Exact convergence rates to derivatives of local time for some self-similar Gaussian processes

In this article, for some $d-$dimensional Gaussian processes $$X=\{X_t=(X_t^1,\cdots ,X_t^d):t\ge 0\},$$ whose components are i.i.d. $1-$dimensional self-similar Gaussian process with Hurst index $H\in (0,1)$, we consider the asymptotic behavior of approximation of its $\mathit{k}-$th derivatives of local time under certain mild conditions, where $\mathit{k}=(k_1,\cdots ,k_d)$ and $k_{\mathrm{\ell }}$’s are non-negative real numbers. We will give a derivative version of the limit theorems for functional of Gaussian processes and use this result to get the asymptotic behaviors.