Limit theorems for random walks

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 $B^{\alpha}$.