III: 01, 1-23, LNM 88 (1969)
ARTZNER, Philippe
Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (
Measure theory,
Functional analysis)
Exposition of three notes by L.~Schwartz (
CRAS 265, 1967 and
266, 1968) showing that some classes of maps between spaces $\ell^p$ and $\ell^q$ transform Gaussian cylindrical measures into Radon measures. The result turns out to be an extension of Minlos' theorem
Comment: Self-contained and detailed exposition, possibly still useful
Keywords: Radonifying mapsNature: Exposition
Retrieve article from Numdam
III: 02, 24-33, LNM 88 (1969)
AZÉMA, Jacques;
DUFLO, Marie;
REVUZ, Daniel
Mesure invariante des processus de Markov récurrents (
Markov processes)
A condition similar to the Harris recurrence condition is studied in continuous time. It is shown that it implies the existence (up to a constant factor) of a unique $\sigma$-finite excessive measure, which is invariant. The invariant measure for a time-changed process is described
Comment: This is related to several papers by the same authors on recurrent Markov processes, and in particular to
201Keywords: Recurrent potential theory,
Invariant measuresNature: Original Retrieve article from Numdam
III: 03, 34-92, LNM 88 (1969)
CAIROLI, Renzo
Étude probabiliste d'un problème de Dirichlet (
Several parameter processes)
This paper aims at a better understanding of separately harmonic functions with respect to a product of two Markov processes, which provide one of the main examples of two parameter martingales (the other one being the Brownian sheet). Here the recently published work of J.~Walsh (
Ann. Inst. Fourier, 18, 1968) on the Dirichlet problem for biharmonic functions was discussed and reinterpreted
Comment: An early step in the theory of two parameter martingales. See also Cairoli,
Publ. Inst. Stat. Univ. Paris, 15, 1966
Keywords: Dirichlet problem,
Biharmonic functionsNature: Original Retrieve article from Numdam
III: 04, 93-96, LNM 88 (1969)
DELLACHERIE, Claude
Une application aux fonctionnelles additives d'un théorème de Mokobodzki (
Markov processes)
Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,
Probabilité et potentiels, Chap. II,
27). They are used here to prove that every continuous additive functional of a Markov process has a ``perfect'' version
Comment: See also
203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
III: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (
Descriptive set theory,
Markov processes,
General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
III: 06, 115-136, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires II (
Descriptive set theory,
Markov processes)
Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (
Fund. Math.,
5, 1924) can be extended to an abstract set-up to show that a non-semi-polar Borel set contains a non-semi-polar compact set
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972. More recent proofs no longer depend on ``rabotages'': Dellacherie-Meyer,
Probabilités et potentiel, Appendix to Chapter IV
Keywords: Sierpinski's ``rabotages'',
Semi-polar setsNature: Original Retrieve article from Numdam
III: 07, 137-142, LNM 88 (1969)
HUBER, Catherine
Un aspect de la loi du logarithme itéré pour des variables aléatoires indépendantes et équidistribuées (
Independence)
A refinement of the classical law of the iterated logarithm for i.i.d. random variables, under regularity assumptions on the tail of the common distribution
Keywords: Law of the iterated logarithmNature: Original Retrieve article from Numdam
III: 08, 143-143, LNM 88 (1969)
MEYER, Paul-André
Un lemme de théorie des martingales (
Martingale theory)
The author apparently believed that this classical and useful remark was new (it is often called ``Hunt's lemma'', see Hunt,
Martingales et Processus de Markov, Masson 1966, p.47)
Keywords: Almost sure convergenceNature: Well-known Retrieve article from Numdam
III: 09, 144-151, LNM 88 (1969)
MEYER, Paul-André
Un résultat de théorie du potentiel (
Potential theory,
Markov processes)
Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded
Comment: See Meyer,
Processus de Markov, Lecture Notes in M.
26Keywords: Green potentials,
Dual semigroupsNature: Original Retrieve article from Numdam
III: 10, 152-154, LNM 88 (1969)
MEYER, Paul-André
Un résultat élémentaire sur les temps d'arrêt (
General theory of processes)
This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times
Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront
Keywords: Stopping times,
Accessible times,
Previsible timesNature: Original Retrieve article from Numdam
III: 11, 155-159, LNM 88 (1969)
MEYER, Paul-André
Une nouvelle démonstration des théorèmes de section (
General theory of processes)
The proof of the section theorems has improved over the years, from complicated-false to complicated-true, and finally to easy-true. This was a step on the way, due to Dellacherie (inspired by Cornea-Licea,
Z. für W-theorie, 10, 1968)
Comment: This is essentially the definitive proof, using a general section theorem instead of capacity theory
Keywords: Section theorems,
Optional processes,
Previsible processesNature: Original Retrieve article from Numdam
III: 12, 160-162, LNM 88 (1969)
MEYER, Paul-André
Rectification à des exposés antérieurs (
Markov processes,
Martingale theory)
Corrections are given to the talk
202 by Cartier, Meyer and Weil and to the talk
106 by Meyer
Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov
Keywords: Time reversal,
Stochastic integralsNature: Correction Retrieve article from Numdam
III: 13, 163-174, LNM 88 (1969)
MEYER, Paul-André
Les inégalités de Burkholder en théorie des martingales, d'après Gundy (
Martingale theory)
A proof of the famous Burkholder inequalities in discrete time, from Gundy,
Ann. Math. Stat.,
39, 1968
Keywords: Burkholder inequalitiesNature: Exposition Retrieve article from Numdam
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
