We have a random walk which starts from the origin at time $n=0$. At each time step, $n$, we take a step of length $1$ in a random direction $\Theta_n \sim \text{Uniform}(0,2\pi) $. We have to show that $\mathbb{E}(\tau)<\infty$.
Now I know that if we can show that $\exists N\in \mathbb{N}$ such that $\mathbb{P}(\tau \leq n+N| \mathcal{F_n})>0$ then $\mathbb{E}(\tau)<\infty$. The problem is I am not sure how we could do this since we are dealing with continuous random directions (rather than discrete directions on a lattice where we know that the probability of making a certain number of consecutive steps in the same direction is positive)?