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

TL;DR

This paper investigates the precise convergence rates of derivatives of local time for certain self-similar Gaussian processes, providing limit theorems and asymptotic behaviors under mild conditions.

Contribution

It introduces a derivative version of limit theorems for Gaussian process functionals, extending understanding of local time derivatives for self-similar Gaussian processes.

Findings

Derived exact convergence rates for derivatives of local time.

Established limit theorems for derivatives of local time.

Analyzed asymptotic behaviors under mild conditions.

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.