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In this talk we will present a new urn model consisting balls of infinite
but countably many colors which we index by the integers. We will consider
two special replacement matrices, one arriving from the right shift
operator, and the other arriving from the simple symmetric random walk on
the one dimensional integer lattice. We show using martingale techniques
that in both the cases the proportion of colors converges weakly almost
surely to a standard normal distribution after an appropriate centering
and scaling by $\sqrt{\log n}$. This shows that even though the associated
Markov chain has different qualitative properties, namely one is transient
and the other is null recurrent, the infinite color urn models have same
asymptotic behavior. This is in sharp contrast to what is generally
observed for finite color urn models.
As an anecdote we will present a new proof of an well known number
theoretic
result first derived by Euler relating the Euler constant and Riemann Zeta
function.
[This is a joint work with Antar Bandyopadhyay]
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