On the paper ``Weak convergence of some classes of martingales with jumps''

TL;DR

This paper extends existing results on martingales with jumps by providing a maximal inequality for stochastic integrals with respect to integer-valued measures, leading to a new tightness criterion based on entropy methods.

Contribution

It introduces a maximal inequality for stochastic integrals with jumps and establishes a tightness criterion using entropy techniques, extending prior work by Nishiyama.

Findings

Maximal inequality for stochastic integrals with jumps

Tightness criterion based on quadratic modulus and entropy conditions

Application of entropy techniques to martingale convergence

Abstract

This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685--712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes.