# Relationship between signal amplitude and input output correlation

I have a linear system with input $x(t)$ and output $y(t)$ given by $$y(t) = \int_0^\infty K(t')x(t-t')dt',$$ where $K(t)$ is a known kernel, with some parameters.

The functional form of $K(t)$ is simple enough such that when $x(t)$ is an Ornstein-Uhlenbeck process, I can analytically compute the input-output correlation $$\langle x(t)y(t) \rangle.$$

My question is: how is this related to the mean or maximum output amplitude $\langle|y(t)|\rangle$ or $\max|y(t)|$? For example, can we say something like, the amplitude is maximal, for a given set of parameters, when the correlation is maximal?