Consider a chain $(X_{t})$ on state-space $I$, and fix $A\subseteq I$. There are many different constructions of new chains whose state space is (exactly or roughly) just $A$, and it’s important not to confuse them. Three elementary constructions are described here. Anticipating the definition of reversible from Chapter 3, it is easy to check that if the original chain is reversible then each new chain is reversible.

This is the chain $(Y_{n})$ defined by

$S_{0}=T_{A}=\min\{t\geq 0:X_{t}\in A\}$ |

$S_{n}=\min\{t>S_{n-1}:X_{t}\in A\}$ |

$Y_{n}=X_{S_{n}}.$ |

The chain $(Y_{n})$ has state space $A$ and transition matrix

$\bar{P}_{A}(i,j)=P_{i}(X_{T_{A}}=j),\ i,j\in A.$ |

From the ergodic theorem (Theorem 2.1) it is clear that the stationary distribution $\pi_{A}$ of $(Y_{t})$ is just $\pi$ conditioned on $A$, that is

$\pi_{A}(i)=\pi_{i}/\pi(A),\ i\in A.$ | (2.27) |

This is the chain with state space $A$ and transition matrix $\hat{P}_{A}$ defined by

$\displaystyle\hat{P}_{A}(i,j)$ | $\displaystyle=$ | $\displaystyle P(i,j),\ i,j\in A,i\neq j$ | ||

$\displaystyle\hat{P}_{A}(i,i)$ | $\displaystyle=$ | $\displaystyle 1-\sum_{j\in A,j\neq i}P(i,j),\ i\in A.$ |

In general there is little connection between this chain and the original chain $(X_{t})$, and in general it is not true that the stationary distribution is given by (2.27). However, when the original chain is reversible, it is easy to check that the restricted chain does have the stationary distribution (2.27).

This chain has state space $I^{*}=A\cup\{a\}$ where $a$ is a new state. We interpret the new chain as “the original chain with states $A^{c}$ collapsed to a single state $a$”. Warning. In later applications we switch the roles of $A$ and $A^{c}$, i.e. we collapse $A$ to a single state $a$ and use the collapsed chain on states $I^{*}=A^{c}\cup\{a\}$. The collapsed chain has transition matrix

$\displaystyle p^{*}_{ij}$ | $\displaystyle=$ | $\displaystyle p_{ij},\ i,j\in A$ | ||

$\displaystyle p^{*}_{ia}$ | $\displaystyle=$ | $\displaystyle\sum_{k\in A^{c}}p_{ik},\ i\in A$ | ||

$\displaystyle p^{*}_{ai}$ | $\displaystyle=$ | $\displaystyle\frac{1}{\pi(A^{c})}\sum_{k\in A^{c}}\pi_{k}p_{ki},\ i\in A$ | ||

$\displaystyle p^{*}_{aa}$ | $\displaystyle=$ | $\displaystyle\frac{1}{\pi(A^{c})}\sum_{k\in A^{c}}\sum_{l\in A^{c}}\pi_{k}p_{% kl}.$ |

The collapsed chain has stationary distribution $\pi^{*}$ given by

$\pi^{*}_{i}=\pi_{i},i\in A;\ \ \pi^{*}_{a}=\pi(A^{c}).$ |

Obviously the ${\bf P}$-chain started at $i$ and run until $T_{A^{c}}$ is the same as the ${\bf P}^{*}$-chain started at $i$ and run until $T_{a}$. This leads to the general collapsing principle

To prove a result which involves the behavior of the chain only up to time $T_{A^{c}}$, we may assume ${A^{c}}$ is a singleton.

For we may apply the singleton result to the ${\bf P}^{*}$-chain run until time $T_{a}$, and the same result will hold for the ${\bf P}$-chain run until time $T_{A^{c}}$.

It is important to realize that typically (even for reversible chains) all three constructions give different processes. Loosely, the chain restricted to $A$ “rebounds off the boundary of $A^{c}$ where the boundary is hit”, the collapsed chain “exits $A^{c}$ at a random place independent of the hitting place”, and the chain watched only on $A$ “rebounds at a random place dependent on the hitting place”.

Generated on Mon Jun 2 14:23:48 2014 by LaTeXML