Functional limit theorems for some self-similar Gaussian processes in critical and subcritical cases

In this paper, under certain conditions, we investigate the asymptotic behavior of $\{{\int }_0^t{f}_{\alpha }\left(n^H(X_s-\lambda )\right)ds,t\ge 0\}$, where $f_{\alpha }$ is the density of symmetric $\alpha$-stable random variables with $\alpha \in (0,2)$ and $X=\left\{X_t,t\ge 0\right\}$ is some self-similar Gaussian process with index $H\in (0,1)$. We mainly focus on the critical case $H(2\alpha +1)=1$ and the subcritical case $H(2\alpha +1)<1$. This work will extend the corresponding results in Hong et al. (2024) and may give another definition for the fractional derivative of local times of the Gaussian process $X$.