A bound in the stable ($\alpha$), $1 < \alpha \le 2$, limit theorem for associated random variables with infinite variance

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 $1 < \alpha < 2$.