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I hope you can help me formalize some things about the shift operator. So let $(\theta _{n})_{n\geq0}$ be the shift operator - that is $\theta _{n}\omega(k)=\omega(k+n)$. I'm using the Durrett setup where $X_n=\omega_n$ which makes $X_n\circ\theta_k=X_{n+k}$, obviously (I hope).

Now let $T_A=\inf\{n\gt0|X_n\in A\}$ be the hitting time of $A$, one would expect $T_A\circ\theta_n$ to be "n times" after $X_n$ hits $A$. How do I show it? And how do I show it's even a stopping time? (that might follow from the answer to the first question I guess).

Now let $\mathcal{F}_n =\sigma(\{X_k|0\le k \le n\})$ one would then expect that $\theta_k^{-1}(\mathcal{F}_n)\subseteq\mathcal{F}_{n+k}$ but how do I show that? (I know I can "exchange function and sigma", but I guess I'm not sure what the preimage of a function by the shift operator is, formally).

Let me know if I haven't been precise enough with my questions or given enough information.

Best regards,

Edit: Thanks David. I think you made a minor error (but helped me do it right) since: $$ T_{A} \circ \theta_{k}\left(\omega\right)=\inf\left\{ m>0\,|\, X_{m}\circ\theta_{k}\left(\omega\right)\in A\right\} \\=\inf\left\{ m>0\,|\, X_{m+k}\left(\omega\right)\in A\right\} \\=\inf\left\{ n>k\,|\, X_{n}\left(\omega\right)\in A\right\} -k$$ - one really has to be careful when substituting i guess.

Question update: Is it possible to get for a general stopping time $S$ that $S\circ\theta_{n}$ is a stopping time - actually I more specifically would like an argument that gives $\{S\circ \theta _k = n-k \}\in\mathcal{F}_n$

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up vote 2 down vote accepted

Actually, $T_A$ act on the sequence $\{X_n(\omega)\}_{n\geq 0}$. Hence we have for a fixed $\omega$: $$T_A(\theta_n\{X_k(\omega)\})=T_A(\{X_{n+k}(\omega)\})=\inf\{j>0,X_{n+j}\in A\}=\inf\{k>n,X_k\in A\}.$$ For an integer $N$, we have $$\{T_A\circ\theta_n=N\}=\begin{cases}\emptyset &\mbox{ if }N\leq n,\\ \bigcap_{j=1}^{N-1}\{X_k\notin A\}\cap \{X_N\in A\}&\mbox{ otherwise}. \end{cases}$$ The latter set is in $\mathcal F_N$ as a (finite) intersection of such sets.

Let $n$ and $k$ fixed integers. Let $X$ the sequence $(X_0,X_1,\dots,X_n,0,\dots)$. We have $$\theta_k(\mathcal F_n)=\theta_k(\sigma(X))=\sigma(\theta_k(X))\subset \sigma(X_k,\dots,X_{n+k})\subset \mathcal F_{n+k}.$$

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Thank you very much for your answer, I like the simplicity. Isn't it correct (or it might be that Durrett just defines it for such in case you probably wouldn't know) that the shift operator works only on omegas but since x is the coordinate map it's working here. It's kind of a rookie question but I haven't done much "hardcore" math - what about theta^-1 ? Is it the same as what you wrote? –  Henrik Aug 1 '12 at 21:33
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