Abstract
We show an invariance principle for the empirical measure of a stationary strongly mixing sequence indexed by the unit ball of some Sobolev space Hs. We also obtain invariance principle and law of iterated logarithm for the local time of Markov processes indexed by Hs.
We note that the regularity condition s > d/2 in the first framework for random variables with values in a compact riemannian manifold E becomes s > d/2-1 in the continuous case of the brownian motion on E.
Keywords
- Brownian Motion
- Markov Process
- Invariance Principle
- Empirical Measure
- Iterate Logarithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your 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.
Bibliography
A. De Acosta. Existence and convergence of probability measures on Banach spaces. Trans. Amer. Math. Soc. 152, pp. 273–298 (1970).
S. Albeverio, R., Hoegh-Krohn, L. Streit. Energy forms, Hamiltonian and distorted Brownian paths. J. of Math. Phys. 18, no 5, pp. 907–917 (1977).
R.M. Battacharya. On the functionnal central limit theorem and the law of the iterated logarithm for Markov processes. Zeit. für Wahr. und Verw. Gebiete 60, pp. 185–201 (1982).
J.R. Baxter, G.A. Brosamler. Energy and the law of the iterated logarithm. Math. Scand. 38, pp. 115–136 (1976).
E. Bolthausen. On the asymptotic behaviour of the empirical random field of the Brownian motion. Stoch. Pr. and their Appl. 16, pp. 199–204,(1983).
R. Carmona. Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger. Séminaire de probabilité XIII, Strasbourg 1977–1978, L.N.M. 721, pp. 557–569 (1979).
D. Dacunha-Castelle, D. Florens. Choix du paramètre de discrétisation pour estimer le paramètre d'une diffusion. C.R.A.S. Série I, Paris, t.299, pp. 65–69 (1984).
H. Dehling, W. Philipp. Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. of Prob. 10, pp. 689–701 (1982).
P. Doukhan. Fonctions d'Hermite et statistiques des processus mélangeants (Submitted to publication, 1985).
P. Doukhan, J.R. Leon, F. Portal. Principe d'invariance faible pour la mesure empirique d'une suite de variables aléatoires dépendantes. (Submitted to publication, 1985).
P. Doukhan, J.R. Leon, F. Portal. Calcul de la vitesse de convergence dans le théorème central limite vis à vis des distances de Prohorov, Dudley et Levy dans le cas de variables aléatoires dépendantes. Prob. and Math. Stat. VI.2, 1985.
X. Fernique. Régularité des trajectoires des fonctions aléatoires gaussiennes. L.N.M. 480, Springer (1975).
D. Florens. Théorème de limite centrale des fonctionnelles de diffusions. C.R.A.S. Série I, Paris, t. 299, pp. 995–998 (1984).
E. Giné. Invariant test for uniformity on compact riemannian manifolds based on Sobolev norms. Ann. of Stat. 3, pp. 1243–1266 (1975).
N. Ikeda, S. Watanabe. Stochastic differential equations and diffusion processes. North-Holland, Tokyo (1981).
K. Ito, H.P. Mc Kean. Diffusion processes and their sample paths. Springer Verlag, Berlin (1974).
J. Kuelbs. Kolmogorov law of the iterated logarithm for Banach space valued random variables. Illinois J. of Math. 21, pp. 784–800 (1977).
J. Kuelbs, R. Lepage. The law of the iterated logarithm for Brownian motion in a Banach space. Trans. of the Amer. Math. Soc. 185, pp.253–264 (1973).
S. Minakshisundaram, A. Pleijel. Some properties of the eigenfunctions of the Laplace-operator on riemannian manifolds. Can. J. of Math. 1, pp. 242–256 (1943).
L. Nirenberg. Pseudodifferential operators. Proc. of Symp. in pure Math. XVI. Global Analysis, A.M.S. Providence, pp. 149–167 (1970).
M. Rosenblatt. Markov processes. Springer Verlag, New York (1971).
R.T. Seeley. Complex powers of an elliptic operator. Proc. Symp. in pure Math. X, A.M.S., Providence, pp. 288–307 (1968).
G. Szegö. Orthogonal polynomials. A.M.S. Providence (1939).
H. Tamura. Asymptotic formulas with sharp remainder estimates for eigenvalues of elliptic operators of second order. Duke Math. J. 49, pp. 87–119 (1982).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag
About this paper
Cite this paper
Doukhan, P., Leon, J.R. (1986). Invariance principles for the empirical measure of a mixing sequence and for the local time of markov processes. In: Fernique, X., Heinkel, B., Meyer, PA., Marcus, M.B. (eds) Geometrical and Statistical Aspects of Probability in Banach Spaces. Lecture Notes in Mathematics, vol 1193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077096
Download citation
DOI: https://doi.org/10.1007/BFb0077096
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16487-6
Online ISBN: 978-3-540-39826-4
eBook Packages: Springer Book Archive
