Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums $S_N(t)=\sum_{i=1}^N e^{tX_i}$ as $t\to\infty$, $N\to\infty$, where $(X_i)$ are i.i.d.\ random variables. Two cases are considered: (A) $\esssup X_i=0$ and (B) $\esssup X_i=\infty$. We assume that the function $h(x)=-\log P(X_i>x)$ (case B) or $h(x)=-\log P(X_i>-1/x)$ (case A) is regularly varying at $\infty$ with index $1<\varrho<\infty$ (case B) or $0<\varrho<\infty$ (case A). The appropriate growth scale of $N$ relative to $t$ is of the form $e^{\lambda H_0(t)}$ ($0<\lambda<\infty$), where the rate function $H_0(t)$ is a certain asymptotic version of the function $H(t)=\log E [e^{tX_i}]$ (case B) or $H(t)=-\log E [e^{tX_i}]$ (case A). We have found two critical points, $\lambda_{1}<\lambda_{2}$, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For $0<\lambda<\lambda_{2}$, under the slightly stronger condition of normalized regular variation of $h$ we prove that the limit laws are stable, with characteristic exponent $\alpha=\alpha(\varrho,\lambda)\in(0,2)$ and skewness parameter $\beta\equiv1$.