We consider a random walk $S_{ \tau}$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{ \tau}$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{ \alpha}$. We also prove asymptotic formula for the transition function of $S_{ \tau}$ similar to the P\'{ o}lya's asymptotic formula for $B^{ \alpha}$.
Categories: math.PR