# Formalizing the shift operator

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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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 $$\theta_k(\mathcal F_n)=\theta_k(\sigma(X))=\sigma(\theta_k(X))\subset \sigma(X_k,\dots,X_{n+k})\subset \mathcal F_{n+k}.$$