Y. Hu and M. Yor :
“Convergence in law and convergence of moments: An example related to Bessel processes 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
Abstract
People
BibTeX
Infinitesimal increments of hitting times of Bessel processes provide interesting sequences of random variables, which converge in law to a stable random variable, and whose moments, properly normalized, converge to moment type quantities, which are identified in this chapter through the means of mathematical theorems and derivations.
@incollection {key1661495m,
AUTHOR = {Hu, Yueyun and Yor, Marc},
TITLE = {Convergence in law and convergence of
moments: {A}n example related to {B}essel
processes},
BOOKTITLE = {Asymptotic methods in probability and
statistics},
EDITOR = {Szyszkowicz, Barbara},
PUBLISHER = {North-Holland/Elsevier},
ADDRESS = {Amsterdam},
YEAR = {1998},
PAGES = {387--397},
DOI = {10.1016/B978-044450083-0/50027-7},
NOTE = {(Ottawa, ON, 8--13 July 1997). A volume
in honour of Mikl\'os Cs\"org\H{o}.
MR:1661495. Zbl:0933.60021.},
ISBN = {9780444500830},
}
M. Csörgő, Z. Shi, and M. Yor :
“Some asymptotic properties of the local time of the uniform empirical process Bernoulli
5 : 6
(1999 ),
pp. 1035–1058 .
MR
1735784 Zbl
0960.60023 article
Abstract
People
BibTeX
We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for the integral of the square of the local time. It is interesting to note that the corresponding problems for the Wiener process remain open. Properties of \( L^p \) -norms of the local time are studied. We also characterize the joint asymptotics of the local time at a fixed level and the maximum local time.
@article {key1735784m,
AUTHOR = {Cs\"org\H{o}, Mikl\'os and Shi, Zhan
and Yor, Marc},
TITLE = {Some asymptotic properties of the local
time of the uniform empirical process},
JOURNAL = {Bernoulli},
FJOURNAL = {Bernoulli. Official Journal of the Bernoulli
Society for Mathematical Statistics
and Probability},
VOLUME = {5},
NUMBER = {6},
YEAR = {1999},
PAGES = {1035--1058},
DOI = {10.2307/3318559},
NOTE = {MR:1735784. Zbl:0960.60023.},
ISSN = {1350-7265},
}
E. Csáki, Z. Shi, and M. Yor :
“Fractional Brownian motions as ‘higher-order’ fractional derivatives of Brownian local times ,”
pp. 365–387
in
Limit theorems in probability and statistics: Fourth Hungarian colloquium on limit theorems in probability and statistics
(Balatonlelle, Hungary, 28 June–2 July 1999 ),
vol. 1 .
Edited by I. Berkes, E. Csáki, and M. Csörgő .
János Bolyai Mathematical Society (Budapest ),
2002 .
Dedicated to Pál Révész on the occasion of his 65th birthday.
MR
1979974 Zbl
1030.60073 incollection
Abstract
People
BibTeX
Fractional derivatives \( \mathcal{D}^{\gamma} \) of Brownian local times are well defined for all \( \gamma \lt 3/2 \) . We show that, in the weak convergence sense, these fractional derivatives admit themselves derivatives which feature all fractional Brownian motions. Strong approximation results are also developed as counterparts of limit theorems for Brownian additive functionals which feature the fractional derivatives of Brownian local times.
@incollection {key1979974m,
AUTHOR = {Cs\'aki, E. and Shi, Z. and Yor, M.},
TITLE = {Fractional {B}rownian motions as ``higher-order''
fractional derivatives of {B}rownian
local times},
BOOKTITLE = {Limit theorems in probability and statistics:
{F}ourth {H}ungarian colloquium on limit
theorems in probability and statistics},
EDITOR = {Berkes, I. and Cs\'aki, E. and Cs\"org\H{o},
M.},
VOLUME = {1},
PUBLISHER = {J\'anos Bolyai Mathematical Society},
ADDRESS = {Budapest},
YEAR = {2002},
PAGES = {365--387},
NOTE = {(Balatonlelle, Hungary, 28 June--2 July
1999). Dedicated to P\'al R\'ev\'esz
on the occasion of his 65th birthday.
