I am studying the response of a stochastic linear time invariant system to excitation. The time-domain response can be deduced from convolution of the input and system response; if $x(t)$ is the input and $h(t)$ is the impulse response then the output is $y(t) = (x * h) (t) = \int_{-\infty}^{+\infty} x(t)h(t-\tau) d\tau$
The input can be a random variable or could be deterministic (say a sine that is turned on and off periodically) while $h(t)$ is always a random variable. I know the pdf of $\tilde h(f) $ which is the Fourier transform of $h(t)$ to be a Lomax distribution (a special type of Pareto distribution. $f$ represents frequency) I’m investigating the pdf of the peaks of $y(t)$ to see how it’s related to the pdfs of $x(t)$ and $h(t)$. My approach has been to start from the convolution theorem. Take $x(t)$ to be a deterministic function of time to simplify things, take its Fourier transform. Multiply to the pdf of $\tilde h(f)$ which is the Lomax distribution and then take the inverse Fourier transform to find $y(t)$. In other words $y(t) = ifft(\tilde x(f) \ast \tilde h(f))$.
Am I on the right track to find the distribution of the peaks of $y(t)$?