Exact Convergence Rates to Derivatives of Local Time for Some Self-similar Gaussian processes

Minhao Hong ()
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Minhao Hong: Shanghai Maritime University

Journal of Theoretical Probability, 2025, vol. 38, issue 3, 1-32

Abstract: Abstract In this article, for some d-dimensional Gaussian processes $$\begin{aligned} X=\big \{X_t=(X^1_t,\ldots ,X^d_t):t\ge 0\big \}, \end{aligned}$$ X = { X t = ( X t 1 , … , X t d ) : t ≥ 0 } , whose components are i.i.d. 1-dimensional self-similar Gaussian processes with Hurst index $$H\in (0,1)$$ H ∈ ( 0 , 1 ) , we consider the asymptotic behavior of approximation of its $$\varvec{k}$$ k -th derivatives of local time under certain mild conditions, where $$\varvec{k}=(k_1,\ldots ,k_d)$$ k = ( k 1 , … , k d ) and $$k_\ell $$ k ℓ ’s are non-negative real numbers. We will prove limit theorems for functionals of Gaussian processes related to derivatives of local time and use this result to obtain the asymptotic behaviors.

Keywords: Gaussian processes; Hurst index; Derivatives of local time; Method of moments; Primary: 60F05; 60F25; Secondary: 60G15; 60G18 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01431-y

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