Some FKG inequalities for stochastic processes

TL;DR

This paper establishes FKG-type correlation inequalities for various continuous-time stochastic processes using Markov chain approximations, including Levy, Bessel, and conditioned Brownian processes, and characterizes lattice conditions for random walks.

Contribution

It introduces new FKG inequalities for a range of stochastic processes and provides a criterion for lattice conditions in random walks, expanding the theoretical understanding of correlation structures.

Findings

Proved FKG inequalities for Levy processes, Bessel processes, and conditioned Brownian motions.

Developed an approximation method using Markov chains for correlation inequalities.

Provided a necessary and sufficient condition for FKG lattice condition in random walks.

Abstract

This paper is interested in proving correlation inequalities of the FKG-type for various stochastic processes in continuous time. The pivotal tool which yields these correlation inequalities is an approximation with (possibly conditioned) Markov chains and random walks. In particular, we prove FKG inequalities for L\'evy processes, Bessel processes and several conditioned Brownian processes. As a side result, we also provide a necessary and sufficient condition for a random walk distribution in $\bbZ$ to satisfy the well-known ``FKG lattice condition''.