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

Thanh, Le Van

doi:10.1051/ps:2006010

On the Brunk-Chung type strong law of large numbers for sequences of blockwise

m

-dependent random variables

Thanh, Le Van

ESAIM: Probability and Statistics, Tome 10 (2006), pp. 258-268.

Résumé

For a sequence of blockwise $m$ -dependent random variables ${X_{n}, n \geq 1}$ , conditions are provided under which ${lim}_{n \to \infty} (\sum_{i = 1}^{n} X_{i}) / b_{n} = 0$ almost surely where ${b_{n}, n \geq 1}$ is a sequence of positive constants. The results are new even when $b_{n} \equiv n^{r}, r > 0$ . 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.

DOI : 10.1051/ps:2006010

Classification : 60F15
Mots clés : strong law of large numbers, almost sure convergence, blockwise $m$-dependent random variables

@article{PS_2006__10__258_0,
     author = {Thanh, Le Van},
     title = {On the {Brunk-Chung} type strong law of large numbers for sequences of blockwise $m$-dependent random variables},
     journal = {ESAIM: Probability and Statistics},
     pages = {258--268},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     doi = {10.1051/ps:2006010},
     mrnumber = {2219343},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2006010/}
}

TY  - JOUR
AU  - Thanh, Le Van
TI  - On the Brunk-Chung type strong law of large numbers for sequences of blockwise $m$-dependent random variables
JO  - ESAIM: Probability and Statistics
PY  - 2006
SP  - 258
EP  - 268
VL  - 10
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2006010/
DO  - 10.1051/ps:2006010
LA  - en
ID  - PS_2006__10__258_0
ER  -

%0 Journal Article
%A Thanh, Le Van
%T On the Brunk-Chung type strong law of large numbers for sequences of blockwise $m$-dependent random variables
%J ESAIM: Probability and Statistics
%D 2006
%P 258-268
%V 10
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2006010/
%R 10.1051/ps:2006010
%G en
%F PS_2006__10__258_0

Thanh, Le Van. On the Brunk-Chung type strong law of large numbers for sequences of blockwise $m$-dependent random variables. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 258-268. doi : 10.1051/ps:2006010. http://www.numdam.org/articles/10.1051/ps:2006010/

Bibliographie
Cité par

[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. | Zbl

[2] Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales. 3rd ed. Springer-Verlag, New York (1997). | MR | Zbl

[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). | Zbl

[4] M. Loève, Probability Theory I. 4th ed. Springer-Verlag, New York (1977). | MR | Zbl

[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. | Zbl

Cité par Sources :