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How do you calculate the stationary distribution of a Markov chain?
Remember that for discrete-time Markov chains, stationary distributions are obtained by solving π=πP. We have a similar definition for continuous-time Markov chains. Let X(t) be a continuous-time Markov chain with transition matrix P(t) and state space S={0,1,2,⋯}.
What is the stationary distribution of the Markov chain?
The stationary distribution of a Markov chain describes the distribution of Xt after a sufficiently long time that the distribution of Xt does not change any longer. To put this notion in equation form, let π be a column vector of probabilities on the states that a Markov chain can visit.
What is the stationary probability distribution?
The stationary distribution has the interpretation of the limiting distribution when the chain is ergodic. The marginal distribution of a stationary process or stationary time series. The set of joint probability distributions of a stationary process or stationary time series.
What is the limiting distribution of this Markov chain?
The probability distribution π = [ π 0, π 1, π 2, ⋯] is called the limiting distribution of the Markov chain X n if π j = lim n → ∞ P (X n = j | X 0 = i) for all i, j ∈ S, and we have ∑ j ∈ S π j = 1.
Does absorbing Markov chain have steady state distributions?
On the other hand, since the row of each limiting matrix for an absorbing Markov chain is the same, the state distribution after a large number of transitions for an absorbing Markov chain is dependent on the initial state distribution. However, I read somewhere that an absorbing Markov chain can have steady state distributions – which contradicts what I have always believed.
What is a homogeneous Markov chain?
I learned that a Markov chain is a graph that describes how the state changes over time, and a homogeneous Markov chain is such a graph that its system dynamic doesn’t change. Here the system dynamic is something also called transition kernel which means the calculation of the probability from one station to the next station.