On the maximal correlation of some stochastic processes

TL;DR

This paper investigates the maximal correlation coefficient between two stochastic processes, providing explicit formulas for specific processes and extending key inequalities in the field.

Contribution

It derives formulas for maximal correlation in processes like random walks and Lévy processes, and extends important inequalities related to correlation measures.

Findings

Explicit formula for $R(X,Y)$ in random walks using Csáki-Fischer identity.

Expression of $R(X,Y)$ for Lévy processes in terms of Lévy measure and covariance.

Extension of Dembo-Kagan-Shepp-Yu and Madiman-Barron inequalities.

Abstract

We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Cs\'{a}ki-Fischer identity and the lower semicontinuity of the map $\text{Law}(X,Y) \to R(X,Y)$. When $(X,Y)$ is a two-dimensional L\'{e}vy process, we express $R(X,Y)$ in terms of the L\'{e}vy measure of the process and the covariance matrix of the diffusion part of the process. Consequently, for a two-dimensional $\alpha$-stable random vector $(X,Y)$ with $0<\alpha<2$, we express $R(X,Y)$ in terms of $\alpha$ and the spectral measure $\tau$ of the $\alpha$-stable distribution. We also establish analogs and extensions of the Dembo-Kagan-Shepp-Yu inequality and the Madiman-Barron inequality.