Abstract.
We study limiting distributions of exponential sums as t→∞, N→∞, where (X i ) are i.i.d. random variables. Two cases are considered: (A) ess sup X i = 0 and (B) ess sup X i = ∞. We assume that the function h(x)= -log P{X i >x} (case B) or h(x) = -log P {X i >-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H0(t) is a certain asymptotic version of the function (case B) or (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 (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.
Article PDF
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues. Wiley, Chichester, 1987
Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin, 1972
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)
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
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Paperback edition (with additions). Cambridge Univ. Press, Cambridge, 1989
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)
Derrida, B.: Random-energy model: Limit of a family of disordered models. Phys. Rev. Lett., 45, 79–82 (1980)
Eisele, Th.: On a third-order phase transition. Comm. Math. Phys., 90, 125–159 (1983)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th ed. (A. Jeffrey, ed.) Academic Press, Boston, 1994
Hall, P.: A comedy of errors: the canonical form for a stable characteristic function. Bull. London Math. Soc., 13, 23–27 (1981)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd ed. At the University Press, Cambridge, 1952
Ibragimov, I.A., Linnik, Yu.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971
Olivieri, E., Picco, P.: On the existence of thermodynamics for the Random Energy Model. Comm. Math. Phys., 96, 125–144 (1984)
Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin, 1975
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester, 1999
Schlather, M.: Limit distributions of norms of vectors of positive i.i.d. random variables. Ann. Probab., 29, 862–881 (2001)
Zolotarev, V.M.: The Mellin-Stieltjes transformation in probability theory. Theory Probab. Appl., 2, 433–460 (1957)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.
Mathematics Subject Classification (2000): 60G50, 60F05, 60E07
Rights and permissions
About this article
Cite this article
Ben Arous, G., Bogachev, L. & Molchanov, S. Limit theorems for sums of random exponentials. Probab. Theory Relat. Fields 132, 579–612 (2005). https://doi.org/10.1007/s00440-004-0406-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0406-3