III: 14, 175-189, LNM 88 (1969)
MEYER, Paul-André
Processus à accroissements indépendants et positifs (
Markov processes,
Independent increments)
This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift
Comment: The conjecture was proved by Kesten (see
503) who actually knew of the problem through this talk. See also
502Keywords: Subordinators,
Polar setsNature: Exposition
Retrieve article from Numdam
III: 15, 190-229, LNM 88 (1969)
MORANDO, Philippe
Mesures aléatoires (
Independent increments)
This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (
Acta Math. Acad. Sci. Hung., 7, 1956 and
8, 1957) and Kingman (
Pacific J. Math., 21, 1967)
Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,
Probabilités et potentiel, Chapter XIII, end of \S4
Keywords: Random measures,
Independent incrementsNature: Exposition,
Original additions Retrieve article from Numdam
V: 02, 17-20, LNM 191 (1971)
ASSOUAD, Patrice
Démonstration de la ``Conjecture de Chung'' par Carleson (
Markov processes,
Independent increments)
Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it
Comment: See Chung,
C. R. Acad. Sci. ,
260, 1965, p.4665. For the statement of the problem see Meyer
314. For Kesten's earlier (contrary to a statement in the paper!) probabilistic proof see Bretagnolle
503. See also
Séminaire Bourbaki 21th year,
361, June 1969
Keywords: Subordinators,
Polar setsNature: Exposition Retrieve article from Numdam
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: 07, 37-77, LNM 381 (1974)
DUPUIS, Claire
Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (
Independent increments)
The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (
J. Math. Mech.,
16, 1967) whose methods are generalized here
Keywords: Hausdorff measures,
Lévy processesNature: Original Retrieve article from Numdam
XIII: 10, 132-137, LNM 721 (1979)
SIDIBÉ, Ramatoulaye
Martingales locales à accroissements indépendants (
Martingale theory,
Independent increments)
It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale
Comment: The case of non-stationary increments is considered in
1544. See also the errata sheet of vol. XV
Keywords: Local martingales,
Lévy processesNature: Original Retrieve article from Numdam
XV: 43, 627-631, LNM 850 (1981)
WANG, Jia-Gang
Some remarks on processes with independent increments (
Independent increments)
This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also
1544Keywords: Poisson processes,
Lévy measuresNature: Original Retrieve article from Numdam
XV: 44, 632-642, LNM 850 (1981)
SIDIBÉ, Ramatoulaye
Mesures à accroissements indépendants et P.A.I. non homogènes (
Independent increments)
This is an improved version of
1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable
Nature: New proof of known results Retrieve article from Numdam
