Summary
This work is a continuation of [7], and also considers the more general case of (probability) laws which are not tight. The main results are:
-
1.
The good representation v=e(F) for all tight laws of infinitely divisible Poisson type, in any polish (linear locally convex) space, and a theorem of accompanying law (corollaries 2 and 3 of theorem II).
-
2.
An extension of a theorem of Lee (example following theorem I.2).
-
3.
A formula of P. Levy for any infinitely divisible law, tight following compact sets of hilbertian separable type, in any linear (locally convex) space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- CV:
-
convergence
- E.V.T.:
-
espace vectoriel topologique
- a. c., I. c.:
-
absolument convexe, localement convexe
- U.T.:
-
uniformément tendue (famille de lois ou mesures)
- K:
-
désigne toujours un ensemble compact
- [Μ]:
-
Variation totale de Μ mesure bornée
- ℜ, ℑ:
-
partie réelle, partie imaginaire de
Références
Badrikian, A.: Les éléments aléatoires vectoriels et la théorie de la fonction caractéristique. Thèse à paraÎtre aux annales de l'Institut Henri-Poincaré (1968).
Delporte, J.: Fonctions aléatoires presque sûrement continues sur un intervalle fermé. Ann. Inst. Henri-Poincaré 1–2, 111–215 (1964).
Fernique, X.: Lois indéfiniment divisibles sur l'espace des distributions. Inventiones math. 3, 282–292 (1967).
Lee, P. M.: Infinitely divisible stochastic processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 7, 147–160 (1967).
Parthasarathy, Ranga Rao, and Varadhan: Probability distributions on locally compact abelian groups. Illinois J. Math. 7, 337–369 (1963).
Schwartz, L.: Mesures de Radon sur des espaces topologiques arbitraires. Cours Inst. Henri-Poincaré (1964/65).
Tortrat, A.: Structure des lois indéfiniment divisibles dans un espace vectoriel topologique, in “Symposium on probability methods in analysis”. Berlin-Heidelberg-New York: Springer 1967.
Varadhan, S. R. S.: Limit theorems for sums of independant random variables with values in a Hilbert space. Sankhya 24, 213–238(1962).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tortrat, A. Sur la structure des lois indéfiniment divisibles (classe \(\mathfrak{F}\) (X)) dans les espaces vectoriels X (sur le corps réel). Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 311–326 (1969). https://doi.org/10.1007/BF00531653
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00531653