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 \[ \text{Pr}(\max_{1\leq n<\infty} S_n(\theta)>x)\sim\overline F(x)\sum^\infty_{k=1}\text{E}\theta^\alpha_k. \]


62G32 Statistics of extreme values; tail inference
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