Summary
We consider the set of random measures which consists of measurable mapsω→μ ω from [0, 1] to the set of measures on ℝ. As it is the dual space ofL 1 ([0, 1];C(
)), we can equip this space with the weak* topology. We construct a special random measure μ, which appears as the weak* limit of a sequence of Dirac random measures\(\left( {\delta _{X_n } } \right)_{n \in \mathbb{N}}\), where (X n ) n∈ℕ is a bounded sequence inL p [0, 1], (1≦p<2). The special form of this random measure, which oscillates randomly between twoq-stable standard measures on ℝ with different normalizations (p<q<2) allows us to prove two properties of (X n ) n∈ℕ is equivalent to the unit vector basis ofl q and has no almost symmetric subsequence.
References
[A] Aldous, D.J.: Subspaces ofL 1 via random measures. Trans. Am. Math. Soc.267, 445–463 (1981)
[BR] Berkes, I., Rosenthal, H.P.: Almost exchangeable sequences of random variables. Z. Warscheinlichkeitstheor.70, 473–507 (1985)
[D] Doeblin, W.: Ensembles de puissances d'une loi de probabilité. Stud. Math.9, 71–96 (1940)
[F] Feller, W.: An introduction to probability theory and its applications, vols.I, II. New York: Wiley 1966
[G1] Guerre, S.: Types et suites symétriques dansL p , 1≦p<+∞,p≠2. Isr. J. Math.53, 191–208 (1986)
[G2] Guerre, S.: Sur les suites presques échangeables dansL q , 1≦q<2. Isr. J. Math.56, 361–380 (1986)
[GR] Guerre, S., Raynaud, Y.: On sequences with no almost symmetric subsequence. Loghorn Notes-UT Funct. Anal. Seminar (Austin-Texas, 1985–86)
[L] Levy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthier-Villars (1937)
[LT] Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces, vols. I, II. Berlin Heidelberg New York: Springer 1979
[M] Maurey, B.: Tout sous-espaces deL 1 contient unl p d'après D. Aldous. Séminaire d'Analyse Fonctionnelle. Ecole Polytechnique (1979–80)
[R] Raynaud, Y.: Extracting almost symmetric sequences in r.i. spaces. Math. Proc. Camb. Philos. Soc.104, 303–316 (1988)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guerre-Delabriere, S. An example of an irregular random measure. Probab. Th. Rel. Fields 86, 155–165 (1990). https://doi.org/10.1007/BF01474640
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01474640
Keywords
- Stochastic Process
- Unit Vector
- Probability Theory
- Special Form
- Vector Basis