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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
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