Markov chain

Definition

We begin with a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ . Let $I$ be a countable set, $(X_n: n \in \integers)$ be a collection of random variables taking values in $I$ , $\mv{T} = (t_{ij}: i,j \in I)$ be a stochastic matrix, and $\mv{\lambda}$ be a distribution. We call $(X_n)_{n\ge 0}$ a Markov chain with initial distribution $\mv{\lambda}$ and transition matrix $\mv{T}$ if:

1. $X_0$ has distribution $\mv{\lambda}$ .
2. For $n \ge 0$ , $\mathbb{P}(X_{n+1}=i_{n+1} | X_0 = i_0, \ldots, X_n = i_n) = t_{i_n i_{n+1}}$ .

That is, the next value of the chain depends only on the current value, not any previous values. This is often summed up in the pithy phrase, ``Markov chains have no memory.''

As a special case of (2) we have that $\mathbb{P}(X_{n+1} = i | X_n = j) = t_{ij}$ whenever $i,j \in I$ . The values $t_{ij}$ are therefore called transition probabilities for the Markov chain.

Discussion

Markov chains are arguably the simplest examples of random processes. They come in discrete and continuous versions; the discrete version is presented above.