×

zbMATH — the first resource for mathematics

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.

MSC:
60F15 Strong limit theorems
46B09 Probabilistic methods in Banach space theory
60G50 Sums of independent random variables; random walks
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.