Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- G. PECCATI
**M. YOR**

**Code(s) de Classification MSC:**

- 60F05 Central limit and other weak theorems
- 60F25 $L^p$-limit theorems
- 60G15 Gaussian processes
- 60G17 Sample path properties
- 60G48 Generalizations of martingales

**Résumé:** We present in a unified framework two examples of a random function $\phi
\left( \omega ,s\right) $ on $\Re _{+}$ such that (a) the integral $%
\int_{0}^{\infty }\phi \left( \omega ,s\right) g\left( s\right) ds$ is well
defined and finite (at least, as a limit in probability) for every
deterministic and square integrable function $g$, and (b) $\phi $ does not
belong to $L^{2}\left( \Re ^{+},ds\right) $ with probability one. In
particular, the second example is related to the existence of principal
values of Brownian local times. Our key tools are Hardy's inequality, some
semimartingale representation results for Brownian local times due to Ray,
Knight and Jeulin, and the reformulation of certain theorems of Jeulin-Yor
(1979) and Cherny (2001) in terms of bounded $L^{2}$ operators. We also
establish, in the last paragraph, several weak convergence results.

**Mots Clés:** *Brownian motion ; local times ; principal values ; Hardy inequality*

**Date:** 2001-06-06

**Prépublication numéro:** *PMA-679*