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A partition of a Markov chain state space is strongly lumpable when, for every target block $C$, the sum ${\sum }_{j\in C}{p}_{ij}$ is identical for all source states $i$ in a given source block. These sums define the transition matrix of the block-valued Markov chain, independently of the initial law. Failure of this condition means the projection is not guaranteed to be a homogeneous Markov chain for every initial law, although special initial laws may still produce a Markov process.

A randomized stopping time $\tau$ is a strong stationary time for a Markov chain with stationary distribution $\pi$ when $X_{\tau }$ has distribution $\pi$ and is independent of $\tau$. Equivalently,

Thinning keeps every $L$th state of a Markov chain. The retained chain has transition kernel $K^L$ and the same stationary distribution. It is generally still dependent, and reducing correlation between retained draws does not by itself improve precision per original transition. Keep all post-warmup draws unless storage or later processing requires thinning; assess precision using the effective sample size of a Markov chain.

For a stationary Markov chain with transition matrix $P$ and stationary distribution $\pi$, the reversed transition matrix isIt is stochastic and has the same stationary distribution.

A state is transient when its probability of ever returning is less than one.

The transition probability $P_{ij}$ is the conditional probability that a Markov chain moves to state $j$ on its next step given that its current state is $i$.