Integral limit theorems for lacunary distributions

For an initial distribution $\{p_k\}$ we consider the family of associated distributions that are defined by the probabilities $p_k(s)=p_ke^{sk}/f(s)$, $k=0,\pm1,\cdots $, where $f(s)=\sum_kp_ke^{sk}$ and $(s_-, s_+)$ is the convergence interval of this series. Let $\eta_1(s),\cdots ,\eta_n(s)$ be independent identically distributed random variables with the distribution $\{p_k(s)\}$. We study in detail limit distributions of the sums $\eta_1(s)+\cdots +\eta_n(s)$ as $n\to\infty$ and for various $s\in(s_-, s_+)$, paying the most attention to the case $s\to s_+$.