Summary
In this contribution, we consider the problem of variance estimation in the computation of the invariant measure of a random dynamical system via ergodic simulations. An adaptive estimator of the variance for such simulations is deduced from a general result stating an almost sure central limit theorem for empirical means. We also provide a speed of convergence for this estimator.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
O. Bardou, Contrôle dynamique des erreurs de simulation et d’estimation de processus de diffusion, Ph.D. thesis, Université de Nice-Sophia Antipolis, 2005.
P. Bertail and S. Clémençoon, Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains, Probab. Theory Related Fields (2004).
R.N. Bhattacharya, On the functional central limit theorem and the law of the iterared logarithm for Markov processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 60 (1982), 185–201.
D. Bosq and Yu. Davydov, Local time and density estimation in continuous time, Math. Methods Stat. 8 (1999), no. 1, 22–45.
F. Chaabane, Invariance principles with logarithmic averaging for continuous local martingales, Statist. Probab. Lett. 59 (2002), no. 2, 209–217.
Ĭ.Ī. Gīhman and A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972, Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72.
R.Z. Has’minskiĭ, Stochastic stability of differential equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, Translated from the Russian by D. Louvish.
N.V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York, 1980, Translated from the Russian by A. B. Aries.
Yu.A. Kutoyants, On a hypotheses testing problem and asymptotic normality of stochastic integral, Theory of probability and its applications (1975).
_____, Statistical inference for ergodic diffusion processes., Springer Series in Statistics, Springer-Verlag, 2003.
D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 (2002), no. 3.
F. Maaouia, Principes d’invariance par moyennisation logarithmique pour les processus de Markov, Ann. Probab. 29 (2001), no. 4, 1859–1902.
M. Martinez, Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées, Ph.D. thesis, Université de Marseilles, 2004.
M. Montfort, Cours de statistique mathématique, Economica, 1982.
E. Pardoux and A.Yu. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab. 29 (2001), no. 3.
C. Soize, The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions, Series on Advances in Mathematics for Applied Sciences, vol. 17, World Scientific Publishing Co. Inc., River Edge, NJ, 1994.
D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law., Stochastics Stochastics Rep. 29 (1990), no. 1, 13–36.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bardou, O. (2006). Invariance Principles with Logarithmic Averaging for Ergodic Simulations. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_1
Download citation
DOI: https://doi.org/10.1007/3-540-31186-6_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25541-3
Online ISBN: 978-3-540-31186-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)