Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T12:01:14.795Z Has data issue: false hasContentIssue false
  • English
  • Français

Moment convergence in conditional limit theorems

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Svante Janson
Svante Janson*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante.janson@math.uu.se
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

Keywords

conditional distributionlimit theoremmoment convergenceoccupancyhashingrandom forestbranching process

MSC classification

Primary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 421 - 437
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chassaing, P., and Louchard, G. (1999). Phase transition for parking blocks, Brownian excursion and coalescence. Preprint, Institut Elie Cartan. Available at http://altair.iecn.u-nancy.fr/~chassain/.Google Scholar
Chibisov, D. M. (1972). On the normal approximation for a certain class of statistics. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. I. University of California Press, Berkeley, pp. 153–174.Google Scholar
Flajolet, P., Poblete, P., and Viola, A. (1998). On the analysis of linear probing hashing. Algorithmica 22, 490–515.CrossRefGoogle Scholar
Hipp, C. (1984). Asymptotic expansions for conditional distributions: the lattice case. Prob. Math. Statist. 4, 207–219.Google Scholar
Holst, L. (1979). Two conditional limit theorems with applications. Ann. Statist. 7, 551–557.CrossRefGoogle Scholar
Holst, L. (1979). A unified approach to limit theorems for urn models. J. Appl. Prob. 16, 154–162.CrossRefGoogle Scholar
Holst, L. (1981). Some conditional limit theorems in exponential families. Ann. Prob. 9, 818–830.CrossRefGoogle Scholar
Ismatullaev, Sh. A. (1987). Asymptotic expansions for conditional distributions. Dokl. Akad. Nauk UzSSR 1987, 12–15 (in Russian).Google Scholar
Janson, S. (2001). Asymptotic distribution for the cost of linear probing hashing. To appear in Random Structures Algorithms.CrossRefGoogle Scholar
Knuth, D. E. (1998). The Art of Computer Programming, Vol. 3, Sorting and Searching, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar
Kolchin, V. F. (1984). Random Mappings. Nauka, Moscow (in Russian). English translation: Optimization Software, New York, 1986.Google Scholar
Kolchin, V. F., Sevastyanov, B. A., and Chistyakov, V. P. (1976). Random allocations. Nauka, Moscow (in Russian). English translation: Winston, Washington, DC, 1978.Google Scholar
Kudlaev, E. M. (1984). Conditional limit distributions of sums of random variables. In Teor. Veroyat. Primen. 29, 743–752 (in Russian). English translation: Theory Prob. Appl. 29, 776–786.Google Scholar
Michel, R. (1979). Asymptotic expansions for conditional distributions. J. Multivar. Analysis 9, 393–400.CrossRefGoogle Scholar
Mirakhmedov, Sh. A. (1994). Limit theorems for conditional distributions. Diskret. Mat. 6, 107–132 (in Russian). English translation: Discrete Math. Appl. 4, 519–542.Google Scholar
Pavlov, Yu. L. (1977). Limit theorems for the number of trees of a given size in a random forest. Mat. Sb. 103(145), 392–403, 464 (in Russian). English translation: Math. USSR Sb. 32, 335–345.Google Scholar
Pavlov, Yu. L. (1996). Random Forests. Karelian Research Centre, Russian Academy of Science, Petrozavodsk (in Russian). English translation: VSP, Zeist, 2000.Google Scholar
Pitman, J. (1998). Enumerations of trees and forests related to branching processes and random walks. In Microsurveys in discrete probability, eds Aldous, D. and Propp, J. American Mathematical Society, Providence, RI, pp. 163–180.CrossRefGoogle Scholar
Portnoy, S. (1977). Asymptotic efficiency of minimum variance unbiased estimators. Ann. Statist. 5, 522–529.CrossRefGoogle Scholar
Rényi, A. (1962). Three new proofs and a generalization of a theorem of Irving Weiss. Magyar Tud. Akad. Mat. Kutató Int. Közl. 7, 203–214.Google Scholar
Steck, G. P. (1957). Limit theorems for conditional distributions. Univ. California Publ. Statist. 2, 237–284.Google Scholar
Swensen, A. R. (1983). A note on convergence of distributions of conditional moments. Scand. J. Statist. 10, 41–44.Google Scholar
Weiss, I. (1958). Limiting distributions in some occupancy problems. Ann. Math. Statist. 29, 878–884.CrossRefGoogle Scholar
Wendel, J. G. (1975). Left-continuous random walk and the Lagrange expansion. Am. Math. Monthly 82, 494–499.CrossRefGoogle Scholar
Zabell, S. L. (1980). Rates of convergence for conditional expectations. Ann. Prob. 8, 928–941.CrossRefGoogle Scholar
Zabell, S. L. (1993). A limit theorem for expectations conditional on a sum. J. Theoret. Prob. 6, 267–283.CrossRefGoogle Scholar