# The inequality of the form $\mathbb{P}_x(\sigma=k)\leq (1-p)^k$, where $\sigma$ is a stopping time

Let $X$ be a metric space and $\Omega=X^{\mathbb{N}}$. Consider the sequence $(X_n)_{n\geq 0}$ given by $$X_n(\omega)=x_n\;\;\;(\omega=(x_0,x_1,...),\;n\in\mathbb{N_0}).$$ Let $P:X\times\mathcal{B}(X)\to\left[0,1\right]$ be a stochastic kernel. Then, for every $x\in X$ we can define a probability measure $\mathbb{P}_x$ on $\mathcal{B}(X^{\mathbb{N}})$ such that $(X_n)_{n\geq 0}$ is a homogeneus Markov chain on the probalility space $(X, \mathcal{B}(X^{\mathbb{N}}), \mathbb{P}_x)$ starting at the point $x$ and $P$ is its transition function.

Let $A\in \mathcal{B}(X)$ and define the stopping time $\tau= \inf\{n\geq 1: X_n\in A\}$ and $$\tau(1)=\tau,\;\;\; \tau(k+1)=\tau(k) + 1_{\left\{\tau(k)<\infty\right\}}(\tau \circ T_{\tau(k)})\;\;\;(k\in\mathbb{N}),$$ where $T_n(x_1,x_2,...)=(x_n,x_{n+1},...)$.

Now, let et $B\in \mathcal{B}(X^{\mathbb{N}})$ and assume that there is $p>0$ such that $\mathbb{P}_x(B)\geq p$, for $x\in A$. Define $$\widehat{\tau}=\inf\{n\geq 1: X_n\in A,\; (X_{k+n})_{k\geq 0}\in B\}.$$ Finally, define $$\sigma=\inf\{k\geq 1: \widehat{\tau}=\tau(k)\}.$$

My question is:

How to prove that $\mathbb{P}_x(\sigma=k)\leq (1-p)^{k-1}$, for $k\in \mathbb{N}$? I have tried to use the Strong Markov Property. Is it necessary to assume that $\tau<\infty$ $\mathbb{P}_x$ a.e.?

I will be grateful for any help.

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