scholarly journals Bounds for tail probabilities of weighted sums of independent gamma random variables

1990 ◽  
pp. 147-166 ◽  
Author(s):  
Persi Diaconis ◽  
Michael D. Perlman
Keyword(s):  
Random Variables ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Bounds For Tail Probabilities

scholarly journals On the Kesten-type inequality for randomly weighted sums with applications to an operational risk model

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1879-1888
Author(s):  
Yishan Gong ◽  
Yang Yang ◽  
Jiajun Liu
Keyword(s):  
Value At Risk ◽  
Risk Model ◽  
Cell Model ◽  
Type Inequality ◽  
Random Variables ◽  
Operational Risk ◽  
Weighted Sums ◽  
Asymptotic Formulas ◽  
Tail Probabilities ◽  
Randomly Weighted Sums

This paper considers the randomly weighted sums generated by some dependent subexponential primary random variables and some arbitrarily dependent random weights. To study the randomly weighted sums with infinitely many terms, we establish a Kesten-type upper bound for their tail probabilities in presence of subexponential primary random variables and under a certain dependence among them. Our result extends the study of Chen [5] to the dependent case. As applications, we derive some asymptotic formulas for the tail probability and the Value-at-Risk of total aggregate loss in a multivariate operational risk cell model.

Inequalities for the tail probabilities of weighted sums of independent random variables with applications to rates of convergence to zero

1982 ◽  
Vol 93 (1-2) ◽  
pp. 111-121
Author(s):  
J. E. A. Dunnage
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Special Case

SynopsisWe obtain inequalities for where Wn = anlX1 + … + annXn, the Xr being independent random variables and the Mn being certain truncated means. We then use these inequalities to study the rate at which this probability tends to zero as N→ ∞, noting that in the special case Wn = (X1 + … + Xn)/n, we obtain the estimate given by L. E. Baum and M. Katz which they show is, in a sense, best possible.A desire to find an inequality which would lead to the result of Baum and Katz was, indeed, the impetus behind this paper.

Weighted sums of subexponential random variables and their maxima

2005 ◽  
Vol 37 (2) ◽  
pp. 510-522 ◽  
Author(s):  
Yiqing Chen ◽  
Kai W. Ng ◽  
Qihe Tang
Keyword(s):  
Ruin Probability ◽  
Random Variables ◽  
Direct Application ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Weighted Sum ◽  
Asymptotic Estimates ◽  
Tail Probabilities ◽  
Inflation Factor ◽  
A Company

Let {Xk, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1nwkXk and the maximum of weighted sums max1≤m≤n∑k=1mwkXk, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk.

scholarly journals Weighted sums of subexponential random variables and their maxima

2005 ◽  
Vol 37 (02) ◽  
pp. 510-522 ◽  
Author(s):  
Yiqing Chen ◽  
Kai W. Ng ◽  
Qihe Tang
Keyword(s):  
Ruin Probability ◽  
Random Variables ◽  
Direct Application ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Weighted Sum ◽  
Asymptotic Estimates ◽  
Tail Probabilities ◽  
Inflation Factor ◽  
A Company

Let {X k , k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {w k , k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑ k=1 n w k X k and the maximum of weighted sums max1≤m≤n ∑ k=1 m w k X k , subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss X k during the kth year, with a discount or inflation factor w k .

scholarly journals Tail probabilities for weighted sums of products of normal random variables

1998 ◽  
Vol 58 (2) ◽  
pp. 239-244 ◽  
Author(s):  
B. Gail Ivanoff ◽  
N.C. Weber
Keyword(s):  
Random Variables ◽  
Tail Probability ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Sums Of Products ◽  
Distributional Limits

Weighted sums of products of independent normal random variables arise naturally as distributional limits for various statistics. This note investigates the rate at which the tail probability of these sums approaches zero.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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