= Constant nonzero drift is impossible on a finite Markov chain
{title2=$Pf-f\not\equiv-\delta\quad(\delta>0)$}
For a real function $f$ on a finite state space, a state minimizing $f$ satisfies $Pf\geq f$, so strictly negative drift cannot hold at every state. Equivalently, telescoping the <expected value> would give $\mathbb E_x f(X_n)=f(x)-n\delta$, contradicting boundedness of $f$. Constant negative drift is useful before a target is reached; it must be restricted to the states outside that target.
Back to article page
