Abstract
It is well-known that the Kolmogorov SLLN (non-i.i.d. case) fails for pair-wise independent random variables. However, as shown in the paper it can be saved even for orthogonal random variables if one allows permutations. We prove it in the setup of Banach space valued random variables.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Billingsley, P. (1968). Convergence of Probability Measures, John Wiley, New York.
Chobanyan, S. (1994). Convergence almost sure of rearranged series in Banach spaces and associated inequalities, In Hoffmann-Joørgensen, J., Kuelbs J., and Markus, M. B. (eds.), Probability in Banach Spaces 9, Birkhäuser, Boston.
Chobanyan S., and Mandrekar V. (1998). Almost everywhere convergence and SLLN under rearrangements, In Kartatzas, I., Rajput, B., and Taqqu, M. S. (eds.), Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943–1995, Birkhäuser, Boston.
Chung, K. L. (1951). The strong law of large numbers. Proc. Second Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, pp. 341–352.
Cörgo, S., Tandori, K., and Totik V. (1983). On the strong law of large numbers for pairwise independent random variables. Acta Math. Hung. 42, 319–330.
Hoffmann-Joørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52, 159–186.
Loéve, M. (1962). Probability Theory, Van Nostrand, New Jersey.
Prohorov, Yu. V. (1950). On the strong law of large numbers. Izv. Akad. Nauk SSSR, Ser. Matem. 14, 523–536.
Rights and permissions
About this article
Cite this article
Chobanyan, S., Mandrekar, V. On Kolmogorov SLLN Under Rearrangements for “Orthogonal” Random Variables in a B-Space. Journal of Theoretical Probability 13, 135–139 (2000). https://doi.org/10.1023/A:1007734910044
Published:
Issue Date:
DOI: https://doi.org/10.1023/A:1007734910044