Almost Surely Finite Stopping Time Inequality

Question

Assume $\tau$ is a $\mathcal{F}_n$- stopping time such that there exists a positive integer $m$ and some $\epsilon>0$ such that for all $n$

$$\mathbb{P}(\tau\leq n+m \,\, \vert \mathcal{F}_n) >\epsilon$$

Show that for every positive integer $k$

$$\mathbb{P}(\tau>km)\leq(1-\epsilon)^k$$

This was really helpful:

Since $\tau$ is $\mathcal{F}_n$-measurable, the set $ (\tau>n)\in\mathcal{F}_n$, so

$$\mathbb{P}(\tau \leq n+ m, \tau>n)>\epsilon\mathbb{P}(\tau >n)$$

Since $\mathbb{P}(n<\tau \leq n+ m) = \mathbb{P}(\tau>n) - \mathbb{P}(\tau>n+m),$ combining we get

$$\mathbb{P}(\tau>n+m)\leq(1-\epsilon)\mathbb{P}(\tau>n)$$

Here's were I get lost The referenced link then says "iterating on multiples of k, we get" $$\mathbb{P}(\tau>km)\leq(1-\epsilon)^k$$

How do I show this last bit? Any help?

Answer 1

Hint: Your argument shows that $\Bbb P(\tau>km)\le(1-\epsilon)\Bbb P(\tau>(k-1)m)$ for $k=1,2,\ldots$. (Take your $n$ equal to $k-1$.) Proceed recursively.