I have a trouble understanding the random walk, where $/xi_1,...,/xi_n$ is iid integer valued rv with the probability mass function $f(x)$.
I want to get the expression $p(x,y) = f(y-x)$.
$p(x,y)= P(X_1 = y | X_0 = x) = P(X_0 + \xi_1 = y | X_0 = 1) = P(\xi_1 = y-X_0 | X_0=1) = f(y-x)$.
I could understand up to the last equality. Using the definition of conditional probability, $P(\xi_1=y-X_0 | X_0=1) = P(\xi=y-X_0 and X_0=1)/P(X_0=1)$ but I cannot connect to the final result mathematically rigorously.
I came into a Markov chain class without much exposure to probability theory so I think I lack some understanding that was assumed.
Any help would be appreciated thanks!