Find the steady state matrix X of the absorbing Markov chain...

Find the steady state matrix $X$ of the absorbing Markov chain with matrix of transition probabilities $P$.
$$P=\left[\begin{array}{llll}
0.7 & 0 & 0.2 & 0.1 \\
0.1 & 1 & 0.5 & 0.6 \\
0 & 0 & 0.2 & 0.2 \\
0.2 & 0 & 0.1 & 0.1
\end{array}\right]$$

Key Concepts

Absorbing Markov Chain

An absorbing Markov chain is a type of Markov chain that includes at least one absorbing state, a state that, once entered, cannot be left. In these chains, every other state (called transient states) eventually leads to an absorbing state with probability one, which is essential for evaluating long-term behavior.

Transition Probability Matrix

A transition probability matrix is a square matrix used to describe the probabilities of moving from one state to another in a Markov chain. Each entry in the matrix represents the probability of a one-step transition from one state to another, and each row typically sums to one.

Canonical Form (of an Absorbing Markov Chain)

The canonical form of an absorbing Markov chain reorganizes the transition probability matrix so that the absorbing states are listed last. This form partitions the matrix into blocks, where one block (Q) corresponds to transitions among transient states and another block (R) corresponds to transitions from transient states to absorbing states, thereby simplifying the computation of absorption probabilities.

Fundamental Matrix

The fundamental matrix, defined as N = (I - Q)?¹, where Q is the submatrix representing transitions among transient states, is a crucial concept for absorbing Markov chains. It provides the expected number of times the process is in each transient state before absorption, aiding in the analysis of long-term behavior.

Steady State (Limiting) Matrix

In the context of absorbing Markov chains, the steady state or limiting matrix refers to the limit of the transition matrix as the number of steps tends to infinity. This matrix details the probabilities with which the chain will be absorbed in each absorbing state when starting from any state, effectively summarizing the long-term behavior of the system.

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