Necessary and Sufficient Conditions for FKG Inequalities in Stochastic Processes

Alapfogalmak

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.

Kivonat

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.

Statisztikák

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Idézetek

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Főbb Kivonatok

Some FKG inequalities for stochastic processes

by Alexandre Le... : arxiv.org 10-01-2024

https://arxiv.org/pdf/2407.13871.pdf

Mélyebb kérdések

What are some potential applications of the FKG inequalities established in this paper beyond the examples provided?

The FKG inequalities, as established in this paper, have a wide range of potential applications beyond the specific examples of Lévy processes, Bessel processes, and conditioned Brownian processes. One significant area of application is in statistical mechanics, particularly in the study of phase transitions and critical phenomena. The FKG inequality can be utilized to demonstrate the existence of phase transitions in models such as the Ising model and percolation theory, where the correlation structure of spins or connectivity can be analyzed.
Additionally, these inequalities can be applied in the field of stochastic domination, where they help in establishing bounds on the probabilities of events in various stochastic models. This is particularly useful in the analysis of random walks and Markov chains, where one can derive results about the long-term behavior of these processes.
In the realm of finance, FKG inequalities can be employed in risk assessment and portfolio optimization, where the association of random variables can lead to better understanding of the dependencies between asset returns. Furthermore, in machine learning, particularly in the context of probabilistic graphical models, the FKG inequalities can assist in understanding the relationships between variables and improving inference algorithms.

Can the results be extended to more general stochastic processes beyond Lévy processes, Bessel processes, and conditioned Brownian processes?

Yes, the results presented in this paper can potentially be extended to more general classes of stochastic processes. The key to such extensions lies in the underlying structure of the processes and the conditions that ensure the association of random variables. For instance, processes that exhibit certain monotonicity properties or those that can be approximated by Markov chains may also satisfy the FKG inequalities.
Moreover, the framework established in this paper, particularly the use of conditional distributions and the necessary and sufficient conditions for association, can be adapted to other stochastic processes such as jump processes, diffusion processes, and even more complex systems like interacting particle systems. The generalization would require careful consideration of the specific characteristics of these processes, such as their transition mechanisms and the nature of their state spaces.

Are there any connections between the FKG lattice condition and other correlation inequalities or notions of association in the literature?

The FKG lattice condition is closely related to several other correlation inequalities and notions of association found in the literature. One prominent connection is with the concept of positive association, which states that for any two increasing functions, the expected product of their values is at least the product of their expected values. This notion is foundational in the study of dependent random variables and is often used in the context of stochastic processes.
Additionally, the FKG condition is linked to the concept of "downward FKG" inequalities, which focus on the behavior of random variables conditioned on certain events. This connection is particularly relevant in the study of percolation and spin systems, where conditioning on specific configurations can lead to different correlation structures.
Moreover, the FKG lattice condition has implications for the study of log-concavity and log-convexity of probability measures, which are important in various fields such as combinatorics and optimization. The interplay between these concepts enriches the understanding of correlation inequalities and provides a robust framework for analyzing dependencies in random systems.