Strong Gaussian approximation for cumulative processes

TL;DR

This paper proves optimal convergence rates for multivariate cumulative processes, providing exponential inequalities and applications to stopped sums and birth-death processes.

Contribution

It introduces the strongest known Gaussian approximation results with optimal rates for multivariate cumulative processes.

Findings

Established optimal logarithmic convergence rates in the strong invariance principle.

Derived exponential probabilistic inequalities of Komlós-Major-Tusnády type.

Applied results to stopped sums and birth-death processes.

Abstract

We establish optimal logarithmic rates of convergence in the strong invariance principle for multivariate cumulative processes in the Smith's sense. Exponential probabilistic inequalities of Koml\'{o}s-Major-Tusn\'{a}dy type are obtained. Provided examples include applications to stopped sums and birth and death processes.