Abstract
For any sequence {a k } with sup \(\frac{1}{n}\sum {_{k = 1}^n \left| {a_k } \right|^q } < \infty \) for some q>1, we prove that \(\frac{1}{n}\sum {_{k = 1}^n {\text{ }}a_k X_k } \) converges to 0 a.s. for every {X n } i.i.d. with E(|X 1|)<∞ and E(X 1)=0; the result is no longer true for q=1, not even for the class of i.i.d. with X 1 bounded. We also show that if {a k } is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {X n { with finite absolute pth moment for some p> 1,\(\frac{1}{n}\sum {_{k = 1}^n {\text{ }}a_k X_k } \) converges a.s.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
references
Aaronson, J. (1977). On the ergodic theory of non-integrable functions and infinite measure spaces. Israel J. Math. 27, 163-173.
Akcoglu, M. A. (1975). A pointwise ergodic theorem in Lp spaces. Canada J. Math. 27, 1075-1082.
Assani, I. (1997). Strong laws for weighted sums of independent identically distributed random variables. Duke Math. J. 88, 217-246.
Assani, I. (1997). Convergence of the p-series for stationary sequences. New York J. Math. 3A, 15-30.
Assani, I. (1998). A weighted pointwise ergodic theorem. Ann. Inst. H. Poincaré Probab. Statist. 34, 139-150.
Chow, Y. S. (1965). Local convergence of martingales and the law of large numbers. Ann. Math. Stat. 36, 552-558.
Chow, Y. S., and Lai, T. L. (1973). Limiting behavior of weighted sums of independent random variables. Ann. Probab. 1, 810-824.
Chung, K. L. (1947). Note on some strong laws of large numbers. Amer. J. Math. 69, 189-192.
Cohen, G., and Lin, M. (2003). Laws of large numbers with rates and the one-sided ergodic Hilbert transform. Illinois J. Math. 47, 997-1031.
Cuzick, J. (1995). A strong law for weighted sums of i.i.d. random variables. J. Theoret. Probab. 8, 625-636; Cuzick, J.Erratum. J. Theoret. Probab. 14 (2001).
Derriennic, Y., and Lin, M. (2001). Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123, 93-130.
Jamison, B., Orey, S., and Pruitt, W. (1965). Convergence of weighted averages of independent random variables. Z. Wahrsch. Verw. Gebiete 4, 40-44.
Jones, R. L. (1977). Inequalities for the ergodic maximal function. Studia Math. 60, 111-129.
Krengel, U. (1985). Ergodic Theorems, de Gruyter, Berlin.
Landers, D., and Rogge, L. (1997). Laws of large numbers for uncorrelated Cesaro uniformly integrable random variables. Sankhyma Ser. A 59, 301-310.
Lin, M., Olsen, J., and Tempelman, A. (1999). On modulated ergodic theorems for Dunford–Schwartz operators. Illinois J. Math. 43, 542-567.
Marcinkiewicz, J., and Zygmund, A. (1937). Sur les fonctions indépendentes. Fund. Math. 29, 60-90.
Petersen, K. (1983). Ergodic Theory, Cambridge University Press, Cambridge.
Pruitt, W. (1966). Summability of independent random variables. J. Math. Mech. 15, 769-776.
Rudolph, D. (1994). A joinings proof of Bourgain's return time theorem. Ergodic Theory Dynam. Systems 14, 197-203.
Ryll-Nardzewski, C. (1975). Topics in Ergodic Theory, Springer Lecture Notes in Math., Vol.472, pp. 131-156.
Sawyer, S. (1966). Maximal inequalities of weak type. Ann. Math. 84, 157-174.
Stout, W. (1974). Almost Sure Convergence, Academic Press, New York.
Tempelman, A. (1974). Ergodic theorems for amplitude modulated homogeneous random fields. Lithuanian J. Math. 14, 221-229 (in Russian; English translation (1975): Lith. Math. Transl. 14, 698–704).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baxter, J., Jones, R., Lin, M. et al. SLLN for Weighted Independent Identically Distributed Random Variables. Journal of Theoretical Probability 17, 165–181 (2004). https://doi.org/10.1023/B:JOTP.0000020480.84425.8d
Issue Date:
DOI: https://doi.org/10.1023/B:JOTP.0000020480.84425.8d