MR:1979974. Zbl:1030.60073.},
ISBN = {9639453013},
}
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
Abstract
People
BibTeX
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.
@incollection {key2106848m,
AUTHOR = {Peccati, Giovanni and Yor, Marc},
TITLE = {Hardy's inequality in \$L^2([0,1])\$ and
principal values of {B}rownian local
times},
BOOKTITLE = {Asymptotic methods in stochastics: {F}estschrift
for {M}ikl\'os Cs\"org\H{o}},
EDITOR = {Horv\'arth, Lajos and Szyszkowicz, Barbara},
SERIES = {Fields Institute Communications},
NUMBER = {44},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {49--74},
NOTE = {(Ottawa, 23--25 May 2002). MR:2106848.
Zbl:1074.60029.},
ISSN = {1069-5265},
ISBN = {9780821835616},
}
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
Abstract
People
BibTeX
We generalize and give new proofs of four limit theorems for quadratic functionals of Brownian motion and Brownian bridge, recently obtained by Deheuvels and Martynov [2003] by means of Karhunen–Loéve expansions. Our techniques involve basic tools of stochastic calculus as well as classical theorems about weak convergence of Brownian functionals. We establish explicit connections with occupation times of Bessel processes, Poincaré’s Lemma and the class of quadratic functionals of Brownian local times studied in [Peccati and Yor 2001].
@incollection {key2106849m,
AUTHOR = {Peccati, Giovanni and Yor, Marc},
TITLE = {Four limit theorems for quadratic functionals
of {B}rownian motion and {B}rownian
bridge},
BOOKTITLE = {Asymptotic methods in stochastics: {F}estschrift
for {M}ikl\'os Cs\"org\H{o}},
EDITOR = {Horv\'arth, Lajos and Szyszkowicz, Barbara},
SERIES = {Fields Institute Communications},
NUMBER = {44},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {75--87},
NOTE = {(Ottawa, 23--25 May 2002). MR:2106849.
Zbl:1071.60017.},
ISSN = {1069-5265},
ISBN = {9780821835616},
}
F. Hirsch, B. Roynette, and M. Yor :
“Kellerer’s theorem revisited 347–363
in
Asymptotic laws and methods in stochastics: A volume in honour of Miklós Csörgő on the occasion of his 80th birthday
(Ottawa, 3–6 July 2012 ).
Edited by D. Dawson, R. Kulik, M. Ould Haye, B. Szyszkowicz, and Y. Zhao .
Fields Institute Communications 76 .
Fields Institute (Toronto ),
2015 .
MR
3409839 Zbl
1368.60045 incollection
Abstract
People
BibTeX
Kellerer’s theorem asserts the existence of a Markov martingale with given marginals, assumed to increase in the convex order. It is revisited here, in the light of previous papers by Hirsch–Roynette and by G. Lowther.
@incollection {key3409839m,
AUTHOR = {Hirsch, Francis and Roynette, Bernard
and Yor, Marc},
TITLE = {Kellerer's theorem revisited},
BOOKTITLE = {Asymptotic laws and methods in stochastics:
{A} volume in honour of {M}ikl\'os {C}s\"org\H{o}
on the occasion of his 80th birthday},
EDITOR = {Dawson, D. and Kulik, R. and Ould Haye,
M. and Szyszkowicz, B. and Zhao, Y.},
SERIES = {Fields Institute Communications},
NUMBER = {76},
PUBLISHER = {Fields Institute},
ADDRESS = {Toronto},
YEAR = {2015},
PAGES = {347--363},
DOI = {10.1007/978-1-4939-3076-0_18},
NOTE = {(Ottawa, 3--6 July 2012). MR:3409839.
Zbl:1368.60045.},
ISSN = {1069-5265},
ISBN = {9781493930753},
}
