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

Abstract

In this article, for some $d-$dimensional Gaussian processes \[X=\big\{X_t=(X^1_t,\cdots,X^d_t):t\ge0\big\},\] 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 $\boldsymbol{k}-$th derivatives of local time under certain mild conditions, where $\boldsymbol{k}=(k_1,\cdots,k_d)$ and $k_\ell$'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.