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If you have a state space $S$, usually I think of a Markov chain $X_n$ on it as $X_n$ takes values in $S$ and satisfies the obvious Markov property and so on. In Durrett's book, he says one should instead consider $S^{\mathbb{N}}$ with $X_n(\omega) = \omega_n$.

So in this new state space, am I meant to consider $\omega \in S^{\mathbb{N}}$ as consisting of all the states the original Markov chain can be in? Is it a realisation of the process as it's run through time? I don't understand this.

He also defines a shift operator $\theta_n$ which acts as $\theta_n \omega (m) = \omega (m+n)$. So $X_j \circ \theta_n(\omega) = X_j(\theta_n(\omega)) = \omega_{j+n}$, which I am a bit unsure about.

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You will get better answers if you ask more specific questions. For example, what about the shift operator are you unsure about? – Byron Schmuland Feb 25 '12 at 14:52
Apologies, I think I was "unsure" about it since I didn't understand the whole set up. – Court Feb 26 '12 at 12:59
No need to apologize. I think user11867 gave you a very good answer. Is there anything that you still don't get? – Byron Schmuland Feb 26 '12 at 13:58
up vote 3 down vote accepted

A discrete time, $S$-valued stochastic process is a collection of $S$-valued random variables, $X=\{X_n:n\in\mathbb{N}\}$. For each fixed $\omega$, we then get a sequence in $S$, namely, the sequence $\{X_n(\omega)\}_{n=1}^\infty$. In this way, we can think of the process $X$ as a function that sends $\omega$ to a sequence in $S$. The set of all sequences in $S$ is denoted by $S^\mathbb{N}$. So we can think of $X$ as a map, $X:\Omega\to S^{\mathbb{N}}$. More specifically, it is the map defined by \begin{equation} (X(\omega))_n = X_n(\omega). \end{equation}

Now, one way to build a variety of stochastic processes is to start with a probability space, $(\Omega,\mathcal{F},P)$, and then define various functions $X:\Omega\to S^\mathbb{N}$, $Y:\Omega\to S^\mathbb{N}$, $Z:\Omega\to S^\mathbb{N}$, and so on. An alternative way to build processes, however, is to start with just a measurable space, $(\Omega,\mathcal{F})$, and only one function, $X:\Omega\to S^\mathbb{N}$. We can then give this one function, $X$, a variety of different properties by defining many different probability measures on $(\Omega,\mathcal{F})$, such as $P_1$, $P_2$, $P_x$, $P_\mu$, and so on.

It is this second approach which is quite common in the study of Markov processes, and it is the one which Durrett is following. More specifically, he starts with the measurable space $(\Omega,\mathcal{F})$, where $\Omega=S^\mathbb{N}$ and $\mathcal{F}=\mathcal{S}^\mathbb{N}$, and with the one function $X:\Omega\to S^\mathbb{N}$, where $X$ is the identity. Since $X$ is the identity, we have $X(\omega)=\omega$. Recalling that both sides of this equality are sequences, we may write this equality component-wise, giving $(X(\omega))_n =\omega_n$. This, in turn, may be rewritten as $X_n(\omega)=\omega_n$. In other words, $X_n$ is the function from $S^\mathbb{N}$ to $S$ which maps a sequence to its $n$-th term.

The shift-operator, $\theta_n$, is a map from $S^\mathbb{N}$ to $S^\mathbb{N}$. It takes a sequence and chops off the first $n$-terms. More specifically, if $\omega\in S^{\mathbb{N}}$, then $\theta_n(\omega)$ is the sequence defined by $(\theta_n(\omega))_m = \omega_{n+m}$. This is why we have $$ (X_j\circ\theta_n)(\omega) = X_j(\theta_n(\omega)) = (\theta_n(\omega))_j = \omega_{n+j}. $$

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