Analytical Properties of a Generating Function for a Number of Renewals

Let $\{t_i\}$ be a renewal process, $N_t=\max(n:t_n<t)$. Some analytic properties, such as analyticy within the circle $|z|<{1/r}$, of the function $\Pi _t (z)=\sum _{k=0}^\infty z^k P\{N_t=k\}$ and others are proved.