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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 maps

Nature: Exposition

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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 201

Keywords: Recurrent potential theory, Invariant measures

Nature: Original

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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 functions

Nature: Original

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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 623

Keywords: Additive functionals, Perfection

Nature: Original

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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 sections

Nature: Original

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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 sets

Nature: Original

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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 logarithm

Nature: Original

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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 convergence

Nature: Well-known

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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. **26**

Keywords: Green potentials, Dual semigroups

Nature: Original

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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 times

Nature: Original

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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 processes

Nature: Original

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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 integrals

Nature: Correction

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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 inequalities

Nature: Exposition

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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 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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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 increments

Nature: Exposition, Original additions

Retrieve article from Numdam

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

Retrieve article from Numdam

III: 02, 24-33, LNM 88 (1969)

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 201

Keywords: Recurrent potential theory, Invariant measures

Nature: Original

Retrieve article from Numdam

III: 03, 34-92, LNM 88 (1969)

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 (

Comment: An early step in the theory of two parameter martingales. See also Cairoli,

Keywords: Dirichlet problem, Biharmonic functions

Nature: Original

Retrieve article from Numdam

III: 04, 93-96, LNM 88 (1969)

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,

Comment: See also 203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

Retrieve article from Numdam

III: 05, 97-114, LNM 88 (1969)

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,

Keywords: Sets with countable sections

Nature: Original

Retrieve article from Numdam

III: 06, 115-136, LNM 88 (1969)

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 (

Comment: See Dellacherie,

Keywords: Sierpinski's ``rabotages'', Semi-polar sets

Nature: Original

Retrieve article from Numdam

III: 07, 137-142, LNM 88 (1969)

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 logarithm

Nature: Original

Retrieve article from Numdam

III: 08, 143-143, LNM 88 (1969)

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,

Keywords: Almost sure convergence

Nature: Well-known

Retrieve article from Numdam

III: 09, 144-151, LNM 88 (1969)

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,

Keywords: Green potentials, Dual semigroups

Nature: Original

Retrieve article from Numdam

III: 10, 152-154, LNM 88 (1969)

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 times

Nature: Original

Retrieve article from Numdam

III: 11, 155-159, LNM 88 (1969)

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,

Comment: This is essentially the definitive proof, using a general section theorem instead of capacity theory

Keywords: Section theorems, Optional processes, Previsible processes

Nature: Original

Retrieve article from Numdam

III: 12, 160-162, LNM 88 (1969)

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 integrals

Nature: Correction

Retrieve article from Numdam

III: 13, 163-174, LNM 88 (1969)

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,

Keywords: Burkholder inequalities

Nature: Exposition

Retrieve article from Numdam

III: 14, 175-189, LNM 88 (1969)

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 502

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

III: 15, 190-229, LNM 88 (1969)

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 (

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,

Keywords: Random measures, Independent increments

Nature: Exposition, Original additions

Retrieve article from Numdam