Densities of some Poisson $ \mathrm{T}$-martingales and random covering numbers

Julien Barral; Aihua Fan

doi:10.1016/j.crma.2004.01.027

Probability Theory/Mathematical Analysis

Densities of some Poisson

T

-martingales and random covering numbers

Presented by: Yves Meyer

Julien Barral ¹; Aihua Fan ^2,³

¹ Équipe “Complex”, INRIA Rocquencourt, 78153 Le Chesnay cedex, France
² Department of Mathematics, Wuhan University, 430072, Wuhan, China
³ LAMFA, UMR 6140 CNRS, Université de Picardie, 33, rue Saint Leu, 80039 Amiens, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 571-574.

Abstract
Résumé

The asymptotic behavior of the logarithm of the density of some $T$ -martingales (in the sense of Kahane theory (Chinese Ann. Math. Ser. B 8 (1) (1987) 1–12)) is described in detail even in absence of statistical self-similarity. Poisson intensities of the form Lebesgue⊗μ on $ℝ \times ℝ_{+}^{*}$ are involved in the construction of these martingales. We prove that there are three possible behaviors according to the fact that $\bar{α} = \lim \sup_{ϵ \to 0} {(- \log ϵ)}^{- 1} \int_{[ϵ, 1)} ℓ d μ (ℓ)$ is zero, positive and finite, or infinite. This problem is closely related to the asymptotic behaviors of covering numbers in Poisson covering of the line and Dvoretzky covering of the circle.

Le comportement asymptotique du logarithme de la densité de certaines $T$ -martingales (au sens de la théorie de Kahane (Chinese Ann. Math. Ser. B 8 (1) (1987) 1–12)) est décrit de façon précise même en l'absence d'auto-similarité en loi. La construction de ces martingales fait intervenir des intensités de Poisson de la forme Lebesgue⊗μ sur $ℝ \times ℝ_{+}^{*}$ . Nous montrons qu'il y a trois comportements possibles selon que $\bar{α} = \lim \sup_{ϵ \to 0} {(- \log ϵ)}^{- 1} \int_{[ϵ, 1)} ℓ d μ (ℓ)$ est nul, strictement positif et fini ou infini. Cette question est intimement liée au comportements asymptotiques des nombres de recouvrements dans le recouvrement de Poisson pour la droite et le recouvrement de Dvoretzky pour le cercle.

Received: 2004-01-07
Accepted: 2004-01-27
Published online: 2004-03-17

DOI: 10.1016/j.crma.2004.01.027

Author's affiliations:

Julien Barral ¹; Aihua Fan ^{2, 3}

¹ Équipe “Complex”, INRIA Rocquencourt, 78153 Le Chesnay cedex, France
² Department of Mathematics, Wuhan University, 430072, Wuhan, China
³ LAMFA, UMR 6140 CNRS, Université de Picardie, 33, rue Saint Leu, 80039 Amiens, France

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     author = {Julien Barral and Aihua Fan},
     title = {Densities of some {Poisson} $ \mathrm{T}$-martingales and random covering numbers},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2004},
     doi = {10.1016/j.crma.2004.01.027},
     language = {en},
}

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Julien Barral; Aihua Fan. Densities of some Poisson $ \mathrm{T}$-martingales and random covering numbers. Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 571-574. doi : 10.1016/j.crma.2004.01.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.027/

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