Probability of Absorption

Topics: Markov Chain - Absorbing Set

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(definition)

In a Markov chain, given a $k$ an absorbing state, the probability of reaching $k$ when we start from another state $i$ is called the probability of absorption to $k$. This probability is denoted by $f_{ik}$.

(theorem)

The probability of absorption to $k$ when starting at $i$ is given by:

$$f_{ik}=\sum _{j=0}{p}_{ij}{f}_{jk}$$

…where $f_{kk}=1$.

(theorem)

Let $k_1,k_2,\dots ,k_s$ be all absorbing states in a Markov chain. Then, for a fixed state $i$, it follows that:

$$\sum _{c=1}^s{f}_{i{k}_c}=1$$

In other words, the result of adding up all possible absorption probabilities when starting at a given state $i$ is simply $1$.

(observation)

Note that we’ll have $f_{ik}=0$ if the state $i$ is absorbing and different from $k$.