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IV: 01, 1-27, LNM 124 (1970)

**CAIROLI, Renzo**

Une inégalité pour martingales à indices multiples et ses applications (Several parameter processes)

This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions

Comment: The next landmark in the theory is Cairoli-Walsh,*Acta. Math.*, **134**, 1975. For the modern results, see Imkeller, *Two Parameter Processes and their Quadratic Variation,* Lect. Notes in M. **1308**, 1989

Keywords: Two-parameter martingales, Maximal inequality, Almost sure convergence

Nature: Original

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IV: 02, 28-36, LNM 124 (1970)

**CARTIER, Pierre**

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,*Problèmes combinatoires de commutation et réarrangements,* LN **85**, 1969

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

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IV: 03, 37-46, LNM 124 (1970)

**CHERSI, Franco**

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (*Studia Math.*, **33**, 1968) is a converse to Doob's inequality: for a positive martingale such that $X_n\leq cX_{n-1}$, the integrability of $\sup_n X_n$ implies boundedness in $L\log^+L$. All martingales satisfy this condition on regular filtrations

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

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IV: 04, 47-59, LNM 124 (1970)

**DACUNHA-CASTELLE, Didier**

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

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IV: 05, 60-70, LNM 124 (1970)

**DELLACHERIE, Claude**

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,*Probabilités et Potentiel,* Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer 906

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

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IV: 06, 71-72, LNM 124 (1970)

**DELLACHERIE, Claude**

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump*from * a semi-polar set; at the first hitting time of any finely closed set $F$, either the process does not jump, or it jumps from outside $F$

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

Nature: Original

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IV: 07, 73-75, LNM 124 (1970)

**DELLACHERIE, Claude**

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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IV: 08, 76-76, LNM 124 (1970)

**DELLACHERIE, Claude**

Un lemme de théorie de la mesure (Measure theory)

A lemma used by Erdös, Kesterman and Rogers (*Coll. Math.,* **XI**, 1963) is reduced to the fact that a sequence of bounded r.v.'s contains a weakly convergent subsequence

Keywords: Convergence in norm, Subsequences

Nature: Original proofs

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IV: 09, 77-107, LNM 124 (1970)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Intégrales stochastiques par rapport aux martingales locales (Martingale theory, Stochastic calculus)

This is a continuation of Meyer 106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in 106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in 106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality

Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

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IV: 10, 108-131, LNM 124 (1970)

**FUCHS, Aimé**; **LETTA, Giorgio**

L'inégalité de Kullback. Application à la théorie de l'estimation (Information theory, Mathematical statistics)

This paper brings together, and shows the mutual relation of a number of classical definitions, centering on Shannon's mutual information of two probability measures, $G(\lambda|\mu)=\int \log {d\lambda\over d\mu}\,d\mu$

Comment: To be asked from the authors

Keywords: Kullback inequality, Cramer-Rao inequality, Sufficient statistics

Nature: Original

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IV: 11, 132-132, LNM 124 (1970)

**SAM LAZARO, José de**

Théorème de Stone et espérances conditionnelles (Ergodic theory)

It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations

Comment: This remark arose from the work on flows in Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Flows, Spectral representation

Nature: Original

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IV: 12, 133-150, LNM 124 (1970)

**MEYER, Paul-André**

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance*Z. für W-theorie,* **2**, 1964), under the very restrictive assumption that the ``event'' has a non-zero probability to occur at fixed times. The general theory is due to Krylov and Yushkevich (*Trans. Moscow Math. Soc.*, **13**, 1965), a deep paper difficult to read and to apply in concrete cases. Hoffmann-Jørgensen (*Math. Scand.*, **24**, 1969) developed the theory under simple and efficient axioms. It is shown that a regenerative set defined axiomatically is the same thing as the set of returns of a strong Markov process to a fixed state, or the range of a subordinator

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (*Th. Prob. Appl.*, **16**, 1971) and Maisonneuve, *Systèmes Régénératifs,* *Astérisque * **15**, 1974

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

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IV: 13, 151-161, LNM 124 (1970)

**MAISONNEUVE, Bernard**; **MORANDO, Philippe**

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

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IV: 14, 162-169, LNM 124 (1970)

**MEYER, Paul-André**

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in*Ann. Math. Stat.*, **36**, 1965, and the inequalities are extensions to martingales of the Borel-Cantelli lemma and the strong law of large numbers. For martingales with bounded jumps, exponential bounds are given (Neveu, *Martingales à temps discret,* gives a better one)

