Theorem (Convergence to Equilibrium — discrete)

Let $\{X_{i}\}^{\infty}_{i=0}$ be a MC with an arbitrary initial distribution $\lambda$ and transition matrix $P$. If $P$ is

1. irreducible
2. aperiodic
3. has an invariant distribution (or is positive recurrent) then $$\sup_{j\in S}|P(X_{n}=j)-\pi_{j}|\to0,\mbox{ as }n\to\infty$$.

Theorem (Convergence to Equilibrium — continuous)

Let $\{X_{t}:t\ge0\}\sim\mbox{Markov}(\lambda,Q)$. If the jump chain $\{Y_{n}:n\ge0\}$ is irreducible, and $Q$ has an invariant distribution $\pi$, then for all $i,j\in I$, $$\lim_{t\to\infty}p_{ij}(t)=\pi_{j}$$