$\ell^{\infty}$ Poisson invariance principles from two classical Poisson limit theorems and extension to non-stationary independent sequences

TL;DR

This paper develops invariance principles for $\, ext{L}^\infty$ Poisson processes, extending classical Poisson limit theorems to non-stationary independent sequences and paving the way for dependent data analysis.

Contribution

It introduces a method to derive $\, ext{L}^\infty$ Poisson invariance principles from classical theorems and extends these results to non-stationary independent sequences.

Findings

Explicit construction of Poisson processes from i.i.d. sums

Weak convergence to scaled Poisson processes for non-stationary sequences

Framework for extending results to dependent data

Abstract

The simple L\'evy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to associated data in the first place.