Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\) , in which the primary random variables X ₁, …, X _n are real valued, independent and subexponentially distributed, while the random weights ?? ₁, …, ?? _n are nonnegative and arbitrarily dependent, but independent of X ₁, …, X _n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of ?? ₁ X ₁, …, ?? _n X _n , which complies with the principle of a single big jump. An application to capital allocation is proposed.     

Tail asymptotics of randomly weighted sums of dependent strong subexponential random variables

Lithuanian Mathematical Journal - Following [C. Yu, Y. Wang, and D. Cheng, Tail behavior of the sums of dependent and heavy-tailed random variables, J. Korean Stat. Soc., 44(1):12–27, 2015],...     

WOD随机变量加权和的完全收敛性

宽象限相依变量(简称WOD变量)是一类包含独立变量,负相协变量(简称NA变量),负象限相依变量(简称NOD变量)和推广的负象限相依变量(简称END变量)在内的非常广泛的相依变量.本文利用WOD变量的Rosenthal型矩不等式和随机变量的截尾技术,在一般的条件下建立了WOD变量加权和的完全收敛性.所得结果推广了若干相依变量的相应结果.     

The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a _n X _n of a weighted series of a sequence {X _n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some in 0<1, ¦a _n ¦ < and E¦X _n ¦ < ; if a _n =z ⁿ for some ¦z¦<1 then it suffices that E(log¦X _n ¦)₊<. Examples show that these sufficient conditions are not necessary. For mutually independent {X _n } necessary conditions can be given: the a.s. absolute convergence of X _n z ⁿ (all ¦z¦<1) then implies E(log¦X _n ¦)₊ < , while if the X _n are non-negative stable r.v.s. of index , ¦a _n X _n ¦< if and only if ¦a _n ¦ < .     

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

END随机变量阵列加权和的完全收敛性

利用END变量的R0senthal型矩不等式,研究了END随机阵列加权和的完全收敛性,给出了证明完全收敛性的一些充分条件.另外,还给出了证明完全收敛性的一个必要条件.所得结果推广了独立变量和若干相依变量的相应结果.     

Complete and complete moment convergence for weighted sums of widely orthant dependent random variables

In this paper, we establish a complete convergence result and a complete moment convergence result for weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results for weighted sums of extended negatively orthant dependent random variables are also obtained, which generalize and improve the related known works in the literature.     

Limit theorems for sums of dependent random variables

Lithuanian Mathematical Journal -     

Complete convergence for weighted sums of LNQD random variables

In this paper, we study the complete convergence for weighted sums of linearly negative quadrant dependent (LNQD) random variables based on the exponential bounds. In addition, we present some complete convergence for arrays of rowwise LNQD random variables.     

Limit theorems for sums of dependent random variables

Summary In Lai and Stout [7] the upper half of the law of the iterated logarithm (LIL) is established for sums of strongly dependent stationary Gaussian random variables. Herein, the upper half of the LIL is established for strongly dependent random variables {X _i} which are however not necessarily Gaussian. Applications are made to multiplicative random variables and to f(Z _i ) where the Z _i are strongly dependent Gaussian. A maximal inequality and a Marcinkiewicz-Zygmund type strong law are established for sums of strongly dependent random variables X _i satisfying a moment condition of the form E¦S _a,n ¦^pg(n), where , generalizing the work of Serfling [13, 14].Research supported by the National Science Foundation under grant NSF-MCS-78-09179Research supported by the National Science Foundation under grant NSF-MCS-78-04014     

H可积下的相依随机变量和的完全收敛性质

在满足H可积的条件下,利用随机变量的截尾方法,以及相关引理,给出了行内两两NQD序列以及p混合条件的随机组列部分和的完全收敛定理以及强大数定理．