Pólya urn model

Example

Let $a \ge 1$ and $b \ge 1$ be integers. An urn holds $a$ red balls and $b$ black balls. At each step one draws a ball uniformly at random and notes its colour. Then one puts it back together with one more ball of the same colour. Let $R_n$ and $B_n$ be the numbers of red and black balls after $n$ steps, so $R_n + B_n = a + b + n$. Then $(R_n, B_n)$ is a Markov chain, since the next draw depends only on the current contents. From $(r, k)$ it moves to $(r + 1, k)$ with probability $r/(r + k)$ and to $(r, k + 1)$ otherwise.

The fraction $M_n = R_n / (a + b + n)$ of red balls is a martingale: $$\mathbb{E}[M_{n+1} \mid R_n, B_n] = \frac{R_n}{R_n + B_n} \cdot \frac{R_n + 1}{R_n + B_n + 1} + \frac{B_n}{R_n + B_n} \cdot \frac{R_n}{R_n + B_n + 1} = \frac{R_n}{R_n + B_n} = M_n.$$ It takes values in $[0, 1]$, so it converges almost surely, and its limit has the beta distribution with parameters $a$ and $b$.