Limit theorems for random walks

We consider a random walk Sτ 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 τ appropriately scaled random walk Sτ converges in the Skorohod space to the symmetric α-stable process Bα. We also prove asymptotic formula for the transition function of Sτ similar to the Pólya's asymptotic formula for Bα.