Doubly uniform complete law of large numbers for independent point processes

TL;DR

This paper establishes a uniform law of large numbers for independent point processes and related functions, accommodating jumps and lacking Markov or martingale properties, under certain regularity and moment conditions.

Contribution

It introduces a complete convergence law of large numbers for point processes with monotonic increments, extending applicability beyond Markov or martingale frameworks.

Findings

Proves uniform complete convergence for independent point processes.

Extends results to functions of two parameters with monotonicity.

Applicable to point processes with jumps, under regularity conditions.

Abstract

We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for the random variables taking values in functions of $2$ parameters which share similar monotonicity properties as the increments of monotone functions. The assumptions for the main result are the H\"older continuity on the expectations as well as moment conditions, while the sample functions may contain jumps. In particular, we can apply the results to point processes (counting processes) which lack Markov or martingale type properties.