## 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:

 6.2e+21 Asymptotic distribution theory in statistics
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### References:

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