n})_{n\ge 1}$ is any sequence $\hat{S}_{1}$, $\hat{S}_{2},\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat{S}_{n}$ and $S_{n}$ have the same law. Under the assumption that the law of $\hat{S}_{1}$ belongs to the domain of attraction of a stable law with finite mean, we prove a functional limit theorem for the \emph{decoupled renewal counting process} $\hat{N}(t)=\sum_{n\ge 1}1_{\{\hat{S}_{n}\le t\}}$, $t\ge 0$, after proper scaling, centering and normalization. We also study the asymptotics of $\log \mathbb{P}\{\min_{n\ge 1}\hat{S}_{n}>t\}$ as $t\to\infty$ under varying assumptions on the law of $\hat{S}_{1}$. In particular, we recover the assertions which were previously known in the case when $\hat{S}_{1}$ has an exponential law. These results, which were formulated in terms of an infinite Ginibre point process, served as an initial motivation for the present work. Finally, we prove strong law of large numbers type results for the sequence of decoupled maxima $M_{n}=\max_{1\le k\le n}\hat{S}_{k}$, $n\in\mathbb{N}$, and the related first passage time process $\hatτ(t)=\inf\{n\in\mathbb{N}: M_{n}>t\}$, $t\ge 0$. In particular, we provide a tail condition on the law of $\hat{S}_{1}$ in the case when the latter has finite mean but infinite variance that implies $\lim_{t\to\infty}t^{-1}\hatτ(t)=\lim_{t\to\infty}t^{-1}\mathbb{E}\hatτ(t)=0$. In other words, $t^{-1}\hatτ(t)$ may exhibit a different limit behavior than $t^{-1}τ(t)$, where $τ(t)$ denotes the level-$t$ first passage time of $(S_{n})_{n\ge 1}$.