## On the maximum of randomly weighted sums with regularly varying tails.(English)Zbl 1090.62046

Summary: Consider the randomly weighted sums $$S_n(\theta)= \sum^n_{k=1}\theta_kX_k$$, $$n=1,2,\dots$$, where $$\{X_k,k = 1,2,\dots\}$$ is a sequence of independent, real-valued random variables with common distribution $$F$$, whose right tail is regularly varying with exponent $$-\alpha<0$$, and $$\{\theta_k,k = 1,2,\dots\}$$ is a sequence of positive random variables, independent of $$\{X_k, k = 1,2,\dots\}$$. Under a suitable summability condition on the upper endpoints of $$\theta_k$$, $$k = 1,2,\dots$$, we prove that $\text{Pr}(\max_{1\leq n<\infty} S_n(\theta)>x)\sim\overline F(x)\sum^\infty_{k=1}\text{E}\theta^\alpha_k.$

### MSC:

 62G32 Statistics of extreme values; tail inference
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### References:

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