# Stationary distribution

2010 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]

A probability distribution for a homogeneous Markov chain that is independent of time. Let $\xi ( t)$ be a homogeneous Markov chain with set of states $S$ and transition probabilities $p _ {ij} ( t) = {\mathsf P} \{ \xi ( t) = j \mid \xi ( 0) = i \}$. A stationary distribution is a set of numbers $\{ {\pi _ {j} } : {j \in S } \}$ such that

$$\tag{1 } \pi _ {j} \geq 0 ,\ \sum _ {j \in S } \pi _ {j} = 1,$$

$$\tag{2 } \sum _ {i \in S } \pi _ {i} p _ {ij} ( t) = \pi _ {j} ,\ j \in S ,\ t > 0.$$

The equalities (2) signify that a stationary distribution is invariant in time: If ${\mathsf P} \{ \xi ( 0) = i \} = \pi _ {i}$, $i \in S$, then ${\mathsf P} \{ \xi ( t) = i \} = \pi _ {i}$ for any $i \in S$, $t > 0$; moreover, for any $t, t _ {1} \dots t _ {k} > 0$, $i _ {1} \dots i _ {k} \in S$,

$${\mathsf P} \{ \xi ( t _ {1} + t) = i _ {1} \dots \xi ( t _ {k} + t) = i _ {k} \} =$$

$$= \ {\mathsf P} \{ \xi ( t _ {1} ) = i _ {1} \dots \xi ( t _ {k} ) = i _ {k} \} .$$

If $i \in S$ is a state of the Markov chain $\xi ( t)$ for which the limits

$$\lim\limits _ {t \rightarrow \infty } p _ {ij} ( t) = \pi _ {j} ( i) \geq 0,\ \ j \in S ,\ \ \sum _ {j \in S } \pi _ {j} ( i) = 1 ,$$

exist, then the set of numbers $\{ {\pi _ {j} ( i) } : {j \in S } \}$ satisfies (2) and is a stationary distribution of the chain $\xi ( t)$( see also Transition probabilities).

The system of linear equations (2) relative to $\{ \pi _ {j} \}$, given the supplementary conditions (1), has a unique solution if the number of classes of positive states of the Markov chain $\xi ( t)$ is equal to 1; if the chain has $k$ classes of positive states, then the set of its stationary distributions is the convex hull of $k$ stationary distributions, each of which is concentrated on one class (see Markov chain, class of positive states of a).

Any non-negative solution of the system (2) is called a stationary measure; a stationary measure can exist also when (1) and (2) are not compatible. For example, a random walk on $\{ 0, 1 ,\dots \}$:

$$\xi ( 0) = 0,\ \ \xi ( t) = \xi ( t- 1) + \eta ( t),\ \ t = 1, 2 \dots$$

where $\eta ( 1) , \eta ( 2) \dots$ are independent random variables such that

$${\mathsf P} \{ \eta ( i) = 1 \} = p,\ \ {\mathsf P} \{ \eta ( i) = - 1 \} = 1- p,\ \ 0 < p < 1,$$

$$i = 1, 2 \dots$$

does not have a stationary distribution, but has a stationary measure:

$$\pi _ {j} = \left ( \frac{p}{1-} p \right ) ^ {j} ,\ \ j = 0, \pm 1 ,\dots .$$

One of the possible probabilistic interpretations of a stationary measure $\{ \pi _ {j} \}$ of a Markov chain $\xi ( t)$ with set of states $S$ is as follows. Let there be a countable set of independent realizations of $\xi ( t)$, and let $\eta _ {t} ( i)$ be the number of realizations for which $\xi ( t) = i$. If the random variables $\eta _ {0} ( i)$, $i \in S$, are independent and are subject to Poisson distributions with respective means $\pi _ {i}$, $i \in S$, then for any $t > 0$ the random variables $\eta _ {t} ( i)$, $i \in S$, are independent and have the same distributions as $\eta _ {0} ( i)$, $i \in S$.

#### References

 [C] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304 [K] S. Karlin, "A first course in stochastic processes" , Acad. Press (1966) MR0208657 Zbl 0315.60016 Zbl 0226.60052 Zbl 0177.21102