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Tail behavior of random sums of negatively associated increments. (English) Zbl 1207.60036
Summary: Let \(\{X, X_k, k \geqslant 1\}\) be a sequence of negatively associated random variables with a common distribution function and finite expectation and let \(\tau \) be a nonnegative integer-valued random variable independent of \(\{X_k, k \geqslant 1\}\). In this paper, we give unified form for the asymptotic behavior of the random sums \(S_\tau = \sum _{k=1}^\tau X_k\) in the case of \(\lim _{x\to +\infty }\frac{P(X>x)}{P(\tau > x)} = C \in [0, +\infty ]\). The results extend earlier results of A. Aleškevi\c viené, R. Leipus and J. Šiaulus [Extremes 11, No. 3, 261–279 (2008; Zbl 1164.60006)].

MSC:
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
62P05 Applications of statistics to actuarial sciences and financial mathematics
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