Let Bi be deterministic symmetric m ×m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested in conditions for the “typical norm” of the random matrix SN = N ∑ i=1 ξiBi to be of order of 1. An evident necessary condition is E{S2 N} 1 O(1)I, which, essentially, translates to N ∑ i=1 B i 1 I; a natural conjecture is that the l...

Received: date / Revised version: date Abstract. We study limiting distributions of exponential sums SN (t) = ∑N i=1 e tXi as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess supXi = 0 and (B) ess supXi = ∞. 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...