Approximations to inverse moments of double-indexed weighted sums

Asymptotic Approximations of Ratio Moments Based on Dependent Sequences

“…On the other hand, independent sequence is a m-WOD sequence with g(n) = 1. So by taking max 1≤i≤n w ni = O(1) (i.e., λ = 0) and β = 0 in Theorem 3, we have (10) with λ = 0 and β = 0, which implies Theorem 2.3 of Yang et al [12] for nonnegative independent sequences. Furthermore, by using Theorems 2 and 3, we obtain Corollary 1 which does not contain the parameter a.…”

“…Third, let us consider a general weighted inverse moment model. Yang et al [12] obtained the inverse moment result (1), where X n = 1 σ n ∑ n i=1 Z i is replaced by a general weighted case X n = ∑ n i=1 w ni Z i , and {w ni , 1 ≤ i ≤ n, n ≥ 1} is a triangular array of non-negative weights. Li et al [13] studied this general weighted case of inverse moment under nonnegative widely orthant dependent (WOD) random variables.…”

“…Shi et al [14] used the inverse moment method to consider the ratio models such as Z i X n and X k X n , where X k = ∑ k i=1 Z i for 1 ≤ k ≤ n. They obtained some asymptotic expressions for the means of E Z i X n and E X k X n . For all a > 0, Yang et al [12] investigated a general ratio X n a+Y n , where X n = ∑ n i=1 w ni Z i , Y n = ∑ n i=1 Z i and {w ni } are non-negative weights. Some asymptotic expressions for ratios E( X n a+Y n ) and E( X n a+Y n ) 2 were presented in Yang et al [12].…”

“…For all a > 0, Yang et al [12] investigated a general ratio X n a+Y n , where X n = ∑ n i=1 w ni Z i , Y n = ∑ n i=1 Z i and {w ni } are non-negative weights. Some asymptotic expressions for ratios E( X n a+Y n ) and E( X n a+Y n ) 2 were presented in Yang et al [12]. To proceed the study of inverse moment models and ratio models, we give some definitions of dependent random variables in the next subsection.…”

“…where Y n = ∑ n i=1 Z i . Yang et al [12] studied the inverse moment model (2) and ratio model (3) based on the independent sequence {Z n , n ≥ 1} and weighted condition max 1≤i≤n w ni = O(1). Li et al [13] also studied this inverse moment model (2) based on WOD sequence {Z n , n ≥ 1} and weighted condition max 1≤i≤n w ni = O(1).…”