Theory of Probability and Mathematical Statistics
A bound in the Stable (α), 1 < α ≤ 2, limit theorem for Associated random variables with infinite variance
M. Sreehari
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Abstract: Consider a stationary sequence $\{X_n\}$ of associated random variables with the common distribution function $F$ which is in the domain of non-normal attraction of the normal law or in the domain of attraction of a symmetric stable ($\alpha$) law for $\alpha <2$. Louhichi and Soulier [13] proved the central limit theorem when $F$ has infinite variance and also proved the stable limit theorem for $\{X_n\}$ when $F$ is in the domain of normal attraction of the stable law. The aim of this article is to obtain bounds for the rate of convergence in the stable ($\alpha$) limit theorem for $1 < \alpha \le 2$ when $F$ is in the domain of non-normal attraction. We consider also the rate of convergence problem when $F$ is in the domain of normal attraction of a stable ($\alpha$) law for $1 < \alpha < 2$.
Keywords: Associated random variables, Central limit theorem, Stable limit theorem, Rate of convergence, non-normal attraction, Berry-Ess\'{e}en type bound
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