Limit theorems for decoupled renewal processes

TL;DR

This paper establishes limit theorems for decoupled renewal processes, including functional central limit theorems and laws of the iterated or single logarithm, under regular variation assumptions on the jump distribution.

Contribution

It provides new limit theorems for decoupled renewal processes with regularly varying tail distributions, extending classical results to this setting.

Findings

Proves a functional central limit theorem for decoupled renewal processes.

Establishes laws of the iterated and single logarithm for these processes.

Applies results to the number of atoms in a determinantal point process with Mittag-Leffler kernel.

Abstract

The decoupled standard random walk is a sequence of independent random variables $(\hat S_n)_{n\geq 1}$, in which $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Denote by $\hat N(t)$ the number of elements of the decoupled standard random walk which do not exceed $t$. The random process $(\hat N(t))_{t\geq 0}$ is called decoupled renewal process. Under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of nonpositive index larger than $-1$ we prove a functional central limit theorem in the Skorokhod space equipped with the $J_1$-topology for the decoupled renewal processes, properly scaled, centered and normalized. Also, under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of index $-\alpha$, $\alpha\in [0,1)\cup (1,2)$ or the distribution of $\hat S_1$ belongs to the domain of attraction of a normal distribution we prove a law of the iterated or single logarithm for $\hat N(t)$, again properly normalized and centered. As an application, we obtain a law of the single logarithm for the number of atoms of a determinantal point process with the Mittag-Leffler kernel, which lie in expanding discs.