Pitman-Rogers

Finite State Space Markov Chains Let $S$ be a finite set and $X_n$ be a Markov chain taking values in $S$ with transition matrix $P$. Let $S'$ be another finite set with $|S'| < |S|$ and $\varphi : S \to S'$, which we will assume is onto without loss of generality.

Moving to Continuous Time The technicalities of the Pitman-Rogers theorem are easiest to handle for discrete time Markov chains taking values in a discrete state space. But generalizing the result is not particularly difficult.

Modulus of Standard $d$-dimensional Brownian Motion Let $(B_t)_{t \geq 0}$ be a standard $d$-dimensional Brownian motion. It is a well known fact that the one-dimensional process $t \mapsto |B_t|$ is a Bessel process of dimension $d$.