Let {X k,i ; i ≥ 1, k ≥ 1} be an array of i.i.d. random variables and let {pn; n ≥ 1} be a sequence of positive integers such that n/pn is bounded away from 0 and ∞. For Wn = max 1≤i

Let {X n ; n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set S n = X 1 + · · · + X n , n ≥ 1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0 < p < 2, ∞ n=1 n=11 n |Sn| n 1/p . (Telephone: 1 -352-273-2983, FAX: 1-352-392-5175) space (Ω, F, P). As usual, let S n = n k=1 X k , n ≥ 1 denote their partial sums. If 0 < p < 2 and if X is a real-valued random variable (that is, if B = R), then lim n→∞ S n n 1/p = 0 almost surely (a.s.)This is the celebrated Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers (SLLN); see Kolmogoroff [8] for p = 1 and Marcinkiewicz and Zygmund [11] for p = 1. The classical Kolmogorov SLLN in real separable Banach spaces was established by Mourier [14]. The extension of the Kolmogorov-Marcinkiewicz-Zygmund SLLN to B-valued random variables is independently due to Azlarov and Volodin [1] and de Acosta [3]. Theorem 1.1. (Azlarov and Volodin [1] and de Acosta [3]). Let 0 < p < 2 and let {X n ; n ≥ 1} be a sequence of independent copies of a B-valued random variable X. Then lim n→∞ S n n 1/p = 0 a.s. if and only if E X p < ∞ and S n n 1/p → P 0. De Acosta [3] also provides a remarkable characterization of Rademacher type p Banach spaces. (Technical definitions such as B being of Rademacher type p will be reviewed below.) Specifically, de Acosta [3] proved the following theorem. Theorem 1.2. (de Acosta [3]). Let 1 ≤ p < 2. Then the following two statements are equivalent: (i) The Banach space B is of Rademacher type p. (ii) For every sequence {X n ; n ≥ 1} of independent copies of a B-valued variable X, lim n→∞ S n n 1/p = 0 a.s. if and only if E X p < ∞ and EX = 0. At the origin of the current investigation is the following recent and striking result by Hechner and Heinkel [4]. Theorem 1.3. (Hechner and Heinkel [4]). Suppose that B is of stable type p (1 < p < 2) and let {X n ; n ≥ 1} be a sequence of independent copies of a B-valued variable X with EX = 0. Then ∞ n=1 1 n E S n n 1/p < ∞ (1.1) if and only if ∞ 0 P 1/p ( X > t)dt < ∞. (1.2)

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [ll], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal 1171, and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established. 963

Concentrating mainly on independent and identically distributed (i.i.d.) real-valued parent sequences, we give an overview of first-order limit theorems available for bootstrapped sample sums for Efron's bootstrap. As a light unifying theme, we expose by elementary means the relationship between corresponding conditional and unconditional bootstrap limit laws. Some open problems are also posed

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