Math 6040 • Probability

At time \(n = 0\), an urn contains \(B_{0} = 1\) blue, and \(R_{0} = 1\) red ball. At each time
\(n = 1, 2, 3, \ldots\), a ball is chosen at random from the urn and returned to the urn,
together with a new ball of the same color. Let \(B_{n}\) and \(R_{n}\) be the number of blue,
respectively, red balls in the urn after the \(n\)-th turn of this procedure. (Note that

\[
B_{n} + R_{n} = n + 2.
\]
Denote by \(\mathcal{F}_{n} = \sigma(B_{j}, 0 \le j \le n) = \sigma(R_{j}, 0 \le j \le n)\), \(n \ge 0\), the

natural filtration of the process. Let

\[
M_{n} = \frac{B_{n}}{B_{n} + R_{n}} = \frac{B_{n}}{n+2}
\]

be the proportion of blue balls in the urn just after time \(n\).
\begin{enumerate}

\item
Show that \((M_{n})_{n \ge 0}\) is a martingale in the filtration \((\mathcal{F}_{n})_{n \ge 0}\).


\item
Show that \(P(B_{n} = k) = 1/(n + 1)\) for all integers \(k \in [1, n + 1]\).
(Hint: Write down the probability of choosing \(k\) blue and \(n - k\) red balls in
whatever fixed order [hence, ending up with \(k + 1\) blue and \(n - k + 1\) red balls].)


\item
Show that \(M_{\infty} = \lim_{n \to \infty} M_{n}\) exists almost surely and compute its distribution.
(Hint: use part (b).)

\end{enumerate}