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!
Your equations are not correct. They should be as follows: $$ \begin{align} p(x,y)&\triangleq \mathbb{P}(X_1=y|X_0=x)\\ &= \mathbb{P}(X_0+\Xi_1=y|X_0=x)\\ &=\mathbb{P}(\Xi_1=y-x|X_0=x)\\ &\overset{(a)}{=}\mathbb{P}(\Xi_1=y-x)\\ &=f_\Xi(y-x), \end{align} $$ where $(a)$ follows since $\Xi_1$ is independent of $X_0$.