= Markov chain obtained from independent innovations
{c}
{title2=$Z_{n+1}=h(Z_n,\xi_{n+1})$}
Let $\xi_1,\xi_2,\ldots$ 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>.
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