מושגי ליבה
This paper establishes necessary and sufficient conditions for various stochastic processes, including Lévy processes, Bessel processes, and conditioned Brownian processes, to satisfy the FKG inequality, a fundamental correlation inequality in probability theory. The key tool is an approximation of these processes using Markov chains and random walks.
תקציר
The paper focuses on proving FKG-type correlation inequalities for stochastic processes in continuous time. The main results are:
Theorem 1.2 provides a necessary and sufficient condition for a d-dimensional Lévy process to satisfy the FKG inequality on the functional space D([0,T], Rd). This extends a previous result for the 1-dimensional Brownian motion.
Theorem 1.7 shows that the conditional distribution of a Markov chain trajectory satisfies the FKG inequality under two mild assumptions: (i) the transition kernel has "unfavorable crossings" and (ii) the conditioning set is max/min-stable. This allows the authors to deduce the FKG inequality for various conditioned Brownian processes and Bessel processes.
Proposition 1.11 fully characterizes the 1-dimensional lattice random walks that satisfy the FKG lattice condition, which is a stronger property than the FKG inequality itself. This complements the results on conditional association.
The proofs rely on classical results about association of random variables, weak convergence of measures, and properties of Markov chains and random walks. The paper provides a comprehensive treatment of FKG inequalities for stochastic processes, with applications in various domains.