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For a double array of random elements {Vmn;m ≥ 1, n ≥1} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which k−1r mn ∑um i=1∑vn j=1(Vij−E(Vij|Fij )) → 0 in Lr (0 < r < 2). The weak law results provide conditions for k −1rmn ∑Tm i=1∑τn j=1(Vij − E(Vij|Fij )) → 0 in probability where {Tm;m ≥ 1} and {τn; n ≥ 1} are sequences of positive integer-valued random variables, {kmn;m ≥ 1, n ≥ 1} is an array of positive integers.
The sharpness of the results is illustrated by examples.
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참고문헌 (10)
참고문헌 신청
A. Adler / 1997 / A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces / Statist. Probab. Lett. 32 (2) : 167 ~ 174
D. Landers / 1987 / Laws of large numbers for pairwise independent uniformly integrable random variables / Math. Nachr. 130 : 187 ~ 192
F. S. Scalora / 1961 / Abstract martingale convergence theorems / Pacific J. Math. 11 : 347 ~ 374
G. Pisier / 1975 / Martingales with values in uniformly convex spaces / Israel J. Math. 20 (3) : 326 ~ 350
G. Pisier / 1986 / Probabilistic methods in the geometry of Banach spaces, Probability and analysis / Springer : 167 ~ 241
D. Landers / 1987 / Laws of large numbers for pairwise independent uniformly integrable random variables / Math. Nachr. 130 : 187 ~ 192
F. S. Scalora / 1961 / Abstract martingale convergence theorems / Pacific J. Math. 11 : 347 ~ 374
G. Pisier / 1975 / Martingales with values in uniformly convex spaces / Israel J. Math. 20 (3) : 326 ~ 350
G. Pisier / 1986 / Probabilistic methods in the geometry of Banach spaces, Probability and analysis / Springer : 167 ~ 241
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