$Y_0, Y_1,Y_2,\dots$ are independent and identically distributed random non-negative integer outcomes. Let X_0 = Y_0 $
Let $X_0 = Y_0$ and $X_n = X_{n-1} - Y_n$ if $X_{n-1}>0$, else $X_n = X_{n-1} + Y_n$.
Show $X_1,X_2,\dots$ is a homogeneous Markov chain. Any suggestions?
Edit: Here is my approach. I am not sure how sound it is but it's hopefully on the right path:
To prove it's time homogeneous, need to show :
$P(X_{n+1}=j\mid X_n=i) = P(X_1=j\mid X_0=i)$ (by definition)
$P(X_1=j\mid X_0=i)=P(X_1=X_0-Y_1=j\mid X_0=i)=P(i-Y_1=j\mid i)$
if $(i>0)$ $P(Y_1=j-i)$, else $P(Y_1=i-j)$ which are equals.
Similarly for $P(X_{n+1}=j\mid X_n=i)$ =>
if $(i>0)$ $P(Y_n=j-i)$ else $P(Y_n=i-j)$
Because $Y_1,Y_n$ are IID though, these probabilities are the same.
Define the function $f(x,y)=x+\left({\bf 1}_{(-\infty,0]}(x)-{{\bf 1}_{(0,\infty)}(x)}\right)y.$ Then $X_n=f(X_{n-1},Y_n)$ for $n>0$, and if $({\cal F}_n)_{n\geq 0}$ is the filtration generated by $(X_n)_{n\geq 0}$, then $$\mathbb{P}(X_n\in B\mid {\cal F}_{n-1})= \mathbb{P}(f(X_{n-1},Y_n)\in B\mid {\cal F}_{n-1})=\mu(X_{n-1},B),$$ where we define the kernel $\mu(x,B)=\mathbb{P}(f(x,Y)\in B)$ for any random variable $Y$ distributed like every $Y_n$. This shows that $(X_n)_{n\geq 0}$ is Markov with transition kernel $\mu$.