THE PROCESS:
Consider an urn which initially has $w$ white balls and $b$ black balls. Make a sequence of draws from the urn as follows. Draw one of the balls in the urn at random, then put this ball back into the urn and add another ball to the urn of the same color of the one just drawn. Continue to draw and add balls in this manner indefinitely. Let $X_n$ = 1 if the $n^{th}$ ball drawn is white and $X_n$ = 0 if it is black.
Let $W_n$ and $B_n$ be the number of white and black balls in the Polya urn after the n-th stage of the process. Thus, $W_0$ = $w$ and $B_0$ = $b$. Is $Y_n$ = ($W_n$, $B_n$) a Markov process? If so identify the state space and give a formula for the transition probabilities.
My work:
$Y_n$ has to be a Markov chain because by the formal definition -they are processes where the distribution of the future steps depends only on the present state of the process.
It's obvious that $$P(Y_{n+1}=j/Y_{n}=i) = P(Y_{n+1}=j/Y_{n}=i,Y_{n-1}=i')$$
because I don't need any information from $Y_{n-1}$ to calculate $P(Y_{n+1})$ because if I know $Y_n$ that's enough. Is my reasoning enough?
state-space will be something of the form (w,b) -----> (w+n,b) and (w,b)--->(w,b+n) = all the combinations. And the transition matrix will be a n*n matrix?
Note that always $W_n + B_n = w + b + n$, since one ball is added at a time. So one need only track the number of white balls (or black balls). Each draw is independent of the past, so you dd indeed get a Markov chain. Precisely, for an event $H$ depending on the history (ie $W_0, ..., W_1$), we have $$ \Pr( W_{n+1} = y \mid W_n = x, \: H ) = \Pr( W_{n+1} = y \mid W_n = x ) =: p_n(x,y). $$ Note that this is a time-inhomogenous Markov chain. We need $y \in \{x,x+1\}$. We have $$ p_n(x,x) = \Pr(\text{draw black ball} \mid W_n = x) = 1 - x/(w+b+n). $$