On the Brunk-Chung type strong law of large numbers for sequences of blockwise m-dependent random variables

Le Van Thanh

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 258-268
  • ISSN: 1292-8100

Abstract

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For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which lim n ( i = 1 n X i ) / b n = 0 almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples.

How to cite

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Van Thanh, Le. "On the Brunk-Chung type strong law of large numbers for sequences of blockwise m-dependent random variables." ESAIM: Probability and Statistics 10 (2006): 258-268. <http://eudml.org/doc/249751>.

@article{VanThanh2006,
abstract = { For a sequence of blockwise m-dependent random variables \{Xn,n ≥ 1\}, conditions are provided under which $\lim_\{n\to\infty\}(\sum_\{i=1\}^nX_i)/b_n=0$ almost surely where \{bn,n ≥ 1\} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples. },
author = {Van Thanh, Le},
journal = {ESAIM: Probability and Statistics},
keywords = {Strong law of large numbers; almost sure convergence; blockwise m-dependent random variables.; strong law of large numbers; blockwise -dependent random variables},
language = {eng},
month = {5},
pages = {258-268},
publisher = {EDP Sciences},
title = {On the Brunk-Chung type strong law of large numbers for sequences of blockwise m-dependent random variables},
url = {http://eudml.org/doc/249751},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Van Thanh, Le
TI - On the Brunk-Chung type strong law of large numbers for sequences of blockwise m-dependent random variables
JO - ESAIM: Probability and Statistics
DA - 2006/5//
PB - EDP Sciences
VL - 10
SP - 258
EP - 268
AB - For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which $\lim_{n\to\infty}(\sum_{i=1}^nX_i)/b_n=0$ almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples.
LA - eng
KW - Strong law of large numbers; almost sure convergence; blockwise m-dependent random variables.; strong law of large numbers; blockwise -dependent random variables
UR - http://eudml.org/doc/249751
ER -

References

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  1. S. Chobanyan, S. Levental and V. Mandrekar, Prokhorov blocks and strong law of large numbers under rearrangements. J. Theoret. Probab.17 (2004) 647–672.  Zbl1062.60028
  2. Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales. 3rd ed. Springer-Verlag, New York (1997).  
  3. V.F. Gaposhkin, On the strong law of large numbers for blockwise independent and blockwise orthogonal random variables. Teor. Veroyatnost. i Primenen. 39 (1994) 804–812 (in Russian). English translation in Theory Probab. Appl.39 (1994) 667–684 (1995).  Zbl0847.60022
  4. M. Loève, Probability Theory I. 4th ed. Springer-Verlag, New York (1977).  Zbl0359.60001
  5. F. Móricz, Strong limit theorems for blockwise m-dependent and blockwise quasiorthogonal sequences of random variables. Proc. Amer. Math. Soc.101 (1987) 709–715.  Zbl0632.60025

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