Kolmogorov's criterion
# Kolmogorov's criterion

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem giving a necessary and sufficient condition for a Markov chain or continuous-time Markov chain to be stochastically identical to its time-reversed version.

## Discrete-time Markov chains

The theorem states that an irreducible, positive recurrent, aperiodic Markov chain with transition matrix P is reversible if and only if its stationary Markov chain satisfies

    $p_{j_{1}j_{2}}p_{j_{2}j_{3}}\cdots p_{j_{n-1}j_{n}}p_{j_{n}j_{1}}=p_{j_{1}j_{n}}p_{j_{n}j_{n-1}}\cdots p_{j_{3}j_{2}}p_{j_{2}j_{1}}$

for all finite sequences of states

    $j_{1},j_{2},\ldots ,j_{n}\in S.$

Here $p_{ij}$ are components of the transition matrix P, and S is the state space of the chain.

### Example

Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,

    $p_{ij}p_{jl}p_{lk}p_{ki}=p_{ik}p_{kl}p_{lj}p_{ji}.$

## Proof

Let $X$ be the Markov chain and denote by $\pi$ its stationary distribution (such exists since the chain is positive recurrent).

If the chain is reversible, the equality follows from the relation $p_{ji}={\frac {\pi _{i}p_{ij}}{\pi _{j}}}$.

Now assume that the equality is fulfilled. Fix states $s$ and $t$. Then

    ${\text{P}}(X_{n+1}=t,X_{n}=i_{n},\ldots ,X_{0}=s|X_{0}=s)$$=p_{si_{1}}p_{i_{1}i_{2}}\cdots p_{i_{n}t}$$={\frac {p_{st}}{p_{ts}}}p_{ti_{n}}p_{i_{n}i_{n-1}}\cdots p_{i_{1}s}$$={\frac {p_{st}}{p_{ts}}}{\text{P}}(X_{n+1}$$=s,X_{n}$$=i_{1},\ldots ,X_{0}=t|X_{0}=t)$.

Now sum both sides of the last equality for all possible ordered choices of $n$ states $i_{1},i_{2},\ldots ,i_{n}$. Thus we obtain $p_{st}^{(n)}={\frac {p_{st}}{p_{ts}}}p_{ts}^{(n)}$ so ${\frac {p_{st}^{(n)}}{p_{ts}^{(n)}}}={\frac {p_{st}}{p_{ts}}}$. Send $n$ to $\infty$ on the left side of the last. From the properties of the chain follows that $\lim _{n\to \infty }p_{ij}^{(n)}=\pi _{j}$, hence ${\frac {\pi _{t}}{\pi _{s}}}={\frac {p_{st}}{p_{ts}}}$ which shows that the chain is reversible.

## Continuous-time Markov chains

The theorem states that a continuous-time Markov chain with transition rate matrix Q is, under any invariant probability vector, reversible if and only if its transition probabilities satisfy

    $q_{j_{1}j_{2}}q_{j_{2}j_{3}}\cdots q_{j_{n-1}j_{n}}q_{j_{n}j_{1}}=q_{j_{1}j_{n}}q_{j_{n}j_{n-1}}\cdots q_{j_{3}j_{2}}q_{j_{2}j_{1}}$

for all finite sequences of states

    $j_{1},j_{2},\ldots ,j_{n}\in S.$

The proof for continuous-time Markov chains follows in the same way as the proof for discrete-time Markov chains.

