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Sur la Lp convergence des martingales liées au recouvrement

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Fan Al Hua
Fan Al Hua*
Affiliation:
Université de Cergy-Pontoise
*
∗Postal address: Department of Mathematics, Université de Cergy-Pontoise, 8, Le Campus, 95033 Cergy-Pontoise, France.
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Abstract

In a coverage problem introduced by Dvoretzky (1956) the circle is covered with lengths The associated indexed martingale can be considered as a set of random densities with respect to Lebesgue measure, of which the limit is a random measure. The total masses of the set constitute a new martingale. From a theorem of Shepp (1972), this martingale does not converge in a space if . If 0 < α < 1 it converges in all spaces , and in this paper we demonstrate that it converges in all spaces Lp for p > 2. We also obtain quite precise estimates for the moments of the total mass of the limit measure, showing that the characteristic function of this total mass is an entire function of order 1/(1 − α). Thus in some sense the mass is distributed in a bounded domain.

Keywords

COVERAGE PROBLEMS ON THE CIRCLEMARTINGALE CONVERGENCE

MSC classification

Primary: 60G46: Martingales and classical analysis
Secondary: 60G48: Generalizations of martingales
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 668 - 678
Copyright
Copyright © Applied Probability Trust 1995 

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References

Références

[1] Dvoretzky, A. (1956) On covering a circle by randomly placed arcs. Proc. Nat. Acad. Sci. USA 42, 199–203.Google Scholar
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