Abstract
This paper provides an overview of results pertaining to moment convergence for certain ratios of random variables involving sums, order statistics and extreme terms in the sense of modulus. Most of the literature on this matter originates from Darling (1952) who gave a criterion for the convergence in probability to 1 of the ratio of the maximum to the sum in case of nonnegative random variables.
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H. Albrecher, S. A. Ladoucette, and J. L. Teugels, “Asymptotics of the sample coefficient of variation and the sample dispersion,” Technical Report 2006-04, University Centre for Statistics, Department of Mathematics, Catholic University of Leuven, Belgium, 2006.
H. Albrecher and J. L. Teugels, “Asymptotic analysis of a measure of variation,” Theory of Probability and Mathematical Statistics vol. 74 pp. 1–9, 2006.
D. Z. Arov and A. A. Bobrov, “The extreme terms of a sample and their role in the sum of independent variables,” Theory of Probability and its Applications vol. 5 pp. 377–396, 1960.
J. Beirlant, Y. Goegebeur, J. Segers, and J. L. Teugels, Statistics of Extremes: Theory and Applications, Wiley: Chichester, 2004.
V. E. Bening and V. Y. Korolev, Generalized Poisson Models and their Applications in Insurance and Finance, Modern Probability and Statistics 7, Brill: Leiden, 2002.
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press: Cambridge, 1987.
N. H. Bingham and J. L. Teugels, “Conditions implying domains of attraction.” In Proc. Sixth Conf. Probab. Th. (Braşov, 1979), pp. 23–34, Ed. Acad. R. S. România, Bucharest, 1981.
A. A. Bobrov, “The growth of the maximal summand in sums of independent random variables,” Math. Sb. Kievskogo Gosuniv. vol. 5 no. 2 pp. 15–38, 1954. (in Russian).
L. Breiman, “On some limit theorems similar to the arc-sin law,” Theory of Probability and its Applications vol. 10 no. 2 pp. 323–331, 1965.
T. L. Chow and J. L. Teugels, “The sum and the maximum of i.i.d. random variables.” In Proceedings of the Second Prague Symposium on Asymptotic Statistics (Hradec Králové, 1978), pp. 81–92. North-Holland, Amsterdam, 1979.
H. Cohn and P. Hall, “On the limit behavior of weighted sums of random variables,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 59 no. 3 pp. 319–331, 1982.
D. A. Darling, “The role of the maximum term in the sum of independent random variables,” Transactions of the American Mathematical Society vol. 73 pp. 95–107, 1952.
J. Dubourdieu, “Remarques relatives à la théorie mathématique de l’assurance-accidents,” Bulletin Trimestriel de l’Institut Actuaires Français vol. 44 pp. 79–146, 1938.
M. Dwass, “Extremal processes,” Annals of Mathematical Statistics vol. 35 pp. 1718–1725, 1964.
W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, Second edn, Wiley: New York, 1971.
A. Fuchs, A. Joffe, and J. L. Teugels, “Expectation of the ratio of the sum of squares to the square of the sum: exact and asymptotic results,” Theory of Probability and its Applications vol. 46 no. 2 pp. 243–255, 2001.
J. Grandell, Mixed Poisson Processes, Monographs on Statistics and Applied Probability 77, Chapman & Hall: London, 1997.
P. Hall, “On the extreme terms of a sample from the domain of attraction of a stable law,” Journal of the London Mathematical Society vol. 18 no. 2 pp. 181–191, 1978.
H.-C. Ho and T. Hsing, “On the asymptotic joint distribution of the sum and maximum of stationary normal random variables,” Journal of Applied Probability vol. 33 no. 1 pp. 138–145, 1996.
R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit, Modern Actuarial Risk Theory, Kluwer: Boston, 2001.
S. A. Ladoucette, “Asymptotic behavior of the moments of the ratio of the random sum of squares to the square of the random sum,” Statistics & Probability Letters, 2007. (in press)
S. A. Ladoucette and J. L. Teugels, “Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures,” Publications de l’Institut Mathématique, Nouv. Sér. vol. 80 no. 94 pp. 219–240, 2006.
S. A. Ladoucette and J. L. Teugels, “Reinsurance of large claims,” Journal of Computational and Applied Mathematics vol. 186 no. 1 pp. 163–190, 2006.
J. Lamperti, “On extreme order statistics,” Annals of Mathematical Statistics vol. 35 pp. 1726–1736, 1964.
B. F. Logan, C. L. Mallows, S. O. Rice, and L. A. Shepp, “Limit distributions of self-normalized sums,” Annals of Probability vol. 1 no. 5 pp. 788–809, 1973.
T. Mack, “Schadenversicherungsmathematik,” Verlag Versicherungswirtschaft e.V., Karlsruhe, 1997.
R. A. Maller and S. I. Resnick. “Limiting behaviour of sums and the term of maximum modulus,” Proceedings of the London Mathematical Society vol. 49 no. 3 pp. 385–422, 1984.
D. L. McLeish and G. L. O’Brien, “The expected ratio of the sum of squares to the square of the sum,” Annals of Probability vol. 10 no. 4 pp. 1019–1028, 1982.
G. L. O’Brien, “A limit theorem for sample maxima and heavy branches in Galton–Watson trees,” Journal of Applied Probability vol. 17 no. 2 pp. 539–545, 1980.
H. H. Panjer and G. E. Willmot, Insurance Risk Models, Society of Actuaries: Schaumburg, 1992.
W. E. Pruitt, “The contribution to the sum of the summand of maximum modulus,” Annals of Probability vol. 15 no. 3 pp. 885–896, 1987.
H. J. Rossberg, Über das asymptotische Verhalten der Rand -und Zentralglieder einer Variationsreihe I. Publication of Mathematics Institute Hungarica Academic Science, Ser. A vol. 8 pp. 463–468, 1963.
R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley: New York, 1980.
B. Smid and A. J. Stam, “Convergence in distribution of quotients of order statistics,” Stochastic Processes and Their Application vol. 3 no. 3 pp. 287–292, 1975.
J. L. Teugels and G. Vanroelen, “Convergence of ratios and differences of two order statistics,” Publications de l’Institut Mathématique Nouv. Sér. vol. 71(85) pp. 113–122, 2002.
A. Thépaut. “Une nouvelle forme de réassurance: le traité d’excédent du coût moyen relatif (ECOMOR),” Bulletin Trimestriel de l’Institut Actuaires Français vol. 49 pp. 273–343, 1950.
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Sophie A. Ladoucette is supported by the grant BDB-B/04/03 of the Catholic University of Leuven.
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Ladoucette, S.A., Teugels, J.L. Asymptotics for Ratios with Applications to Reinsurance. Methodol Comput Appl Probab 9, 225–242 (2007). https://doi.org/10.1007/s11009-007-9020-z
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DOI: https://doi.org/10.1007/s11009-007-9020-z
Keywords
- Limit theorems
- Functions of regular variation
- Domain of attraction of a stable law
- Order statistics
- Sum of i.i.d. random variables
- Dominance of summands
- Moments