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A batch-ballot problem and applications

Published online by Cambridge University Press:  14 July 2016

Haim Mendelson
Haim Mendelson*
Affiliation:
The University of Rochester
*
∗Postal address: The Graduate School of Management, The University of Rochester, Rochester NY 14627, U.S.A.
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Abstract

Let (X1, X2, · ··, Xn) be a vector of discrete splittable and interchangeable random variables, and let Sj = Σji = 1Xi. This paper is concerned with probabilities of the form P{Sj < jb – d for j = 1, 2, 3, ···, n | Sn = r}. We focus on applications of these probabilities, study their properties and suggest how they may be computed. The applications are from the areas of computer science, economics, operations management, finance and risk theory.

Keywords

BALLOT THEOREMINTERCHANGEABLE RANDOM VARIABLESHASHINGMARKET MECHANISMSSAFETY STOCK
Type
Research Paper
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 144 - 157
Copyright
Copyright © Applied Probability Trust 1982 

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References

Bertrand, J. L. F. (1887) Calcul des probabilités. Solution d'un problème. C.R. Acad. Sci. Paris 105, 369.Google Scholar
Blake, I. F. and Konheim, A. G. (1977) Big buckets are (are not) better! J. Assoc. Comput. Mach. 24, 591–606.Google Scholar
Bühlman, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag, Berlin.Google Scholar
Garman, M. (1976) Market microstructure. J. Financial Economics 3, 256–275.Google Scholar
Gould, J. P. (1978) Inventories and stochastic demand: equilibrium models of the firm and industry. J. Business 51, 1–42.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Discrete Distributions. Houghton Mifflin, Boston.Google Scholar
Keilson, J. and Gerber, H. (1971) Some results for discrete unimodality. J. Amer. Statist. Soc. 66, 386–389.Google Scholar
Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
Keilson, J. and Sumita, U. (1980) Uniform stochastic ordering.Google Scholar
Kim, E. H. (1978) A mean-variance theory of optimal capital structure and corporate debt capacity. J. Finance 33, 45–64.Google Scholar
Knuth, D. E. (1973) The Art of Computer Programming, Vol. 3 (Sorting and Searching). Addison-Wesley, Reading, Ma.Google Scholar
Knuth, D. E. (1974) Computer science and its relation to mathematics. Amer. Math. Monthly 81, 323–343.CrossRefGoogle Scholar
Kraus, A. and Litzenberger, R. (1973) A state-preference model of optimal financial leverage. J. Finance 28, 911–922.Google Scholar
Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.Google Scholar
Mendelson, H. and Yechiali, U. (1980) A new approach to the analysis of linear probing schemes.Google Scholar
Seal, H. L. (1966) The random walk of a simple risk business. Austin Bulletin 4, 19–28.Google Scholar
Seal, H. L. (1969) Stochastic Theory of a Risk Business. Wiley, New York.Google Scholar
Seal, H. L. (1972) Risk theory and the single-server queue. Mitt. Verein. Schweiz. Versicherungsmath. 171–178.Google Scholar
Stiglitz, J. E. (1972) Some aspects of the pure theory of corporate finance: bankruptcies and take-overs. Bell J. Economics and Management Sci. 3, 458–482.CrossRefGoogle Scholar
Takács, L. (1961) The probability law of the busy period for two types of queueing processes. Operat. Res. 9, 402–407.Google Scholar
Takács, L. (1964) Combinatorial methods in the theory of queues. Rev. Internat. Statist. Inst. 32, 207–219.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
Takács, L. (1976) On fluctuation problems in the theory of queues. Adv. Appl. Prob. 8, 548–583.Google Scholar
Vinso, J. D. (1979) A determination of the risk of ruin. J. Financial and Quantitative Analysis 14, 77–100.CrossRefGoogle Scholar