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A conditional Berry–Esseen inequality

Part of: Distribution theory Combinatorial probability Limit theorems

Published online by Cambridge University Press:  12 July 2019

Thierry Klein ,
Agnés Lagnoux  and
Pierre Petit
Thierry Klein*
Affiliation:
ENAC and Institut de Mathématiques de Toulouse (UMR CNRS 5219)
Agnés Lagnoux*
Affiliation:
Institut de Mathématiques de Toulouse (UMR CNRS 5219)
Pierre Petit*
Affiliation:
Institut de Mathématiques de Toulouse (UMR CNRS 5219)
*
*Postal address: ENAC, 7 Avenue Edouard Belin, F-31400 Toulouse, France. Email address: thierry.klein@math.univ-toulouse.fr
**Postal address: Institut de Mathématiques de Toulouse (UMR CNRS 5219), Université Toulouse 2, 5 Allées Antonio Machado, F-31058 Toulouse, France.
**Postal address: Institut de Mathématiques de Toulouse (UMR CNRS 5219), Université Toulouse 2, 5 Allées Antonio Machado, F-31058 Toulouse, France.
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Abstract

As an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.

Keywords

Berry–Esseen inequalityconditional distributioncombinatorial problemoccupancyhashing with linear probingrandom forestbranching processBose–Einstein statistics

MSC classification

Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory
Secondary: 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 76 - 90
Copyright
© Applied Probability Trust 2019 

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References

Chassaing, P. and Flajolet, P. (2003). Hachage, arbres, chemins & graphes. Gaz. Math., 29–49.Google Scholar
Chassaing, P. and Louchard, G. (2002). Phase transition for parking blocks, Brownian excursion and coalescence. Random Struct. Algorithms 21, 76–119.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Flajolet, P., Poblete, P. and Viola, A. (1998). On the analysis of linear probing hashing. Algorithmica 22, 490–515.CrossRefGoogle Scholar
Holst, L. (1979). Two conditional limit theorems with applications. Ann. Statist. 7, 551–557.CrossRefGoogle Scholar
Janson, S. (2001). Asymptotic distribution for the cost of linear probing hashing. Random Struct. Algorithms 19, 438–471.CrossRefGoogle Scholar
Janson, S. (2001). Moment convergence in conditional limit theorems. J. Appl. Prob. 38, 421–437.CrossRefGoogle Scholar
Janson, S. (2005). Individual displacements for linear probing hashing with different insertion policies. ACM Trans. Algorithms 1, 177–213.CrossRefGoogle Scholar
Knuth, D. E. (1974). Computer science and its relation to mathematics. Amer. Math. Monthly 81, 323–343.CrossRefGoogle Scholar
Knuth, D. E. (1998). Linear probing and graphs. Algorithmica 22, 561–568.CrossRefGoogle Scholar
Kolchin, V. F. (1986). Random Mappings, Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York.Google Scholar
Loève, M. (1955). Probability Theory: Foundations, Random Sequences. Van Nostrand, New York.Google Scholar
Marckert, J.-F. (2001). Parking with density. Random Struct. Algorithms 18, 364–380.CrossRefGoogle Scholar
Pavlov, Y. L. (1977). Limit theorems for the number of trees of a given size in a random forest. Mat. Sb. (N.S.) 103 (145), 392–403, 464.Google Scholar
Pavlov, Y. L. (1997). Random forests. In Probabilistic Methods in Discrete Mathematics (Petrozavodsk, 1996), VSP, Utrecht, pp. 11–18.Google Scholar
Quine, M. P. and Robinson, J. (1982). A Berry–Esseen bound for an occupancy problem. Ann. Prob. 10, 663–671.CrossRefGoogle Scholar
Wendel, J. G. (1975). Left-continuous random walk and the Lagrange expansion. Amer. Math. Monthly 82, 494–499.CrossRefGoogle Scholar