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Limit theorems for sums of random exponentials. (English) Zbl 1073.60017
Let \(X_1, X_2, \ldots\) be a sequence of independent identically distributed random variables. The authors investigate a limit behaviour of \(S_N(t):=\sum_{i=1}^N e^{tX_i}\) when \(t\) and \(N\) go to \(\infty\). This problem naturally appears when one studies the evolution of branching population, random energy model and risk theory. Two cases are considered: (A) \(\text{ess\,sup}\, X_i=0\) and \(h(x):=-\log P\{X_i>-1/x\}\) regularly varies at \(\infty\); (B) \(\text{ess\,sup}\,X_i=\infty\) and \(h(x):=-\log P\{X_i>x\}\) regularly varies at \(\infty\).
Under appropriate growth assumptions on \(N\) and \(t\) the authors “have found two critical points, below which the law of large numbers and the central limit theorem break down”. If \(h\) is normalized regularly varying, it is proved that appropriate limit laws are stable. In proving the main results the authors make extensive use of the theory of regular variation, in particular, the Kasahara-de Bruijn Tauberian theorem.

60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
60G50 Sums of independent random variables; random walks
Full Text: DOI
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