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On the Modular Value and Fractional Part of a Random Variable

Published online by Cambridge University Press:  27 July 2009

Stephen J. Herschkorn
Stephen J. Herschkorn
Affiliation:
School of Business and RUTCOR, Rutgers University, New Brunswick, New Jersey 08903
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Abstract

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.

When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.

Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 4 , October 1995 , pp. 551 - 562
Copyright
Copyright © Cambridge University Press 1995

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References

1.Drake, A.W. (1967). Fundamentals of applied probability theory. New York: McGraw-Hill.Google Scholar
2.Feller, W. (1971). An introduction to probability theory and its applications, Vol. II. New York: John Wiley and Sons.Google Scholar
3.Katznelson, Y. (1976). An introduction to harmonic analysis, 2nd corrected ed. New York: Dover.Google Scholar
4.Körner, T.W. (1988). Fourier analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
5.Mardia, K.V. (1972). Statistics of directional data. London: Academic Press.Google Scholar
6.Oppenheim, A.V. & Shafer, R.W. (1975). Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
7.Papoulis, A. (1962). The Fourier integral and its applications. New York: McGraw-Hill.Google Scholar
8.Rees, C.S., Shah, S.M., & Stanojević, C.V. (1981). Theory and applications of Fourier analysis. New York: Marcel Dekker.Google Scholar
9.Schatte, P. (1973). Zür Verteilungen der Mantisse in der Gleitkommadarstellung einer Zufallsgröβe. Zeitschrift für Angewandte Mathematik und Mechanik 53: 553–565.CrossRefGoogle Scholar
10.Schatte, P. (1983). On sums modulo 2π of independent random variables. Mathematische Nachrichten 110: 243–262.CrossRefGoogle Scholar
11.Stadje, W. (1984). Wrapped distributions and measurement errors. Metrika 31: 303–317.Google Scholar
12.Zygmund, A. (1959). Trigonometric series, Vol. I, 2nd ed.Cambridge: Cambridge University Press.Google Scholar