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Abstract. We study limiting distributions of exponential sums SN (t) = N i=1 e tX i as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess sup Xi = 0 and (B) ess sup Xi = ∞. We assume that the function h(x) = − log P{Xi > x} (case B) or h(x) = − log P{Xi > −1/x} (case A) is regularly varying at ∞ with index 1 < ̺ < ∞ (case B) or 0 < ̺ < ∞ (case A). The appropriate growth scale of N relative to t is of the form e λH 0 (t) (0 < λ < ∞), where the rate function H0(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, λ1 < λ2, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ2, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α(̺, λ) ∈ (0, 2) and skewness parameter β ≡ 1.

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We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance Vn tends to infinity. In this work, we mainly focus on the opposite case where Vn is bounded, and derive a simple necessary and sufficient condition for convergence of Vn to a finite limit, thus settling a long-standing question raised by Karlin in the seminal paper of 1967. One striking consequence of our result is that the possible limit may only be a positive integer number. Some new conditions for other types of behavior of the variance, like boundedness or convergence to infinity, are also obtained. The proofs are based on the poissonization techniques.

Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\to\infty$, of "typical" $\varGamma\in {\varPi}_n$ with respect to a parametric family of probability measures $\{P_n^r,0