probability theory - Application of Strong Markov Property

Theorem SMP (Strong Markov Property)

Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or
$\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many
values. Then

$$P[\theta _{\tau}X\in A\mid \mathscr F_{\tau}]\overset{\text {a.s on }\left\{ \tau <\infty \right\} }{=}P_{X_\tau}(X\in A)$$

Assume from now on $T=\mathbb{Z}_+$ and that $X$ is canonical. Define the first hitting time of state $y$ as $\tau=\inf \left\{ n\geq 0:X_n=y \right\}$. Now define recursively

$$\tau_y ^{k+1}=\tau_y^k+\tau_y\circ \theta_{\tau^k_y},\;\;\;k\geq 0$$ starting from $\tau^0_y=0$.

Why does the SMP imply the following equality?

$$P_x \left\{ \tau^k_y<\infty,\;\tau_y\circ\theta_{\tau^k_y} \right\}=P_x \left\{ \tau^k_y<\infty \right\} P_y \left\{ \tau_y<\infty \right\}$$

Answer

The missing step has nothing in particular to do with Markov processes - it uses the decomposition $P(A\cap B)=P(B\mid A)P(A)$. Thus

$$\begin{align} P_x \left\{ \tau^k_y<\infty,\;\tau_y\circ\theta_{\tau^k_y} \right\} & =P_x \left\{ \tau_y\circ\theta_{\tau^k_y} \mid \tau _y^k<\infty \right\} P_x \left\{ \tau^k_y<\infty \right\}\\ & =P_x \left\{ \tau^k_y<\infty \right\} P_y \left\{ \tau_y<\infty \right\} \end{align}$$ where the strong Markov property is used in the final transition.