On the maximal correlation of some stochastic processes

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.