Question: Random replacement II. As in the previous problem, consider an urn holding at most N balls, but now they come in two colours, either white

Random replacement II. As in the previous problem, consider an urn holding at most N balls, but now they come in two colours, either white or red. If the urn is non-empty, a ball is picked at random and is or is not replaced according to the outcome of the flip of a fair coin.

If the urn is empty, the coin is flipped to decide whether the urn should be filled again; if so, it is filled with N balls, each of which is white or red depending on the outcomes of further independent coin flips. Let Wn and Rn be the numbers of white and red balls, respectively, after performing this procedure n times. Show that Xn D .Wn;Rn/ is a Markov chain, and determine its asymptotic distribution.

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