V: 03, 21-36, LNM 191 (1971)
BRETAGNOLLE, Jean
Résultats de Kesten sur les processus à accroissements indépendants (
Markov processes,
Independent increments)
The question is to find all Lévy processes for which single points are polar. Kesten's answer (
Mem. Amer. Math. Soc.,
93, 1969) is almost complete and in particular proves Chung's conjecture. The proofs in this paper have been considerably reworked
Comment: See also
502 in the same volume
Keywords: Subordinators,
Polar setsNature: Exposition,
Original additions
Retrieve article from Numdam
VI: 03, 51-71, LNM 258 (1972)
BRETAGNOLLE, Jean
$p$-variation de fonctions aléatoires~: 1. Séries de Rademacher 2. Processus à accoissements indépendants (
Independent increments)
The main result of the paper is theorem III, which gives a necessary and sufficient condition for the sample paths of a centered Lévy process to have a.s. a finite $p$-variation on finite time intervals, for $1<p<2$: the process should have no Gaussian part, and $|x|^p$ be integrable near $0$ w.r.t. the Lévy measure $L(dx)$. The proof rests on discrete estimates on the $p$-variation of Rademacher series. Additional results on $h$-variation w.r.t. more general convex functions are given or mentioned
Comment: This paper improves on Millar,
Zeit. für W-theorie, 17, 1971
Keywords: $p$-variation,
Rademacher functionsNature: Original Retrieve article from Numdam
VIII: 02, 11-19, LNM 381 (1974)
BRETAGNOLLE, Jean
Une remarque sur le problème de Skorohod (
Brownian motion)
The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see
516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XII: 23, 332-341, LNM 649 (1978)
BRETAGNOLLE, Jean;
HUBER, Catherine
Lois empiriques et distance de Prokhorov (
Mathematical statistics)
Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones
Keywords: Empirical distribution function,
Prohorov distanceNature: Original Retrieve article from Numdam
XII: 24, 342-363, LNM 649 (1978)
BRETAGNOLLE, Jean;
HUBER, Catherine
Estimation des densités~: risque minimax (
Mathematical statistics)
A sequel to the preceding paper
1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed
Comment: For a correction see
1360. An improved version appeared in (
Zeit. für W-theorie, 47, 1979)
Keywords: Empirical distribution functionNature: Original Retrieve article from Numdam
XIII: 60, 647-647, LNM 721 (1979)
BRETAGNOLLE, Jean;
HUBER, Catherine
Corrections à un exposé antérieur (
Mathematical statistics)
Two misprints and a more substantial error (in the proof of proposition 1) of
1224 are corrected
Comment: A revised version appeared in (
Zeit. für W-Theorie, 47, 1979)
Keywords: Empirical distribution function,
Prohorov distanceNature: Correction Retrieve article from Numdam
