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On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in rademacher type p Banach spaces

Rosalsky A., Thanh L. V.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

205--222

For a sequence of random elements {Vn, n ≥1} taking values in a real separable Rademacher type p (1 ≤p ≤2) Banach space and positive constants bn↑∞, conditions are provided for the strong law of large numbers ∑ni=1Vi/bn→0 almost surely. We treat the following cases: (i) {Vnn ≥1} is blockwise independent with EVn=0, n≥1, and (ii) {Vn, n≥1} is blockwise p-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of Móricz [12], Móricz et al. [13], and Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative examples or counterexamples are provided.

On stability of trimmed sums

Hu T.-C., Huang C.-Y., Rosalsky A.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

153--172

Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables and let {an, n ≥ 1} and {bn, n ≥ 1} be sequences of constants where 0 < bn ↑ ∞. Let Xn(1), Xn(2),…, Xn(n) be a rearrangement of X1,…, Xn such that |Xn(1)| ≥ |Xn(2)| ≥ … ≥ |Xn(n)|. Consider the sequence of weighted sums Tn = Σni=1 ai Xi, n ≥ 1, and, for fixed r ≥ 1, set T(r)n = Σni=1 ai Xi I(|Xi| ≤ |X(r+1)n|), n ≥ r + 1; i.e., T(r)n is the sum Tn minus the sum of the X(k)n’s multiplied by their corresponding coefficients for k = 1,…, r. The main results provide sufficient and, separately, necessary conditions for b−1n T(r)n − kn → 0 almost surely for some sequence of centering constans {kn, n ≥1}. The current work extends that of Mori [14], [15] wherein an ≡ 1.