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REPLENISHMENT-DEPLETION URN IN EQUILIBRIUM

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L. R. SHENTON^{1} AND K. O. BOWMAN^{2}

^{1} *Department of Statistics, University of Georgia, Athens, GA 30603, U.S.A.*

^{2} *Computer Science and Mathematics Division, Oak Ridge National Laboratory,*

P.O.Box 2008, Oak Ridge, TN 37831-6367, U.S.A.
(Received April 11, 1996; revised January 10, 1997)

**Abstract.**
An urn has balls of colors *C*_{1}
and *C*_{2}. It is replenished (*R*) by balls of both
colors and then depleted by (*D*) the same number; this
constitutes a cycle. When *R* = *D*, the system is closed
and equilibrium will be reached after many cycles. The
ultimate distribution is found only when the
replenishment is the same for each color. Asymptotic
normal and asymptotic binomial distributions arise when
the parameters reach extreme values. For the multicolor
urn an expression is given for the correlation between
the number of balls of any two
colors.

*Key words and phrases*:
Bernard's urn, beta
integral transforms, finite difference calculus,
generating functions, hypergeometric distributions,
hypergeometric functions, moments,
replenishment-depletion urn.

**Source**
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