Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$.
So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then $X_{n+1}=1$ with probability 1.
I would like to know if we can compute $lim_{n \rightarrow \infty} \frac{X_0+X_1+...+X_n}{n+1}$ when $\theta < \frac{1}{2}$? (i guess the limit is $\infty$ when $\theta=\frac{1}{2}$).
Thank you