Question 5
Let $Y_0, Y_1,...$ be a sequence of independent, identically distributed random variable on $Z$ such that
$P(Y_n = 1) = P(Y_n = -1) = 1/2$, $n = 0, 1, ...$
Consider the Stochastic process {$X_n$}$_{n \ge 0}$ given by
$X_n = \frac{Y_n + Y_{n+1}}{2}$, $n = 0, 1, ...$
(5.1) Find the transition probabilities
$p_{jk}(m, n) = P(X_n = k | X_m = j)$ for $m < n$ and $j, k = -1, 0, 1$.
(5.2) Draw the transition probability graph for the process.
(5.3) Show that the Chapman-Kolmogorov equations are not satisfied, and that consequently
{$X_n$}$_{n \ge 0}$ is not a homogeneous Markov chain.