Edgeworth expansion for ergodic diffusions
type: Item
instance of: scholarly article
title: Edgeworth expansion for ergodic diffusions
zbMATH Open document ID: 1154.60018
author: Masaaki Fukasawa
publication date: 2008-10-16T00:00:00Z
review text: The present paper validates the Edgeworth expansion for continuous-time strong Markov processes. Let \(X = (X_t )_{t \geq 0}\) be such a process and \(Z = (Z_t )_{t \geq 0}\) be an \(n\)-additive functional of \(X\). Denote by \(P_\nu \) the law of \(X\) with initial distribution \(\nu\). It is proved that for a given sufficiently smooth function \(A:\mathbb{R}^n \to \mathbb{R}^1\), \[ \sup_B \left| {P_\nu \left[ {\sqrt T (A(Z_T /T) - A(\gamma ))/\sigma \in B} \right] - \int_B {p_T (z)dz} } \right| = O(T^{ - 1})\quad\text{as }T \to \infty, \] where supremum is taken over all Borel sets, \[ p_T(z) = \varphi(z)\left\{ 1 + T^{ - 1/2} \left\{ A_1 z/\sigma + A_2 (z^3 - 3z)/6\sigma ^3 \right\} \right\}, \] \(\varphi\) is the density of the standard normal distribution and \(\gamma \), \(\sigma \), \(A_j \), j=1,2 are constants written explicitly in terms of moments of certain distributions depending on \(X\) and \(Z\). In particular for diffusions of dimension one, verifiable conditions for the validity of the expansion are given on terms of the coefficients of the corresponding stochastic differential equations.
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 5351821
zbMATH Keywords: Edgeworth expansion
zbMATH Keywords: ergodic diffusions
zbMATH Keywords: additive functionals
MaRDI profile type: Publication
full work available at URL: s00440-007-0097-7
OpenAlex ID: W1972651645
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DOI: 10.1007/S00440-007-0097-7
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5351821