Some conditional FKG inequalities for Markov chains and random processes

This paper is interested in proving correlation inequalities of the FKG-type for a broad range of random processes. The two pivotal tools which yield these correlation inequalities are that many discrete-time Markov chains, even under a very general conditioning, satisfy the FKG inequality, and that such inequalities are preserved through weak convergences. In particular, we prove that L\'{e}vy processes, Bessel processes and several conditioned Brownian processes have positive correlations.