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 measuresNature: Original
Retrieve article from Numdam
VII: 19, 198-204, LNM 321 (1973)
MEYER, Paul-André
Limites médiales d'après Mokobodzki (
Measure theory,
Functional analysis)
Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see
304). The same problem is now solved for weak convergences, using a special kind of Banach limits
Comment: The paper contains a few annoying misprints, in particular p.199 line 9
s.c;s. should be deleted and line 17
atomique should be
absolument continu. For a misprint-free version see Dellacherie-Meyer,
Probabiliés et Potentiel, Volume C, Chapter X,
55--57
Keywords: Continuum axiom,
Weak convergence of r.v.'s,
Medial limitNature: Exposition Retrieve article from Numdam
XI: 10, 109-119, LNM 581 (1977)
MEYER, Paul-André
Convergence faible de processus, d'après Mokobodzki (
General theory of processes)
The following simple question of Benveniste was answered positively by Mokobodzki (
Séminaire de Théorie du Potentiel, Lect. Notes in M. 563, 1976): given a sequence $(U^n)$ of optional processes such that $U^n_T$ converges weakly in $L^1$ for every stopping time $T$, does there exist an optional process $U$ such that $U^n_T$ converges to $U_T$? The proof is rather elaborate
Keywords: Weak convergence in $L^1$Nature: Exposition Retrieve article from Numdam
XVI: 27, 314-318, LNM 920 (1982)
LENGLART, Érik
Sur le théorème de la convergence dominée (
General theory of processes,
Stochastic calculus)
Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see
1110Keywords: Stopping times,
Optional processes,
Weak convergence,
Stochastic integralsNature: Original Retrieve article from Numdam
