Some Research On Multiple-Urn Models

The urn model is proposed originally by James Bernoulli in 1713.It is one of the simple model of random walks and has been applied in many fields like physics,biology and lemology.In this thesis,we extend the classic two-urn model to a K-urn case(K ? 2),which involves two parts specifically.In the first part,suppose that there are several red balls and white balls placed in every urn at the beginning.Then we perform the following action in each step: draw a certain number of balls randomly from the first urn to the last urn one by one,note their color,return these balls back and add a multiple number of balls to the next urn,which has the same color distribution as the balls extracted from the previous urn.Repeat the above action infinite times.This thesis proves that the fraction of red balls in each urn converges to a same limit almost surely and the limits of the expectation of these fractions are equal to each other through matrix computation and martingale convergence theory.In the second part,suppose that a specific number of balls are placed arbitrarily in these K urns at the beginning.Then at each step,one ball is chosen at random,removed from the current urn it resides in,and placed in one of the other K-1 urns equally likely.We use the method of stopping times to compute the expected hitting time of moving from an arbitrary given configuration to a different one.Besides,its corollary proves a conjecture recently proposed in Chen et al.[6].