Markov chain obtained from independent innovations ($Z_{n+1}=h(Z_n,{\xi }_{n+1})$)

Let ${\xi }_1,{\xi }_2,\dots$ be independent and identically distributed random variables, independent of an initial state $Z_0$, and let $h$ be a measurable function. The displayed recursion defines a time-homogeneous Markov chain: conditioning on the state history leaves ${\xi }_{n+1}$ with its original probability distribution, so its next-state law depends only on $Z_n$. Its Markov kernel is $K(z,A)=\mathbb{P}(h(z,{\xi }_1)\in A)$. Running sums use $h(z,x)=z+x$; running maxima use $h(z,x)=\max (z,x)$. Independence from the complete past, rather than merely from the current state, supplies the Markov property.