Abstract
Let {X,X n,n∈Z d+ } be a sequence of independent and identically distributed random variables and {a n ,n ∈Z d+ } be a sequence of constants. We examine the almost sure limiting behavior of weighted partial sums of the form ∑|n|⩽N a n X n . Suppose further that eitherEX=0 orE|X|=∞. In most situations these normalized partial sums fail to have a limit, no matter which normalizing sequence we choose. Thus, the investigation lends itself to the study of the limit inferior and limit superior of these sequences. On the way to proving results of this type we first establish several weak laws. These weak laws prove to be of great value in establishing generalized laws of the iterated logarithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, A. (1990). Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean.J. Th. Prob. 3, 587–598.
Adler, A. (1991). Accumulation points of a particular normalized random walk.Proc. of the Sheffield Symp. on Appl. Prob., R. L. Taylor and I. V. Basawa (eds.),IMS Lecture Notes Series 18, 156–158.
Adler, A. (1992). The “fair” games problem for multidimensional indexed random variables. Sankhya Series A, (to appear).
Adler, A., Rosalsky, A., and Taylor, R. L. (1991). A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces.J. Multiv. Anal. 37, 259–268.
Chow, Y. S., and Teicher, H. (1988).Probability Theory: Independence, Interchangeability, Martingales, Second Edition, Springer-Verlag, New York.
Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II, Second Edition, John Wiley, New York.
Fernholz, L. T. (1980). Limit points associated with generalized iterated logarithm laws.Ann. Prob. 8, 390–392.
Klass, M., and Teicher, H. (1977). Iterated logarithm laws for asymmetric random variables barely with or without finite mean.Ann. Prob. 5, 861–874.
Smythe, R. T. (1974). Sums of independent random variables on partially ordered sets.Ann. Prob. 2, 906–917.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Adler, A. One-sided limit theorems for sums of multidimensional indexed random variables. J Theor Probab 5, 707–726 (1992). https://doi.org/10.1007/BF01058726
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01058726