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

Abstract

The simple Lévy 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.

21 pages