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**Approximation of the tail probability of randomly weighted sums and applications.**
*(English)*
Zbl 1271.62030

Summary: Consider the problem of approximating the tail probability of randomly weighted sums \(\sum^n_{i=1}\Theta_iX_i\) and their maxima, where \(\{X_i,i\geq 1\}\) is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation class, and \(\{\Theta_i,\;i\geq 1\}\) is a sequence of nonnegative random variables, independent of \(\{X_i,\;i\geq 1\}\) and satisfying certain moment conditions. Under the assumption that \(\{X_i,\;i\geq 1\}\) has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations:
\[
\text{Pr} \left(\max_{1\leq k\leq n}\sum^k_{i=1}\Theta_iX_i>x\right)\sim \text{Pr} \left (\sum^n_{i=1}\Theta_iX_i>x\right)\sim\sum^n_{i=1}\text{Pr} (\Theta_iX_i>x),
\]
and
\[
\text{Pr}\left(\max_{1\leq k\leq \infty} \sum^k_{i=1}\Theta_i X_i>x\right)\sim \text{Pr}\left( \sum^\infty_{i=1}\Theta_iX_i^+>x\right)\sim\sum^\infty_{i=1}\text{Pr}(\Theta_iX_i>x),
\]
as \(x\to\infty\). In doing so, no assumption is made on the dependence structure of the sequence \(\{\Theta_i,i\geq 1\}\).

### MSC:

62E20 | Asymptotic distribution theory in statistics |

### Keywords:

asymptotics; regular variation; upper tail dependence; ruin probability; stochastic difference equations
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\textit{Y. Zhang} et al., Stochastic Processes Appl. 119, No. 2, 655--675 (2009; Zbl 1271.62030)

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