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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
##### Keywords:
consistent variation; negatively association; random sums
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##### References:
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