The Annals of Probability

Strong law of large numbers for multilinear forms

Anda Gadidov

Full-text: Open access

Abstract

Let $m \geq 2$ be a nonnegative integer and let ${X^{(l)}, X_i^{(l)}}_{i \epsilon \mathbb{N}}, l = 1, \dots, m$, be $m$ independent sequences of independent and identically distributed symmetric random variables. Define $S_n = \Sigma_{1 \leq i_1, \dots, i_m \leq n} X_{i_1}^{(l)} \dots X_{i_m}^{(m)}$, and let ${\gamma_n}_{n \epsilon \mathbb{N}}$ be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For $m = 2$ necessary and sufficient conditions are obtained for the strong law of large numbers $\gamma_n^{-1} S_n \to 0$ a.s. to hold, and for $m > 2$ the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the $X^{(l)}$ .

Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 902-923.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855655

Digital Object Identifier
doi:10.1214/aop/1022855655

Mathematical Reviews number (MathSciNet)
MR1626539

Zentralblatt MATH identifier
0937.60016

Subjects
Primary: 60F15: Strong theorems

Keywords
Strong laws multilinear forms $U$-statistics martingale maximal inequality

Citation

Gadidov, Anda. Strong law of large numbers for multilinear forms. Ann. Probab. 26 (1998), no. 2, 902--923. doi:10.1214/aop/1022855655. https://projecteuclid.org/euclid.aop/1022855655


Export citation

References

  • Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19-42.
  • Cuzick, J., Gin´e, E. and Zinn, J. (1995). Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables. Ann. Probab. 23 292-333.
  • de la Pe na, V. H. and Montgomery-Smith, S. J. (1995). Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23 806-817.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. Gin´e, E. and Zinn, J. (1992a). On Hoffmann-Jørgensen's inequality for U-processes. In Probability in Banach Spaces (R. M. Dudley, M. G. Kahn and J. Kuelbs, eds.) 8 80-91. Birkh¨auser, Boston. Gin´e, E. and Zinn, J. (1992b). Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics. In Probability in Banach Spaces (R. M. Dudley, M. G. Kahn and J. Kuelbs, eds.) 8 273-291. Birkh¨auser, Boston.
  • Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52 159-186.
  • Kwapi´en, S. and Woyczynski, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkh¨auser, Boston. Rosenthal, H. P. (1970a). On the subspaces of Lp p > 2 spanned by sequences of independent random variables. Israel J. Math. 8 273-303. Rosenthal, H. P. (1970b). On the span in Lp of sequences of independent random variables. II. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 149-167. Univ. California Press, Berkeley.
  • Sen, P. K. (1977). Almost sure convergence of generalized U-statistics. Ann. Probab. 5 287-290.
  • Zhang, C.-H. (1996). Strong law of large numbers for sums of products. Ann. Probab. 24 1589-1615.