Abstract
This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of Fn/Gn. As a tool for our work, an Edgeworth expansion for the distribution of Fn/Gn, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs.
Highlights
Two Functionals of Gaussian FieldsLet X = {X (h), h ∈ H} be an isonormal Gaussian process defined on a probability space (Ω, F, P), where H is a real separable Hilbert space, and let { Fn, n ≥ 1} be a sequence of random variables of functionals of infinite-dimensional Gaussian fields associated with X
When we considered the statistical estimation problem of parameters appearing in Stochastic Differential Equations (SDEs) or Stochastic Partial Differential Equations (SPDEs), quite often we encountered statistics of the form
The authors obtain the optimal bound for the convergence rate of the normal approximation for Maximum Likelihood Estimator (MLE) of the parameter occurring in SPDE
Summary
The authors of [12] developed a technique to obtain the upper and lower rates of the convergence on the Kolmogorov distance for a sequence { Fn /Gn } Using both these rates, the authors obtain the optimal bound for the convergence rate of the normal approximation for MLE of the parameter occurring in SPDE. The paper [12] does not provide a complete answer to the optimal rates of convergence on the Kolmogorov distance because the convergence rates in the upper and lower bound derived in [12] may not match each other. The aim of the present work is to find a complete answer for the optimal rate To this end, a set of sufficient conditions will be derived, ensuring that the upper and lower bounds yield an optimal rate for the Kolmogorov distance. Throughout this paper, c (or C) stands for an absolute constant with possibly different values in different places
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