Let $ G(t,\omega)$ be a stochastic process with a.s. continuous paths defined on a probability space $ \Omega$ with $ \omega \in \Omega$ and over a time period $ t\in[0,T]$ .

$ T$ here is a constant, not a random variable. Suppose that the distribution of the process at every time point is known.

I am interested in the distribution of the random variable $ \int_0^T G(t,\omega) \,dt$ . This essentially boils down to finding the limiting distribution of a sum of correlated random variables.

If $ G(t,\omega)$ is a Gaussian process, it can be shown that this integral also has a Gaussian distribution. In addition, in some works on stochastic homogenization, limiting distributions can be derived for this integral for cases when $ G(t,\omega)$ is non-Gaussian but for which its correlation is such that $ G(t,\omega)$ is highly oscillatory making $ G(t_1,\omega)$ and $ G(t_2,\omega)$ almost independent for $ t_1,t_2$ close to each other.

I am wondering if there are any other results available aside from the 2 cases above. Suppose that $ G(t,\omega)$ has the exponential correlation function $ \exp(-\lambda |t_1-t_2|)$ for $ \lambda > 0$ . Can the distribution of $ \int_0^T G(t,\omega) \,dt$ be derived for certain values of $ \lambda$ ? I.e. for $ \lambda$ such that the dependence is weak but not too weak as in that of homogenization?

I’d appreciate if someone can point me to references about this topic. Thanks!