Why is a random walk $(X_n)_{n\in \mathbb{N}_0}$ on $\mathbb{R}^d $(defined below) a time-homogeneous Markov process? In particular, why does $(X_n)$ satisfy requirement 3 of Def17.3?
Definition of $(X_n)_{n\in \mathbb{N}_0}$
Let $(Y_n)_{n\in \mathbb{N}}$ be i.i.d $\mathbb{R}^d$valued random variables and let
$$S^x_0=x,\;\;\;\;\;S^x_n=x+\sum_{k=1}^n Y_k\;\;\;(n\in \mathbb{N})\;\;\;\;x\in \mathbb{R}^d $$
Define probability measures $P_x
on ((\mathbb{R}^d)^{\mathbb{N}_0},(B(\mathbb{R}^d)^{\otimes \mathbb{N}_0}))$
by $P_x=P\circ (S^x)^{−1}$
.
Then the canonical process $$X_n:(\mathbb{R}^d)^{\mathbb{N}_0}\ni x=(x_k)_{k\in \mathbb{N}_0}\mapsto x_n\in \mathbb{R}^d$$
is a Markov chain with distributions $(P_x)_{x\in \mathbb{R}^d}$
.