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The complete convergence of subsequence for sums of independent B-valued random variables. (English) Zbl 1090.60029
From the author’s abstract: “Sufficient and necessary conditions that \(\sum_{k=1}^{\infty} \mathbb P(\| \sum_{i=1}^{n_{k}} X_{i}\| \geq \varepsilon a_{n_{k}}) < \infty\) are derived for the sequence of B-valued i.i.d. random variables \(\{ X_{i} \},\) the strictly increasing subsequence of positive integers \(\{ n_{k} \}\) and the positive monotone sequence of real numbers \(\{ a_{n} \}\) with \(a_{n} \to \infty.\)”
In fact, equivalence is proved under the additional assumption \(\limsup_{k \to \infty} n_{k+1}/\Big(\sum_{i=1}^{k} n_{i}\Big) < \infty\) and some other technical assumptions relating the growth of the subsequence \(a_{n_k}\) to the growth of \(n_{k}.\) This extends some previously known results in the real-valued case [S. Asmussen and T. G. Kurtz, Ann. Probab. 8, 176–182 (1980; Zbl 0426.60026); K.B. Athreya and N. Kaplan, ibid. 4, 38–50 (1976; Zbl 0356.60048) and in: Branching processes. Adv. Probab. relat. Top., Vol. 5, 27–60 (1978; Zbl 0404.60088); O. Nerman, Z. Wahrscheinlichkeitstheorie Verw. Geb. 57, 365–395 (1981; Zbl 0451.60078); A. Gut, Ann. Probab. 13, 1286–1291 (1985; Zbl 0582.60057); Q. Wang and C. Su, J. Math., Wuhan Univ. 11, No. 2, 161–171 (1991; Zbl 0749.60032); D. Deng, Acta Math. Appl. Sin. 16, No. 3, 308–316 (1993; Zbl 0781.60026)]. The main approach is due to the Talagrand’s isoperimetric inequality and entropy estimate.

60F15 Strong limit theorems
46B09 Probabilistic methods in Banach space theory
60G50 Sums of independent random variables; random walks
Full Text: DOI
[1] Asmussen, S.; Kurtz, T.G., Necessary and sufficient conditions for complete convergence in the law of large numbers, Ann. probab., 8, 176-182, (1980) · Zbl 0426.60026
[2] Athreya, K.B.; Kaplan, N., Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process, Ann. probab., 4, 38-50, (1976) · Zbl 0356.60048
[3] Athreya, K.B.; Kaplan, N., Additive property and its applications in branching processes, () · Zbl 0404.60088
[4] Chen, X., The Kolmogorov’s LIL of B-valued random elements and empirical processes, Acta math. sinica, 36, 600-619, (1993) · Zbl 0785.60019
[5] Deng, D., The complete convergence of subsequences for sums of independent random variables, Acta math. appl. sinica, 16, 308-316, (1993) · Zbl 0781.60026
[6] Einmahl, U., Toward a general law of the iterated logarithm in Banach space, Ann. probab., 21, 2012-2045, (1993) · Zbl 0790.60034
[7] Gut, A., On complete convergence in the law of large numbers for subsequence, Ann. probab., 13, 1286-1291, (1985) · Zbl 0582.60057
[8] Hsu, P.L.; Robbins, H., Complete convergence and law of large numbers, Proc. nat. acad. sci. U.S.A., 33, 25-31, (1947) · Zbl 0030.20101
[9] Ledoux, M.; Talagrand, M., Probability in Banach space, (1991), Springer Berlin
[10] Nerman, O., 1981. On the convergence of supercritical general (C-MJ) Branching Processes, Z. Wahrsch. verw. Gebiete 57, 365-395. · Zbl 0451.60078
[11] Wang, Q.; Su, C., Two questions on the complete convergence of subsequence for sums of independent random variables, Math. J. (PRC), 11, 161-171, (1991)
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