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Abstract:
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In this work we will discuss the continuous Polya process originating out of an urn with white and blue balls with random addition of balls at each renewal epoch. The dynamics of the process is driven by a partial differential equation for which, in general a closed form solution is difficult to obtain. We consider a special case of a Geometric(1/2) addition of balls at each renewal instant, which gives rise to a nice closed form solution. In particular, the distribution of the number of white balls (or blue balls) by time t, scaled by a suitable function, approaches a distribution whose moments are given by the famous Catalan numbers. We will also discuss extensions to multicolor cases and the challenges in the general case.
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