Limit theorems for sums of random exponentials. (English) Zbl 1073.60017

Ben Arous, Gérard; Bogachev, Leonid V.; Molchanov, Stanislav A.

Probab. Theory Relat. Fields 132, No. 4, 579-612 (2005).

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.

Reviewer: Aleksander Iksanov (Kiev)

MSC:

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

Keywords:

regular variation; exponential Tauberian theorem; infinitely divisible distributions; stable distributions; central limit theorem; law of large numbers

References:

[1]	Asmussen, S.: Applied Probability and Queues. Wiley, Chichester, 1987 · Zbl 0624.60098
[2]	Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin, 1972 · Zbl 0259.60002
[3]	Bahr, Ann. Math. Statist.,, 36, 299 (1965)
[4]	Ben Arous, G., Bogachev, L.V., Molchanov, S.A.: Limit theorems for random exponentials. Preprint NI03078-IGS, Isaac Newton Institute, Cambridge, 2003. http://www.newton.cam.ac.uk/preprints/NI03078.pdf
[5]	Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Paperback edition (with additions). Cambridge Univ. Press, Cambridge, 1989 · Zbl 0667.26003
[6]	Bovier, Ann. Probab.,, 30, 605 (2002)
[7]	Derrida, Phys. Rev. Lett.,, 45, 79 (1980)
[8]	Eisele, Comm. Math. Phys.,, 90, 125 (1983)
[9]	Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th ed. (A. Jeffrey, ed.) Academic Press, Boston, 1994 · Zbl 0918.65002
[10]	Hall, Bull. London Math. Soc.,, 13, 23 (1981)
[11]	Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd ed. At the University Press, Cambridge, 1952
[12]	Ibragimov, I.A., Linnik, Yu.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971 · Zbl 0219.60027
[13]	Olivieri, Comm. Math. Phys.,, 96, 125 (1984)
[14]	Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin, 1975 · Zbl 0322.60043
[15]	Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester, 1999 · Zbl 0940.60005
[16]	Schlather, i.d. random variables. Ann. Probab.,, 29, 862 (2001)
[17]	Zolotarev, Theory Probab. Appl.,, 2, 433 (1957)