The Chover-type law of the iterated logarithm for certain power series

Let $\{p_n,\ n\geq 0\}$ be a sequence of real numbers with $p_n\sim R(n)$, $R(\cdot)$ a regular varying function with index greater than $-1/\alpha$ $(0<\alpha<2)$. We prove the Chover-type law of the iterated logarithm for the $(J_p)$ power transform of sequence $\{X_n,\,n\geq 0\}$ of independent identically distributed stable random variables with exponent $\alpha$.