×

zbMATH — the first resource for mathematics

On asymptotic expansion in the random allocation of particles by sets. (English) Zbl 1202.60037
Summary: We consider a scheme of equiprobable allocation of particles into cells by sets. An Edgeworth-type asymptotic expansion in the local central limit theorem for the number of empty cells left after allocation of all sets of particles is derived.
MSC:
60F05 Central limit and other weak theorems
60C05 Combinatorial probability
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bartlett, M.S.: The characteristic function of a conditional statistics. J. Lond. Math. 13, 62–67 (1938) · Zbl 0018.22503 · doi:10.1112/jlms/s1-13.1.62
[2] Bhattacharya, R.N., Ranga Rao, R.: Normal Approximation and Asymptotic Expansions. Wiley, New York (1976) · Zbl 0331.41023
[3] Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn. Wiley, New York (1968) · Zbl 0155.23101
[4] Frobenius, G.: Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzenberichte der Preussischen Akademie der Wissenschaften, pp. 829–830 (1910) · Zbl 1264.11013
[5] Gani, J.: Random allocation methods in an epidemic model. In: Cambanis, S., Ghosh, J.K., Karandikar, R.L., Sen, P.K. (eds.) Stochastic Processes: A Festachrift in Honour of Gopinath Kallianpur, pp. 97–106. Springer, New York (1993)
[6] Gani, J.: Random allocation and urn models. J. Appl. Probab. 41A, 313–320 (2004) · Zbl 1054.60046 · doi:10.1239/jap/1082552207
[7] Ivanov, V.A., Ivchenko, G.I., Medvedev, Yu.I.: Discrete problems of the probability theory (a survey). J. Sov. Math. 31(2), 2759–2795 (1985) · Zbl 0579.62001 · doi:10.1007/BF02116601
[8] Harper, L.: Stirling behavior is asymptotically normal. Ann. Math. Stat. 38, 410–414 (1967) · Zbl 0154.43703 · doi:10.1214/aoms/1177698956
[9] Holst, L.: A unified approach to limit theorems for urn models. J. Appl. Probab. 16(1), 154–162 (1979) · Zbl 0396.60027 · doi:10.2307/3213383
[10] Jonson, N.L., Kotz, S.: Urn Models and Their Applications. Wiley, New York (1977)
[11] Kolchin, V.F., Sevastyanov, B.A., Chistyakov, V.P.: Random Allocation. Wiley, New York (1978)
[12] Kotz, S., Balakrishnan, N.: Advances in urn models during the past two decades. In: Advances in Combinatorial Methods and Appl. to Probabl. and Statist., pp. 203–257. Birkhäuser, Basel (1997) · Zbl 0888.60014
[13] Mikhailov, V.G.: Asymptotic normality of the number of empty cells for group allocation of particles. Theory Probab. Appl. 25, 82–90 (1978) · Zbl 0455.60015 · doi:10.1137/1125007
[14] Mikhailov, V.G.: Convergence to multi-dimensional normal law in an equiprobable scheme for group allocation of particles. Math. USSR-St 39, 145–168 (1981) · Zbl 0463.60016 · doi:10.1070/SM1981v039n02ABEH001479
[15] Menezes, A., Van Qorshot, P., Vanstone, S.: Handbook of Applied Cryptology. CRC Press, New York (1997) · Zbl 0868.94001
[16] Mirakhmedov, Sh.A.: The estimations of the closeness to a normal law in the scheme without replacement. Theory Probab. Appl. 30, 427–439 (1985) · Zbl 0574.60028
[17] Mirakhmedov, Sh.A.: Asymptotical analysis of the conditional distributions of the sum of independent random variables and applications in the theory of decomposable statistics. Dissertation for Doctor of Sciences degree. Tashkent, Uzbekistan, 290 p. (1994)
[18] Mirakhmedov, Sh.A.: Limit theorems for conditional distributions. Discrete Math. Appl. 5, 107–132 (1995) · Zbl 0841.11064 · doi:10.1515/dma.1995.5.2.107
[19] Mirakhmedov, Sh.A.: Limit theorems on decomposable statistics in a generalized allocation schemes. Discrete Math. Appl. 6, 379–404 (1996) · Zbl 0868.60021 · doi:10.1515/dma.1996.6.4.379
[20] Mirakhmedov, Sh.M.: Asymptotic normality associated with generalized occupancy problem. Stat. Probab. Lett. 77, 1549–1558 (2007) · Zbl 1124.62006 · doi:10.1016/j.spl.2007.03.035
[21] Mukhin, A.B.: Local limit theorems for the distribution of a sum of independent random vectors. Theory Probab. Appl. 28, 360–366 (1984) · Zbl 0543.60025
[22] Park, C.J.: On the distribution of the number of unobserved elements when m samples of size n are drawn from a finite population. Commun. Stat. A10, 371–383 (1981) · Zbl 0454.62011 · doi:10.1080/03610928108828044
[23] Petrov, V.V.: Limit Theorems of Probability Theory. Oxford University Press, London (1995) · Zbl 0826.60001
[24] Vatutin, V.A., Mikhailov, V.G.: Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory Probab. Appl. 27, 734–743 (1982) · Zbl 0536.60017 · doi:10.1137/1127084
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.