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

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     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},
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     year = {2006},
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     mrnumber = {2219343},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2006010/}
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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. https://www.numdam.org/articles/10.1051/ps:2006010/

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Thành, Lê Vǎn Almost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spaces, Archiv der Mathematik, Volume 122 (2024) no. 2, p. 203 | DOI:10.1007/s00013-023-01929-z
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Thanh, Le Van On the almost sure convergence for dependent random vectors in Hilbert spaces, Acta Mathematica Hungarica, Volume 139 (2013) no. 3, p. 276 | DOI:10.1007/s10474-012-0275-7
Le, Van Thanh; Vu, Ngoc Anh A STRONG LIMIT THEOREM FOR SEQUENCES OF BLOCKWISE AND PAIRWISE m-DEPENDENT RANDOM VARIABLES, Bulletin of the Korean Mathematical Society, Volume 48 (2011) no. 2, p. 343 | DOI:10.4134/bkms.2011.48.2.343
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