Abstract
Let (X k) k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>−1 one defines the following coefficients:
It is shown that for all α∈]0, 1[ the sequenceV n =(1/A α n )∑0⩽k⩽n A α−1 n−k X k converges almost surely toE(X) if and only if ‖X‖1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Déniel and Derriennic (case α=1/2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Acosta, A. (1980). Strong exponential integrability of sums of independent B-valued random vectors.Prob. Math. Stat. 1, 133–150.
de Acosta, A. (1981). Inequalities forB-valued random vectors with applications to the strong law of large numbers.Ann. Prob. 9, 157–161.
Alt, J. C. (1989). Stabilité de sommes pondérées de variables aléatoires vectorielles.J. Multivariate Anal. 29, 137–154.
Chow, Y. S., and Lai, T. L. (1973). Limiting behavior of weighted sums of independent random variables.Ann. Prob. 1, 810–824.
Chow, Y. S., and Teicher, H. (1978).Probability Theory. Springer, Berlin.
Déniel, Y., and Derriennic, Y. (1988). Sur la convergence presque sûre, au sens de Cesaro d'ordre α, 0<α<1, de variables aléatoires indépendantes et identiquement distribuées.Prob. Theory Rel. Fields 79, 629–636.
Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables.Studia Math. 52, 159–186.
Jain, N. C. (1975). Tail probabilities for sums of independent Banach space valued random variables.Z. Wahrsch. verw. Gebiete 33, 155–166.
Kuelbs, J., and Zinn, J. (1979). Some stability results for vector valued random variables.Ann. Prob. 7, 75–84.
Laha, R. G., and Rohatgi, V. K. (1979).Probability Theory. Wiley Sons, New York.
Lorentz, G. G. (1955). Borel and Banach properties of methods of summation.Duke Math. J. 22, 129–141.
Mourier, E. (1953). Les éléments aléatoires dans un espace de Banach.Ann. Inst. H. Poincaré 13, 159–244.
Stout, W. F. (1974). Almost sure convergence. Academic, New York.
Zygmund, A. (1935).Trigonometrical series. Monografje Matematyczne, Warsaw.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Heinkel, B. An infinite-dimensional law of large numbers in Cesaro's sense. J Theor Probab 3, 533–546 (1990). https://doi.org/10.1007/BF01046094
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01046094