8.5.4 In the Ehrenfest urn model (see Chapter 3, Section 3.3.2) for molecular diffusion through a membrane, if there are i particles in urn A, the probability that there will be iC1 after one time unit is 1????i=.2N/, and the probability of i????1 is i=.2N/, where 2N is the aggregate number of particles in both urns. Following Chapter 3, Section 3.3.2, let Yn be the number of particles in urn A after the nth transition, and let Xn D Yn ????N. Let 1X D XnC1 ????Xn be the change in urn composition. The probability law is

We anticipate a limiting process in which the time between transitions becomes small and the number of particles becomes large. Accordingly, let 1t D 1=N and measure fluctuations of a rescaled process in units of order 1= p N. The definition of the rescaled process is

Note that in the duration t D 0 to t D 1 in the rescaled process, there are N transitions in the urns, and a unit change in the rescaled process corresponds to a fluctuation of order p N in the urn composition. Let 1V D VN.tC1=N/????VN.t/
be the displacement in the rescaled process over the time interval 1t D 1=N.
Show that

and that .1V/2 D 1=N, whence

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x Pr{AX = 1|X = x} = F 22N