On decoupled random walks
A talk in the Oberseminar Wahrscheinlichkeitstheorie series by
Oleksandr Iksanov
| Abstract: | We call a decoupled random walk a sequence $\hat S_1$, $\hat S_2,\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Similarly, we call a decoupled renewal process the counting process $(\hat N(t))_{t\geq 0}$ defined by $\hat N(t)=\sum_{n\geq 1}\1_{\{\hat S_n\leq t\}}$. I shall present a functional limit theorem for $(\hat N(t))_{t\geq 0}$, properly scaled, normalized and centered, as $t\to\infty$ under the assumption that the variance of $\hat S_1$ is positive and finite. Also, I shall discuss the asymptotic of $\log \mathbb{P}\{\min_{n\geq 1}\hat S_n>t\}$ as $t\to\infty$ under various assumptions imposed on the distribution of $\hat S_1$. Our interest to the so defined decoupled random walks was caused by their appearance in the particular case when $\hat S_1$ has an exponential distribution of unit mean in the context of infinite Ginibre point processes. The talk is based on a recent joint work with Gerold Alsmeyer and Zakhar Kabluchko (Muenster). Within the CRC this talk is associated to the project(s): B10 |