Local visitation measures for some sequences of random variables with decreasing coefficients

For the sequence of equally distributed random variables $\{x_n,\ n\ge 1\}$, satisfying some mixing condition, the local visitation measures of the sequences $\{x_n/a_n,\ n\ge 1\}$ are considered with various real sequences $\{a_n,\ n\ge 1\}$ tending monotonically to 0. We give sufficient conditions for the measures $\{\sum_{k=1}^n P(x_k/a_k)^{-1},\ n\ge 1\}$ to be the local visitation measures for $\{x_n/a_n,\ n\ge 1\}$ with probability 1.