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II: 07, 123-139, LNM 51 (1968)

**SAM LAZARO, José de**

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,*Th. Prob. Appl.*, **5**, 1960; Sinai, *Th. Prob. Appl.*, **8**, 1963) is the definition of multiple stochastic integrals with respect to a second order process whose covariance satisfies suitable spectral properties

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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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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V: 27, 278-282, LNM 191 (1971)

**SAM LAZARO, José de**; **MEYER, Paul-André**

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Brownian motion, Wiener chaos, Screw-lines

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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IX: 01, 2-96, LNM 465 (1975)

**MEYER, Paul-André**; **SAM LAZARO, José de**

Questions de théorie des flots (7 chapters) (Ergodic theory)

This is part of a seminar given in the year 1972/73. A flow is meant to be a one-parameter group $(\theta_t)$ of 1--1 measure preserving transformations of a probability space. The main topic of this seminar is the theory of filtered flows, i.e., a filtration $({\cal F}_t)$ ($t\!\in\!**R**$) is given such that $\theta_s ^{-1}{\cal F}_t={\cal F}_{s+t}$, and particularly the study of *helixes,* which are real valued processes $(Z_t)$ ($t\!\in\!**R**$) such that $Z_0=0$, which for $t\ge0$ are adapted, and on the whole line have homogeneous increments ($Z_{s+t}-Z_t=Z_t\circ \theta_s$). Two main classes of helixes are considered, the increasing helixes, and the martingale helixes. Finally, a filtered flow such that ${\cal F}_{-\infty}$ is degenerate is called a K-flow (K for Kolmogorov). Chapter~1 gives these definitions and their simplest consequences, as well as the definition of (continuous time) point processes, and the Ambrose construction of (unfiltered) flows from discrete flows as *flows under a function.* Chapter II shows that homogeneous discrete point processes and flows under a function are two names for the same object (Hanen, *Ann. Inst. H. Poincaré,* **7**, 1971), leading to the definition of the Palm measure of a discrete point process, and proves the classical (Ambrose-Kakutani) result that every flow with reasonable ergodicity properties can be interpreted as a flow under a function. A discussion of the case of filtered flows follows, with incomplete results. Chapter III is devoted to examples of flows and K-flows (Totoki's theorem). Chapter IV contains the study of increasing helixes, their Palm measures, and changes of times on flows. Chapter V is the original part of the seminar, devoted to the (square integrable) martingale helixes, their brackets, and the fact that in every K-flow these martingale helixes generate all martingales by stochastic integration. The main tool to prove this is a remark that every filtered K-flow can be interpreted (in a somewhat loose sense) as the flow of a stationary Markov process, helixes then becoming additive functionals, and standard Markovian methods becoming applicable. Chapter VI is devoted to spectral multiplicity, the main result being that a filtered flow, whenever it possesses one martingale helix, possesses infinitely many orthogonal helixes (orthogonal in a weak sense, not as martingales). Chapter VII is devoted to an independent topic: approximation in law of any ergodic stationary process by functionals of the Brownian flow (Nisio's theorem)

Comment: This set of lectures should be completed by the paper of Benveniste 902 which follows it, by an (earlier) paper by Sam Lazaro-Meyer (*Zeit. für W-theorie,* **18**, 1971) and a (later) paper by Sam Lazaro (*Zeit. für W-theorie,* **30**, 1974). Some of the results presented were less original than the authors believed at the time of the seminar, and due acknowledgments of priority are given; for an additional one see 1031. Related papers are due to Geman-Horowitz (*Ann. Inst. H. Poincaré,* **9**, 1973). The theory of filtered flows and Palm measures had a striking illustration within the theory of Markov processes as Kuznetsov measures (Kuznetsov, *Th. Prob. Appl.*, **18**, 1974) and the interpretation of ``Hunt quasi-processes'' as their Palm measures (Fitzsimmons, *Sem. Stoch. Processes 1987*, 1988)

Keywords: Filtered flows, Kolmogorov flow, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures

Nature: Exposition, Original additions

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XII: 21, 265-309, LNM 649 (1978)

**YOR, Marc**; **SAM LAZARO, José de**

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (*Michigan Math. J.* 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in $H^1$. The main result can be stated as follows: given a law $P\in{\cal P}$, the set ${\cal N}$ has the previsible representation property, i.e., ${\cal F}_0$ is trivial and stochastic integrals with respect to elements of ${\cal N}$ are dense in $H^1$, if and only if $P$ is an extreme point of ${\cal P}$. Many examples and applications are given

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XXX: 07, 100-103, LNM 1626 (1996)

**SAM LAZARO, José de**

Un contre-exemple touchant à l'indépendance

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Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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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

V: 27, 278-282, LNM 191 (1971)

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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VI: 09, 109-112, LNM 258 (1972)

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 (

Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer,

Keywords: Prediction theory, Filtered flows

Nature: Original

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IX: 01, 2-96, LNM 465 (1975)

Questions de théorie des flots (7 chapters) (Ergodic theory)

This is part of a seminar given in the year 1972/73. A flow is meant to be a one-parameter group $(\theta_t)$ of 1--1 measure preserving transformations of a probability space. The main topic of this seminar is the theory of filtered flows, i.e., a filtration $({\cal F}_t)$ ($t\!\in\!

Comment: This set of lectures should be completed by the paper of Benveniste 902 which follows it, by an (earlier) paper by Sam Lazaro-Meyer (

Keywords: Filtered flows, Kolmogorov flow, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures

Nature: Exposition, Original additions

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XII: 21, 265-309, LNM 649 (1978)

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XXX: 07, 100-103, LNM 1626 (1996)

Un contre-exemple touchant à l'indépendance

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