## Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type $$p$$ Banach spaces.(English)Zbl 1112.60002

Summary: For a double array of independent random elements $$\{V_{mn}$$, $$m\geq 1$$, $$n\geq 1\}$$ in a real separable Rademacher type $$p$$ $$(1\leq p\leq 2)$$ Banach space, strong and weak laws of large numbers are established for the double sums $$\sum_{i=1}^m \sum_{j=1}^n V_{ij}$$, $$m\geq 1$$, $$n\geq 1$$. For the strong law result, it is assumed that $$EV_{mn}=0$$, $$m\geq 1$$, $$n\geq 1$$ and conditions are provided under which $$\sum_{i=1}^m \sum_{j=1}^n V_{ij}/mn\to 0$$ almost surely as $$\max\{m,n\}\to\infty$$. These conditions for the strong law are shown to completely characterize Rademacher type $$p$$ Banach spaces. The weak law results provide conditions for $$\sum_{i=1}^m \sum_{j=1}^n (V_{ij}-c_{ijmn})/mn @>P>> 0$$ as $$\max\{m,n\}\to\infty$$ or for $$\sum_{i=1}^{T_m} \sum_{j=1}^{\tau_n} (V_{ij}- c_{ijmn})/mn @>P>>0$$ as $$\max\{m,n\}\to\infty$$ to hold where $$c_{ijmn}= E(V_{ij}/(\|V_{ij}\|\leq mn))$$, $$i,j,m,n\geq 1$$ and $$\{T_m$$, $$m\geq 1\}$$ and $$\{\tau_n$$, $$n\geq 1\}$$ are sequences of positive integer-valued random variables. Illustrative examples are provided.

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F15 Strong limit theorems 60F05 Central limit and other weak theorems
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### References:

 [1] DOI: 10.1214/aop/1176997031 · Zbl 0258.60026 · doi:10.1214/aop/1176997031 [2] DOI: 10.2307/2371268 · Zbl 0019.35406 · doi:10.2307/2371268 [3] DOI: 10.1215/S0012-7094-39-00501-6 · Zbl 0021.23501 · doi:10.1215/S0012-7094-39-00501-6 [4] Dunford N., Acta Sci. Math. Szeged 14 pp 1– (1951) [5] Zygmund A., Acta Sci. Math. Szeged 14 pp 103– (1951) [6] DOI: 10.1073/pnas.17.12.656 · Zbl 0003.25602 · doi:10.1073/pnas.17.12.656 [7] Griffiths R.B., Encyclopedia of Statistical Sciences 2 pp 225– (1982) [8] Simon B., Encyclopedia of Statistical Sciences 4 pp 519– (1983) [9] DOI: 10.1063/1.533184 · Zbl 0977.82019 · doi:10.1063/1.533184 [10] Pyke R., Stochastic Analysis: A Tribute to the Memory of Rollo Davidson pp 331– (1973) [11] Taylor R.L., Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces 672 (1978) · Zbl 0443.60004 · doi:10.1007/BFb0063205 [12] DOI: 10.1214/aop/1176996029 · Zbl 0368.60022 · doi:10.1214/aop/1176996029 [13] DOI: 10.1016/0047-259X(78)90080-5 · Zbl 0376.60006 · doi:10.1016/0047-259X(78)90080-5 [14] Adler A., Bull. Inst. Math. Acad. Sinica 20 pp 335– (1992) [15] DOI: 10.1080/03610929108830747 · Zbl 0800.60011 · doi:10.1080/03610929108830747 [16] Woyczyński W.A., Probability on Banach Spaces 4 pp 267– (1978) [17] Giang N.V., Teor. Veroyatnost. i Primenen. 40 pp 213– (1995) [18] DOI: 10.1007/BF01013465 · Zbl 0438.60027 · doi:10.1007/BF01013465 [19] Giang N.V., Acta Math. Vietnam. 14 pp 37– (1989) [20] Fazekas I., Publ. Math. Debrecen 53 pp 149– (1998) [21] Loève M., Probability Theory I, 4. ed. (1977) · doi:10.1007/978-1-4684-9464-8 [22] Hille E., Functional Analysis and Semi-Groups (1957) · Zbl 0033.06501 [23] DOI: 10.1023/A:1022645526197 · Zbl 0884.60007 · doi:10.1023/A:1022645526197 [24] DOI: 10.1215/S0012-7094-76-04323-4 · Zbl 0338.60027 · doi:10.1215/S0012-7094-76-04323-4 [25] DOI: 10.1073/pnas.63.2.266 · Zbl 0186.20302 · doi:10.1073/pnas.63.2.266
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