On the weak laws of large numbers for weighted sums of dependent identically distributed random vectors in Hilbert space
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On the weak laws of large numbers for weighted sums of dependent identically distributed random vectors in Hilbert spaces Vu T. N. Anh1 · Nguyen T. T. Hien2 Received: 22 February 2020 / Accepted: 18 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract This note establishes the weak laws of large numbers with general normalizing sequences for weighted sums of dependent identically distributed random vectors taking values in real separable Hilbert spaces. The dependent structures include pairwise and coordinatewise negative dependence and coordinatewise negative association. The sharpness of the result for the case where the random vectors are coordinatewise negatively associated is illustrated by an example. Keywords Identically distributed · Pairwise and coordinatewise negative dependence · Coordinatewise negatively associated · Hilbert space · Slowly varying function Mathematics Subject Classification 60F05 · 60B12
1 Introduction Weak laws of large numbers with the norming sequences are of the form n1∕p L(n) where 0 < p < 2 and L(⋅) is a slowly varying function was studied by Gut [5, 6], and Matsumoto and Nakata [16]. In this note, by using some results related to slowly varying functions, we establish weak laws of large numbers for sequences of pairwise and coordinatewise negatively dependent identically distributed random vectors as well as for coordinatewise negatively associated identically distributed random vectors in Hilbert spaces with the norming sequences are of the form bn = n1∕p L(n + A) for some A > 0.
* Vu T. N. Anh [email protected] Nguyen T. T. Hien [email protected] 1
Department of Mathematics, Hoa Lu University, Xuan Thanh Street, Ninh Binh City, Ninh Binh Province, Vietnam
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Department of Mathematics, Vinh University, 182 Le Duan Street, Vinh City, Nghe An Province, Vietnam
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V. T. N. Anh, N. T. T. Hien
The concept of pairwise negatively dependent of random variables was introduced by Lehmann [14]. A sequence of (real-valued) random variables {Xi , i ≥ 1} is said to be pairwise negatively dependent (PND) if for all x, y ∈ ℝ and for all i ≠ j,
P(Xi ≤ x, Xj ≤ y) ≤ P(Xi ≤ x)P(Xj ≤ y). Negative association is a stronger concept of dependence, which was introduced by JoagDev and Proschan [7]. A collection {X1 , X2 , … , Xn } of (real-valued) random variables is said to be negatively associated (NA) if for any disjoint subsets I, J of {1, 2, … , n} and any real coordinatewise nondecreasing functions f on ℝ|I| and g on ℝ|J| ,
Cov (f (Xk , k ∈ I), g(Xk , k ∈ J)) ≤ 0 whenever the covariance exists, where |I| denotes the cardinality of I. A sequence {Xn , n ≥ 1} of random variables is said to be negatively associated if every finite subfamily is negatively associated. The limit theorems, including the law of large numbers and the central limit theorem, for random variables taking values in Banach spaces were studied by many authors. We refer to Ledoux and Talagrand [13], Marcus and Woyczyński [15], and Pisier [18, 19] for this top
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