Complete Convergence for Randomly Weighted Sums of Random Variables Satisfying Some Moment Inequalities
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Complete Convergence for Randomly Weighted Sums of Random Variables Satisfying Some Moment Inequalities Ping Yan CHEN Department of Mathematics, Ji’nan University, Guangzhou 510630, P. R. China E-mail : [email protected]
Soo Hak SUNG1) Department of Applied Mathematics, Pai Chai University, Daejeon 35345, South Korea E-mail : [email protected] Abstract For random variables and random weights satisfying Marcinkiewicz-Zygmund and Rosenthal type moment inequalities, we establish complete convergence results for randomly weighted sums of the random variables. Our results generalize those of (Thanh et al. SIAM J. Control Optim., 49, 106–124 (2011), Han and Xiang J. Ineq. Appl., 2016, 313 (2016), Li et al. J. Ineq. Appl., 2017, 182 (2017), and Wang et al. Statistics, 52, 503–518 (2018).) Keywords
Randomly weighted sum, complete convergence, moment inequality
MR(2010) Subject Classification
1
60F15
Introduction
A sequence of random variables {Xn , n ≥ 1} converges completely to the constant θ if ∞
P {|Xn − θ| > ε} < ∞,
∀ε > 0.
n=1
This concept of complete convergence was introduced by Hsu and Robbins [5]. For a sequence of independent and identically distributed (i.i.d.) random variables {X, Xn , n ≥ 1} with EX = 0, they proved that if EX 2 < ∞, then ∞ n P Xk > εn < ∞, ∀ε > 0. n=1
k=1
The converse was proved by Erd˝ os [3]. Baum and Katz [1] extended the Hsu–Robbins–Erd˝os result and established a rate of convergence in the Marcinkiewicz and Zygmund law of large numbers. Theorem 1.1 ([1]) Let r ≥ 1 and 1 ≤ p < 2. Let {X, Xn , n ≥ 1} be a sequence of i.i.d. random variables with EX = 0. Then the following statements are equivalent: Received January 6, 2020, accepted July 1, 2020 The research of Pingyan Chen is supported by the National Natural Science Foundation of China (Grant No. 71471075); the research of Soo Hak Sung is supported by the Pai Chai University research grant in 2020 1) Corresponding author
Chen P. Y. and Sung S. H.
2
(i) E|X|rp < ∞, n ∞ (ii) n=1 nr−2 P {| k=1 Xk | > εn1/p } < ∞, ∀ε > 0. Sung [10] and Wu et al. [15] extended the result of Baum and Katz [1] to weighted sums of ρ∗ -mixing random variables. Theorem 1.2 ([10, 15]) Let r ≥ 1 and 1 ≤ p < 2. Let {X, Xn , n ≥ 1} be a sequence of identically distributed ρ∗ -mixing random variables with EX = 0 and E|X|rp < ∞. Let {ank , 1 ≤ k ≤ n, n ≥ 1} be an array of real numbers satisfying n
|ank |q = O(n)
for some q > rp.
k=1
Then
∞ n=1
n
r−2
P
m 1/p max ank Xk > εn < ∞, 1≤m≤n
∀ε > 0.
k=1
The case r > 1 in Theorem 1.2 is due to Sung [10], and the case r = 1 was proved by Wu et al. [15]. n A number of researches obtained complete convergence for randomly weighted sums k=1 Ank Xk of random variables {Xn , n ≥ 1}, where {Ank } is an array of random variables (called random weights). We refer to Thanh et al. [12] for ρ∗ -mixing random variables with rowwise ρ∗ -mixing random weights, T
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