A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Abstract

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)

A characterization of a new type of strong law of large numbers

“…Furthermore, each of (1.1) and (1.2) implies that lim n→∞ S n n 1/p = 0 a. The current work continues the investigations by Hechner and Heinkel [5] and Li, Qi, and Rosalsky [10] and [11]. More specifically:…”

“…Again, these results are new when B = R; see Theorem 2.5 of Li, Qi, and Rosalsky [10]. Motivated by the results obtained by Li, Qi, and Rosalsky [10], we introduce a new type strong law of large numbers as follows. Definition 1.1.…”

“…Inspired by the above discovery by Hechner and Heinkel [5], Li, Qi, and Rosalsky [10] obtained sets of necessary and sufficient conditions for ∞ n=1 1 n S n n 1/p < ∞ a.s. for the three cases: 0 < p < 1, p = 1, 1 < p < 2. Moreover, Li, Qi, and Rosalsky [10] obtained necessary and sufficient conditions for…”

“…characterizations of a Banach space being of stable type p, properties of stable type p Banach spaces, as well as various relationships between the conditions "Rademacher type p" and "stable type p" may be found in Maurey and Pisier [16], Woyczyński [20], Marcus and Woyczyński [15], Rosiński [19], Pisier [18], and Ledoux and Talagrand [9]. Some of these properties and relationships are summarized in Li, Qi, and Rosalsky [10]. De Acosta [3] also provided a remarkable characterization of Rademacher type p Banach spaces.…”

On the rate of convergence in the strong law of large numbers for martingales

“…For the general p, r satisfying the conditions of Theorem A, Katz [11], and later Baum and Katz [1] proved the equivalence between EjXj p , 1 and (1.3) and Chow [2] established the equivalence between EjXj p , 1 and (1.4). Recently, Li et al [14] established a refined version of the classical Kolmogorov -Marcinkiewicz -Zygmund strong law of large numbers.…”