  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.
