Let $(X_n)_{n≥1}$ be i.i.d random variables such that $X_1 ∼ Binom(r,p)$. Prove or refute that we have a Markov chain, and in the affirmative case give the state space and the transition matrix.
The length of the current run of equal values, that is,
$S_n = \max\limits_k\{X_i = X_n, \forall i = n − 1, . . . , n − k + 1\}$
What I got,
State space is $\mathbb N$
Abbreviate $P(X_i=k)=\binom{r}{k}p^k(1-p)^{r-k}=:p_k$ then,
$S_{n+1}=\begin{cases} S_n, &\text{if}\ X_{n+1}\neq X_{n} \\ S_n+1, &\text{if}\ X_{n+1}= X_{n} \end{cases}$
$P(S_{n+1}=j|S_n=i)=p_{ij}=\begin{cases} \sum\limits_{k=0}^r p^2_k, &\text{if}\ j=i+1 \\ 1-\left(\sum\limits_{k=0}^r p^2_k\right),&\text{if}\ j=1\\ 0,& otherwise \end{cases}$
It looks terrible, can this be correct ?