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Almost sure convergence and complete convergence for the weighted sums of martingale differences. (English) Zbl 0963.60031

Consider a sequence \(\{(D_n,{\mathcal F}_n) :n\geq 1\}\) of martingale differences together with an array \(\{a_{ni}:1\leq i\leq n,n\geq 1\}\) of possible weights. On assuming that the sequence \(\{D_n:n\geq 1\}\) is stochastically bounded by some random variable \(D_0\) and that \(D_0\) satisfies either a suitable moment condition or a certain tail behaviour, the authors prove two strong laws of large numbers for weighted sums \(S_n=\sum^n_{i=1}a_{ni}X_i\), in which the weights \(a_{ni}\) are related to the tail behaviour of \(D_0\) in a particular way. Under similar assumptions, complete convergence is established for partial sums of Banach space valued random variables based on moving averages of a martingale difference sequence. The results extend earlier strong laws or Marcinkiewicz-Zygmund type laws for independent random variables.

MSC:

60F15 Strong limit theorems
60G42 Martingales with discrete parameter
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[1] Yu, K-f, Complete convergence of weighted sums of martingale differences[J], J. Theoret Probab, 3, 339-347 (1990) · Zbl 0698.60035 · doi:10.1007/BF01045165
[2] Choi, Sung S. R., Almost sure convergence theorems of weighted sums of random variable[J], Stocha Analys & Appl, 15, 4, 365-376 (1987) · Zbl 0633.60049 · doi:10.1080/07362998708809124
[3] De-li, Li; Bhaskara, Rao M.; Xiang-chen, Wang, Complete convergence and almost convergence of weighted sums of random variable[J], J Theoret Probab, 8, 49-76 (1995) · Zbl 0814.60026 · doi:10.1007/BF02213454
[4] Teicher, H., Almost certain convergence in double arrays[J], Z wahrsch verw Geb, 69, 331-345 (1985) · Zbl 0548.60028 · doi:10.1007/BF00532738
[5] Thrum, R., A remark on almost sure convergence of weighted sums[J], Probab Theory and Rel Fields, 75, 425-430 (1987) · Zbl 0599.60031 · doi:10.1007/BF00318709
[6] Jing-jun, Liu; Shi-xing, Gan, Strong convergence of weighted sums of random variable[J], Acta Mathematica Sinica, 41, 4, 823-832 (1998) · Zbl 1009.60017
[7] Stout, W. F., Almost sure convergence[M], 128-128 (1972), Wuhan: Academic Press, Wuhan
[8] Deli, Li; Bhaskara, Rao M.; Xiang-chen, Wang, Complete convergence for the moving average processes[J], Statist Probab Lett, 14, 111-114 (1992) · Zbl 0756.60031 · doi:10.1016/0167-7152(92)90073-E
[9] Burton, R. M.; dehling, H., Large deviations for some weakly dependent random processes[J], Statist Probab Lett, 9, 397-401 (1990) · Zbl 0699.60016 · doi:10.1016/0167-7152(90)90031-2
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