Celebratio Mathematica

Y. Hu and M. Yor: “Convergence in law and convergence of moments: An example related to Bessel processes,” pp. 387–397 in Asymptotic methods in probability and statistics (Ottawa, ON, 8–13 July 1997). Edited by B. Szyszkowicz. North-Holland/Elsevier (Amsterdam), 1998. A volume in honour of Miklós Csörgő. MR 1661495 Zbl 0933.60021 incollection

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

G. Peccati and M. Yor: “Hardy’s inequality in $L^{2} ([0, 1])$ and principal values of Brownian local times,” pp. 49–74 in Asymptotic methods in stochastics: Festschrift for Miklós Csörgő (Ottawa, 23–25 May 2002). Edited by L. Horvárth and B. Szyszkowicz. Fields Institute Communications 44. American Mathematical Society (Providence, RI), 2004. MR 2106848 Zbl 1074.60029 incollection

We present in a unified framework two examples of a random function $ \phi(\omega,s) $ on $ \mathfrak{R}_+ $ such that

the integral \[ \int_0^{\infty}\phi(\omega,s)g(s)\,ds \] is well defined and finite (at least, as a limit in probability) for every deterministic and square integrable function $ g $, and
$ \phi $ does not belong to $ L^2([0,\infty) $, $ ds) $ 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.

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

G. Peccati and M. Yor: “Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge,” pp. 75–87 in Asymptotic methods in stochastics: Festschrift for Miklós Csörgő (Ottawa, 23–25 May 2002). Edited by L. Horvárth and B. Szyszkowicz. Fields Institute Communications 44. American Mathematical Society (Providence, RI), 2004. MR 2106849 Zbl 1071.60017 incollection

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

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Marc Yor

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