A note on functional limit theorems for compound Cox processes

TL;DR

This paper improves the functional limit theorem for compound Cox processes, showing their convergence to Lévy processes under realistic conditions, with implications for various jump process models.

Contribution

It establishes a more general weak convergence theorem for compound Cox processes to Lévy processes under less restrictive moment conditions.

Findings

Proves convergence of compound Cox processes to Lévy processes.

Shows convergence to stable Lévy processes and hyperbolic distributions.

Extends existing theorems with more realistic assumptions.

Abstract

An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to L{\'e}vy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to L{\'e}vy processes with variance-mean mixed normal distributions, in particular, to stable L{\'e}vy processes, generalized hyperbolic and generalized variance-gamma L{\'e}vy processes.