Martingale Nature and Laws of the Iterated Logarithm for Markov Processes of Pure-Jump Type

TL;DR

This paper establishes conditions under which certain pure-jump Markov processes are martingales with finite variance and proves the law of the iterated logarithm for their sample paths, advancing understanding of their long-term behavior.

Contribution

It provides new sufficient conditions based on jumping kernels for pure-jump Markov processes to be martingales and proves the law of the iterated logarithm for these processes.

Findings

Pure-jump Markov processes are martingales under specified conditions.

Law of the iterated logarithm is established for these processes.

Finite second moment is confirmed for the processes studied.

Abstract

We present sufficient conditions, in terms of the jumping kernels, for two large classes of conservative Markov processes of pure-jump type to be purely discontinuous martingales with finite second moment. As an application, we establish the law of the iterated logarithm for sample paths of the associated processes.