Let $f(t)$ be a stochastic process. Suppose that I know $F(s)$, which is also a stochastic process and is the Laplace transform of $f(t)$. I also know the values $\langle F(s) \rangle$ and $\langle F^2(s)\rangle$, and I can compute the mean auto-convolution of $f(t)$ based on the relationship \begin{equation} \left\langle f(t) * f(t)\right\rangle = \mathcal{L}^{-1}\left[\left\langle F^2(s)\right\rangle \right]. \end{equation} My question is, if I am interested in an expression like $\langle f^2(t) \rangle$, or $\langle f(t) f(t+\tau)\rangle$ (auto-correlation), is there any way to extract this information from the auto-convolution of $f(t)$? Is there any mathematical theorem that connects the auto-convolution and auto-correlation of a stochastic process, just like there is one for time-continuous deterministic signals?
Side note: we define the variance of a stochastic process as \begin{equation} \text{Var}[f(t)] = \left\langle f^2(t)\right\rangle - \left\langle f(t)\right\rangle^2, \end{equation} so is there an analogous expression for something like \begin{equation} \left\langle f(t) * f(t)\right\rangle - \left\langle f(t)\right\rangle * \left\langle f(t)\right\rangle = ? \end{equation}