Partial mixing and Edgeworth expansion
type: Item
instance of: scholarly article
title: Partial mixing and Edgeworth expansion
zbMATH Open document ID: 1057.62017
author: Nakahiro Yoshida
publication date: 2004-10-28T00:00:00Z
review text: Given a probability space \((\Omega , {\mathcal F}, P)\), consider \({\mathcal C}\) a sub \(\sigma\)-field of \({\mathcal F}\), \(\{ {\mathcal B}_I, I\subset {\mathbb R}_+\}\) an increasing family of sub \(\sigma\)-fields of \({\mathcal F}\), and \((Z_t)_{t\in {\mathbb R}_+}\) a stochastic process such that \(Z_0\) is \({\mathcal B}_{[0]}\vee {\mathcal C}\)-measurable and \(Z_t-Z_s\) is \({\mathcal B}_{[s,t]}\vee {\mathcal C}\)-measurable, \(0\leq s < t\). The author derives asymptotic expansions for the expectation of \(f(Z_T/\sqrt{T})\), allowing the conditional mixing coefficient given \({\mathcal C}\) to have a slower than exponential decay. His general results apply to \(\sigma\)-fields of the form \[ {\mathcal B}_I = \sigma (X_t-X_s : s,t\in I)\vee \sigma (Y_t : t\in I ), \] where \((Y_t)_{t\in {\mathbb R}_+}\) is a \({\mathcal C}\)-conditional \(\epsilon\)-Markov process driven by a process \((X_t)_{t\in {\mathbb R}_+}\) with independent increments. They also apply when \((Z_t)_{t\in {\mathbb R}_+}\) is a solution to a stochastic differential equation with jumps and random coefficients that are functions of \((Y_t)_{t\in {\mathbb R}_+}\). Malliavin calculus and the support theorem for stochastic differential equations with jumps are both used to satisfy the needed conditional Cramér condition on the characteristic function of the increments of \((Z_t)_{t\in {\mathbb R}_+}\).
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 2111580
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: partial mixing
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: support theorem
MaRDI profile type: Publication
full work available at URL: s00440-003-0325-8
OpenAlex ID: W1973628224
cites work: Q4279819
cites work: Asymptotic efficiency of statistical estimators: concepts and higher order asymptotic efficiency
cites work: Asymptotic expansions for the power of distribution free tests in the one-sample problem
cites work: On Edgeworth expansions in the mixture cases
cites work: Q4326424
cites work: On the functional central limit theorem and the law of the iterated logarithm for Markov processes
cites work: On the validity of the formal Edgeworth expansion
cites work: Q4097679
cites work: Q3996311
cites work: The Edgeworth expansion for U-statistics of degree two
cites work: Time series: theory and methods.
cites work: Q4735879
cites work: Q3883242
cites work: On the first-order Edgeworth expansion for a Markov chain
cites work: Mixing: Properties and examples
cites work: Integration by parts for the single jump process
cites work: The bootstrap and Edgeworth expansion
cites work: Asymptotic expansions in the central limit theorem for compound and Markov processes
cites work: Support theorem for jump processes of canonical type
cites work: Existence of the density for a singular jump process and its short time properties
cites work: Q3828838
cites work: Weak limit theorems for stochastic integrals and stochastic differential equations
cites work: Q3336457
cites work: Malliavin calculus, geometric mixing, and expansion of diffusion functionals
cites work: Statistical inference for spatial Poisson processes
cites work: Refinements in asymptotic expansions for sums of weakly dependent random vectors
cites work: Q4365842
cites work: Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model
cites work: Support theorems for a class of anticipating stochastic differentialequations
cites work: Asymptotic expansions and bootstrapping distributions for dependent variables: A martingale approach
cites work: Asymptotic expansions for martingales
cites work: Martingale expansions and second order inference
cites work: More Exact Statement of Limit Theorems for Homogeneous Markov Chains
cites work: Q3795015
cites work: Q3479316
cites work: Q4379507
cites work: Asymptotic expansions for general statistical models. With the assist. of W. Wefelmeyer
cites work: Duality formulas on the Poisson space
cites work: On the existence of smooth densities for jump processes
cites work: Chaotic and variational calculus in discrete and continuous time for the poisson process
cites work: A transfer principle from Wiener to Poisson space and applications
cites work: A different quantum stochastic calculus for the Poisson process
cites work: Exponential convergence of Langevin distributions and their discrete approximations
cites work: Time series regression with long-range dependence
cites work: Théorème de support pour processus à sauts
cites work: Support theorem for jump processes.
cites work: Support of a Marcus equation in dimension \(1\)
cites work: Q5603655
cites work: Probability Theory
cites work: Q5670045
cites work: Bayesian prediction with approximate frequentist validity.
cites work: Higher order asymptotic theory for time series analysis
cites work: Asymptotic theory of statistical inference for time series
cites work: Applications of Malliavin's calculus to time-dependent systems of heat equations
cites work: Information criteria in model selection for mixing processes
cites work: Bounds for the Mixing Rate in the Theory of Stochastic Equations
cites work: On polynomial mixing bounds for stochastic differential equations

Partial mixing and Edgeworth expansion (cites work)
cites work: Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Partial mixing and Edgeworth expansion (cites work)
cites work: Malliavin calculus and asymptotic expansion for martingales

Partial mixing and Edgeworth expansion (DOI)
DOI: 10.1007/S00440-003-0325-8

Partial mixing and Edgeworth expansion (published in)
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete