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VI: 01, 1-34, LNM 258 (1972)
ARTZNER, Philippe
Echantillons et couples indépendants de points aléatoires portés par une surface convexe (Independence)
The general problem is to find conditions under which a convolution equation $\mu{*}\mu=\mu'{*}\mu''$ in $R^n$ implies that $\mu'=\mu''=\mu$. It is shown here that the result is true if the measures are carried by a convex surface which is not too flat
Comment: The results were announced in the note C. R. Acad. Sci., 272, 1971
Keywords: Decomposition of laws
Nature: Original
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VI: 02, 35-50, LNM 258 (1972)
AZÉMA, Jacques
Une remarque sur les temps de retour. Trois applications (Markov processes, General theory of processes)
This paper is the first step in the investigations of Azéma on the dual'' form of the general theory of processes (for which see Azéma (Ann. Sci. ENS, 6, 1973, and 814). Here the $\sigma$-fields of cooptional and coprevisible sets are introduced in a Markovian set-up, and without their definitive names. A section theorem by return times is proved, and applications to the theory of Markov processes are given
Keywords: Homogeneous processes, Coprevisible processes, Cooptional processes, Section theorems, Projection theorems, Time reversal
Nature: Original
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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 functions
Nature: Original
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VI: 04, 72-89, LNM 258 (1972)
CHATTERJI, Shrishti Dhav
Un principe de sous-suites dans la théorie des probabilités (Measure theory)
This paper is devoted to results of the following kind: any sequence of random variables with a given weak property contains a subsequence which satisfies a stronger property. An example is due to Komlós: any sequence bounded in $L^1$ contains a subsequence which converges a.s. in the Cesaro sense. Several results of this kind, mostly due to the author, are presented without detailed proofs
Comment: See 1302 for extensions to the case of Banach space valued random variables. See also Aldous, Zeit. für W-theorie, 40, 1977
Keywords: Subsequences, Central limit theorem, Law of the iterated logarithm
Nature: Exposition
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VI: 05, 90-97, LNM 258 (1972)
CHUNG, Kai Lai
On universal field equations (General theory of processes)
There is a pun in the title, since field'' here is a $\sigma$-field and not a quantum field. The author proves useful results on the $\sigma$-fields ${\cal F}_{T-}$ and ${\cal F}_{T+}$ associated with an arbitrary random variable $T$ in the paper of Chung-Doob, Amer. J. Math., 87, 1965. As a corollary, he can prove easily that for a Hunt process, accessible = previsible
Keywords: Filtrations
Nature: Original
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VI: 06, 98-100, LNM 258 (1972)
KAZAMAKI, Norihiko
Examples on local martingales (Martingale theory)
Two simple examples are given, the first one concerning the filtration generated by an exponential stopping time, the second one showing that local martingales are not preserved under time changes (Kazamaki, Zeit. für W-theorie, 22, 1972)
Keywords: Changes of time, Local martingales, Weak martingales
Nature: Original
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VI: 07, 101-104, LNM 258 (1972)
KAZAMAKI, Norihiko
Krickeberg's decomposition for local martingales (Martingale theory)
It is shown that a local martingale bounded in $L^1$ is a difference of two (minimal) positive local martingales
Keywords: Local martingales, Krickeberg decomposition
Nature: Original
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VI: 08, 105-108, LNM 258 (1972)
KAZAMAKI, Norihiko
Note on a stochastic integral equation (Stochastic calculus)
Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see 1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process
Comment: The author developed the subject further in Tôhoku Math. J. 26, 1974
Keywords: Stochastic differential equations
Nature: Original
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VI: 09, 109-112, LNM 258 (1972)
SAM LAZARO, José de; MEYER, Paul-André
Un gros processus de Markov. Application à certains flots (Markov processes)
In a vague but useful sense, a big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (Trans. Amer. Math. Soc., 131, 1968)
Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer, Probabilités et Potentiel, \no XIV.45
Keywords: Prediction theory, Filtered flows
Nature: Original
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VI: 10, 113-117, LNM 258 (1972)
MAISONNEUVE, Bernard
Topologies du type de Skorohod (General theory of processes)
This paper presents an adaptation of the well known Skorohod topology, to the case of an arbitrary (i.e., non-compact) interval of the line
Keywords: Skorohod topology
Nature: Original
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VI: 11, 118-129, LNM 258 (1972)
MEYER, Paul-André
La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)
The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer, Zeit für X-theorie, 21, 1970
Keywords: Supermartingales, Föllmer measures
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VI: 12, 130-150, LNM 258 (1972)
MEYER, Paul-André
Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)
The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (Invent. Math., 14, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group
Comment: A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Filling scheme, Balayage of measures, Skorohod imbedding
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VI: 13, 151-158, LNM 258 (1972)
MEYER, Paul-André
Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)
The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (Acta Math., 124, 1970), Burkholder-Davis-Gundy (Proc. 6-th Berkeley Symposium, 3, 1972). The technique of proof is now obsolete
Keywords: Burkholder inequalities, Moderate convex functions
Nature: Exposition
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VI: 14, 159-163, LNM 258 (1972)
MEYER, Paul-André
Temps d'arrêt algébriquement prévisibles (General theory of processes)
The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$
Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers
Keywords: Stopping times, Previsible processes
Nature: Original
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VI: 15, 164-167, LNM 258 (1972)
MEYER, Paul-André
Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)
The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set
Keywords: Réduite, Fine topology, Absolute continuity hypothesis
Nature: Original
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VI: 16, 168-172, LNM 258 (1972)
MEYER, Paul-André; WALSH, John B.
Un résultat sur les résolvantes de Ray (Markov processes)
This is a complement to the authors' paper on Ray processes in Invent. Math., 14, 1971: a lemma is proved on the existence of many martingales which are continuous whenever the process is continuous (a wrong reference for it was given in the paper). Then it is shown that the mapping $x\rightarrow P_x$ is continuous in the weak topology of measures, when the path space is given the topology of convergence in measure. Note that a correction is mentioned on the errata page of vol. VII
Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng Ann. Inst. Henri Poincaré 20, 1984
Keywords: Ray compactification, Weak convergence of measures
Nature: Original
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VI: 17, 173-176, LNM 258 (1972)
MOKOBODZKI, Gabriel
Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (Potential theory)
The last four pages of this paper have been omitted by mistake, and appear in the following volume as 729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in Séminaire Bourbaki, 1969-70, 377) are quickly reviewed first, and then applied to the following problem concerning the potential kernel $V$ of a resolvent: given a pair of measures $\lambda\le\mu$ in the sense of balayage, then we have $\lambda V\le \mu V$ in the ordinary sense. The corresponding density (dominated by $1$) does not depend on the resolvent, but only on the potential cones of excessive functions and potentials associated with it, and a way to compute it is indicated
Keywords: Cones of potentials
Nature: Original
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VI: 18, 177-197, LNM 258 (1972)
NAGASAWA, Masao
Branching property of Markov processes (Markov processes)
To be completed
Keywords: Branching processes
Nature: Original
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VI: 19, 198-201, LNM 258 (1972)
RAO, Murali
Doob's decomposition and Burkholder's inequalities (Martingale theory)
The Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (Ann. Math. Stat., 37, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale
Comment: This little-known paper would probably deserve a modern translation in continuous time
Keywords: Burkholder inequalities, Decomposition of supermartingales
Nature: Original
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VI: 20, 202-214, LNM 258 (1972)
REVUZ, Daniel
Le principe semi-complet du maximum (Potential theory)
The problem studied here (and not completely solved) consists in finding potential theoretic characterizations for the recurrent potential operators constructed in the basic paper of Neveu, Ann. Inst. Fourier, 22-2, 1972. It is shown that these operators satisfy suitable maximum principles (as usual, slightly stronger in the discrete case than in the continuous case). The converse is delicate and some earlier work of Kondo (Osaka J. Math., 4, 1967) and Oshima (same journal, 6, 1969) is discussed in this new set-up
Comment: This topic is discussed again in Revuz' book Markov Chains, North-Holland
Keywords: Recurrent potential theory, Maximum principles, Recurrent Markov chains
Nature: Original
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VI: 21, 215-232, LNM 258 (1972)
WALSH, John B.
Transition functions of Markov processes (Markov processes)
Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points
Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()
Keywords: Transition functions, Strong Markov property, Moderate Markov property, Ray compactification
Nature: Original
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VI: 22, 233-242, LNM 258 (1972)
WALSH, John B.
The perfection of multiplicative functionals (Markov processes)
In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure
Keywords: Multiplicative functionals, Perfection, Essential topology
Nature: Original
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VI: 23, 243-252, LNM 258 (1972)
MEYER, Paul-André
Quelques autres applications de la méthode de Walsh (La perfection en probabilités'') (Markov processes)
This is but an exercise on using the method of the preceding paper 622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc
Comment: A correction to this paper is mentioned on the errata list of vol. VII
Keywords: Additive functionals, Return times, Essential topology
Nature: Original
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