Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T21:40:42.621Z Has data issue: false hasContentIssue false

Sur la Lp convergence des martingales liées au recouvrement

Published online by Cambridge University Press:  14 July 2016

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

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1995 

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

Références

[1] Dvoretzky, A. (1956) On covering a circle by randomly placed arcs. Proc. Nat. Acad. Sci. USA 42, 199203.Google Scholar
[2] Kahane, J.-P. (1959) Sur le recouvrement d'un cercle par des arcs disposés au hasard. C. R. Acad. Sci. Paris 248, 184186.Google Scholar
[3] Kahane, J.-P. (1985) Some Random Series of Functions. 2nd edn. Cambridge University Press.Google Scholar
[4] Kahane, J.-P. (1987) Positive martingales and random measures. Chinese Ann. Math. 8B1, 112.Google Scholar
[5] Kahane, J.-P. (1987) Intervalles aléatoires et décomposition des mesures. C. R. Acad. Sci. Paris. 304, 551554.Google Scholar
[6] Lukacs, E. (1983) Developments in Characteristic Function Theory. Griffin, London.Google Scholar
[7] Shepp, L. A. (1972) Covering the circle with random arcs. Israël J. Math, 11, 328345.CrossRefGoogle Scholar