Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T01:58:21.895Z Has data issue: false hasContentIssue false

Random fractals generated by a local Gaussian process indexed by a class of functions

Published online by Cambridge University Press:  05 January 2012

Claire Coiffard*
Affiliation:
LSTA, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France; [email protected]
Get access

Abstract

In this paper, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28 (1974) 174–192]relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arcones, M., The large deviation principle of stochastic processes. II. Theory Probab. Appl. 48 (2003) 1944. CrossRef
Chernoff, H., A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics 23 (1952) 493507. CrossRef
Deheuvels, P. and Lifshits, M.A., On the Hausdorff dimension of the set generated by exceptional oscillations of a Wiener process. Studia. Sci. Math. Hungar. 33 (1997) 75110.
Deheuvels, P. and Mason, D.M., Random fractals generated by oscillations of processes with stationary and independent increments. Probability in Banach Spaces. 9 (1994) 7390. (J. Hoffman-Jørgensen, J. Kuelbs and M.B. Marcus, eds.)
Deheuvels, P. and Mason, D.M., On the fractal nature of empirical increments. Ann. Probab. 23 (1995) 355387. CrossRef
Dindar, Z., On the Hausdorff dimension of the set generated by exceptional oscillations of a two-parameter Wiener process. J. Multivariate Anal. 79 (2001) 5270. CrossRef
K.J. Falconer, The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985).
P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars et Cie (1937)
Mason, D.M., A uniform functional law of the logarithm for a local Gaussian process. Progress in Probability 55 (2003) 135151.
Mason, D.M., A uniform functional law of the logarithm for the local empirical process. Ann. Probab. 32 (2004) 13911418. CrossRef
Orey, S. and Taylor, S.J., How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28 (1974) 174192. CrossRef
Schilder, M., Some asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125 (1966) 6385. CrossRef
Talagrand, M., Sharper bounds for gaussian and empirical processes. Ann. Probab. 22 (1994) 2876. CrossRef
A.W. van der Vaart and A.J. Wellner, Weak convergence and Empirical Processes. Springer, New-York (1996).