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

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IV: 15, 170-194, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in*Séminaire Bourbaki,* **422**, November 1972. See also Chapter XII of Dellacherie-Meyer, *Probabilités et potentiel,* containing new proofs due to Feyel

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 16, 195-207, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 17, 208-215, LNM 124 (1970)

**REVUZ, Daniel**

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see*Ann. Inst. Fourier,* **21**, 1971

Keywords: Recurrent potential theory

Nature: Original

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IV: 18, 216-239, LNM 124 (1970)

**WEIL, Michel**

Quasi-processus (Markov processes)

Excessive measures which are not potentials of measures were shown by Hunt (*Ill. J. Math.*, **4**, 1960) to be associated with a probabilistic object which is a kind of projective limit of Markov processes. Hunt's construction was performed in discrete time only, and is difficult in continuous time because of measure theoretic difficulties (the standard theorem on projective limits cannot be applied). Here the construction is done in full detail

Comment: Further work by M.~Weil on the same subject in 532; see the references there

Keywords: Hunt quasi-processes

Nature: Original

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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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Une inégalité pour martingales à indices multiples et ses applications (Several parameter processes)

This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions

Comment: The next landmark in the theory is Cairoli-Walsh,

Keywords: Two-parameter martingales, Maximal inequality, Almost sure convergence

Nature: Original

Retrieve article from Numdam

IV: 02, 28-36, LNM 124 (1970)

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

Retrieve article from Numdam

IV: 03, 37-46, LNM 124 (1970)

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

Retrieve article from Numdam

IV: 04, 47-59, LNM 124 (1970)

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

Retrieve article from Numdam

IV: 05, 60-70, LNM 124 (1970)

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

Retrieve article from Numdam

IV: 06, 71-72, LNM 124 (1970)

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

Nature: Original

Retrieve article from Numdam

IV: 07, 73-75, LNM 124 (1970)

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

Retrieve article from Numdam

IV: 08, 76-76, LNM 124 (1970)

Un lemme de théorie de la mesure (Measure theory)

A lemma used by Erdös, Kesterman and Rogers (

Keywords: Convergence in norm, Subsequences

Nature: Original proofs

Retrieve article from Numdam

IV: 09, 77-107, LNM 124 (1970)

Intégrales stochastiques par rapport aux martingales locales (Martingale theory, Stochastic calculus)

This is a continuation of Meyer 106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in 106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in 106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality

Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

Retrieve article from Numdam

IV: 10, 108-131, LNM 124 (1970)

L'inégalité de Kullback. Application à la théorie de l'estimation (Information theory, Mathematical statistics)

This paper brings together, and shows the mutual relation of a number of classical definitions, centering on Shannon's mutual information of two probability measures, $G(\lambda|\mu)=\int \log {d\lambda\over d\mu}\,d\mu$

Comment: To be asked from the authors

Keywords: Kullback inequality, Cramer-Rao inequality, Sufficient statistics

Nature: Original

Retrieve article from Numdam

IV: 11, 132-132, LNM 124 (1970)

Théorème de Stone et espérances conditionnelles (Ergodic theory)

It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations

Comment: This remark arose from the work on flows in Sam Lazaro-Meyer,

Keywords: Flows, Spectral representation

Nature: Original

Retrieve article from Numdam

IV: 12, 133-150, LNM 124 (1970)

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

Retrieve article from Numdam

IV: 13, 151-161, LNM 124 (1970)

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

Retrieve article from Numdam

IV: 14, 162-169, LNM 124 (1970)

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

Retrieve article from Numdam

IV: 15, 170-194, LNM 124 (1970)

Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

Retrieve article from Numdam

IV: 16, 195-207, LNM 124 (1970)

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

Retrieve article from Numdam

IV: 17, 208-215, LNM 124 (1970)

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see

Keywords: Recurrent potential theory

Nature: Original

Retrieve article from Numdam

IV: 18, 216-239, LNM 124 (1970)

Quasi-processus (Markov processes)

Excessive measures which are not potentials of measures were shown by Hunt (

Comment: Further work by M.~Weil on the same subject in 532; see the references there

Keywords: Hunt quasi-processes

Nature: Original

Retrieve article from Numdam

IV: 19, 240-282, LNM 124 (1970)

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

Retrieve article from Numdam