Blackwell-type Theorems for Weighted Renewal Functions

TL;DR

This paper extends Blackwell-type theorems to weighted renewal functions for random walks with broad conditions on weights and jumps, using advanced probabilistic tools to analyze asymptotic behaviors.

Contribution

It introduces new asymptotic results for weighted renewal sums under broader conditions on weights and jump distributions, including non-positive weights and jumps.

Findings

Broader conditions on weights and jumps established.

Asymptotic behavior characterized for various jump distribution classes.

Results applicable to non-positive weights and jumps.

Abstract

For a numerical sequence ${a_n}$ satisfying broad assumptions on its "behaviour on average" and a random walk $S_n=\xi_1 +...+\xi_n$ with i.i.d. jumps $\xi_j$ with positive mean $\mu$, we establish the asymptotic behaviour of the sums [\sum_{n\ge 1} a_n \pr (S_n\in[x, x+\D)) \quad as \quad x\to \infty,] where $\D>0$ is fixed. The novelty of our results is not only in much broader conditions on the weights ${a_n}$, but also in that neither the jumps $\xi_j$ nor the weights $a_j$ need to be positive. The key tools in the proofs are integro-local limit theorems and large deviation bounds. For the jump distribution $F$, we consider conditions of four types: (a) the second moment of $\xi_j$ is finite, (b) $F$ belongs to the domain of attraction of a stable law, (c) the tails of $F$ belong to the class of the so-called locally regularly varying functions, (d) $F$ satisfies the moment Cram\'er condition. Regarding the weights, in cases (a)--(c) we assume that ${a_n}$ is a so-called $\psi$-locally constant on average sequence, $\psi(n)$ being the scaling factor ensuring convergence of the distributions of $(S_n - \mu n)/\psi (n)$ to the respective stable law. In case (d) we consider sequences of weights of the form $a_n=b_n e^{qn},$ where ${b_n}$ has the properties assumed about the sequence ${a_n}$ in cases (a)--(c) for $\psi(n)=\sqrt{n}.$