Limit theorems for decoupled renewal processes

TL;DR

This work analyzes decoupled renewal processes arising from a decoupled standard random walk with nonnegative jumps. Under a regularly varying tail for $\mathbb{P}\{\hat S_1>t\}$ with index $-\alpha$ in $[0,1)$, the authors prove a functional central limit theorem in the Skorokhod space $D(\mathbb{R})$ for the centered and normalized renewal process, identifying an explicit Gaussian limit with a computable covariance. They also establish laws of the iterated or single logarithm for $\hat N(t)$ across tail regimes (A)–(D), including a Mittag-Leffler limiting structure when $\alpha\in[0,1)$, and provide a suite of explicit constants. As an application, they derive a law of the single logarithm for the number of atoms of a determinantal point process with the Mittag-Leffler kernel inside expanding discs, linking renewal theory to spatial point processes. The proofs combine finite-dimensional convergence via Lindeberg–Feller, tightness arguments, and a general LIL/LSL framework, with a careful handling of tail regimes and time-change mappings $h_\alpha$. The results extend prior FCLTs by removing a Lipschitz requirement and by covering regularly varying tails beyond finite-variance settings.

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 $-α$, $α\in [0,1)\cup (1,2)$ 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.

Theorems & Definitions (13)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Example 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['main']}