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 integralsNature: Exposition
Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
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-linesNature: Original Retrieve article from Numdam
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 flowsNature: Original Retrieve article from Numdam
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 measuresNature: Exposition,
Original additions Retrieve article from Numdam
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 lawsNature: Original Retrieve article from Numdam
XXX: 07, 100-103, LNM 1626 (1996)
SAM LAZARO, José de
Un contre-exemple touchant à l'indépendance Retrieve article from Numdam
