×

zbMATH — the first resource for mathematics

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.

MSC:
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
60G50 Sums of independent random variables; random walks
PDF BibTeX XML Cite
Full Text: DOI
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
[3] von Bahr, B., Esseen, C.-G.: Inequalities for the rth absolute moment of a sum of random variables, 1 < r < 2. Ann. Math. Statist., 36, 299–303 (1965) · Zbl 0134.36902
[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 · Zbl 1073.60017
[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, A., Kurkova, I., Löwe, M.: Fluctuations of the free energy in the REM and the p-spin SK model. Ann. Probab., 30, 605–651 (2002) · Zbl 1018.60094
[7] Derrida, B.: Random-energy model: Limit of a family of disordered models. Phys. Rev. Lett., 45, 79–82 (1980)
[8] Eisele, Th.: On a third-order phase transition. Comm. Math. Phys., 90, 125–159 (1983) · Zbl 0526.60093
[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, P.: A comedy of errors: the canonical form for a stable characteristic function. Bull. London Math. Soc., 13, 23–27 (1981) · Zbl 0505.60022
[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, E., Picco, P.: On the existence of thermodynamics for the Random Energy Model. Comm. Math. Phys., 96, 125–144 (1984) · Zbl 0585.60098
[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, M.: Limit distributions of norms of vectors of positive i.i.d. random variables. Ann. Probab., 29, 862–881 (2001) · Zbl 1014.60015
[17] Zolotarev, V.M.: The Mellin-Stieltjes transformation in probability theory. Theory Probab. Appl., 2, 433–460 (1957)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.