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.

A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables

2013 ◽  
Vol 18 (4) ◽  
pp. 519-525 ◽  
Author(s):  
Yang Yang ◽  
Kaiyong Wang ◽  
Remigijus Leipus ◽  
Jonas Šiaulys
Keyword(s):  
Random Variables ◽  
Tail Probability ◽  
Weighted Sums ◽  
Asymptotic Formulas ◽  
Dependence Structure ◽  
Insurance Risk ◽  
Risk Models ◽  
Randomly Weighted Sums ◽  
Random Weights ◽  
Heavy Tailed

This paper investigates the asymptotic behavior for the tail probability of the randomly weighted sums ∑k=1nθkXk and their maximum, where the random variables Xk and the random weights θk follow a certain dependence structure proposed by Asimit and Badescu [1] and Li et al. [2]. The obtained results can be used to obtain asymptotic formulas for ruin probability in the insurance risk models with discounted factors.

APPROXIMATION OF THE TAIL PROBABILITIES FOR BIDIMENSIONAL RANDOMLY WEIGHTED SUMS WITH DEPENDENT COMPONENTS

2018 ◽  
Vol 34 (1) ◽  
pp. 112-130
Author(s):  
Xinmei Shen ◽  
Mingyue Ge ◽  
Ke-Ang Fu
Keyword(s):  
Discrete Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Weighted Sums ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Moment Conditions ◽  
Randomly Weighted Sums ◽  
Asymptotic Formulae ◽  
Extended Regular Variation

AbstractLet $\left\{ {{\bi X}_k = {(X_{1,k},X_{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of independent and identically distributed random vectors whose components are allowed to be generally dependent with marginal distributions being from the class of extended regular variation, and let $\left\{ {{\brTheta} _k = {(\Theta _{1,k},\Theta _{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of nonnegative random vectors that is independent of $\left\{ {{\bi X}_k, k \ge 1} \right\}$. Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums $\left( {\sum\nolimits_{k = 1}^n {\Theta _{1,k}} X_{1,k},\sum\nolimits_{k = 1}^n {\Theta _{2,k}} X_{2,k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{1,k}} X_{1,k},{{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{2,k}} X_{2,k})^{\rm \top }$ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on $\left\{ {{\brTheta} _k, k \ge 1} \right\}$. Direct applications of the results to risk analysis are proposed, with two types of ruin probability for a discrete-time bidimensional risk model being evaluated.

scholarly journals A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Yang Yang ◽  
Jun-feng Liu ◽  
Yu-lin Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Finite Time ◽  
Risk Model ◽  
Random Variables ◽  
Weighted Sums ◽  
Financial Risks ◽  
Tail Behavior ◽  
Insurance Risk ◽  
Randomly Weighted Sums

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.

scholarly journals Precise Large Deviations for Random Sums of END Random Variables with Dominated Variation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yu Chen ◽  
Zhihui Qu
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Risk Model ◽  
Random Variables ◽  
Random Sums ◽  
Tail Probabilities ◽  
Precise Large Deviations ◽  
Surplus Process ◽  
End Random Variables ◽  
Renewal Risk

We investigate the precise large deviations for random sums of extended negatively dependent random variables with long and dominatedly varying tails. We find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence. We apply the results to a generalized dependent compound renewal risk model including premium process and claim process and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.

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