On massive sets for subordinated random walks

Abstract

We study massive (reccurent) sets with respect to a certain random walk \(S_\alpha \) defined on the integer lattice \(\mathbb{Z} ^d\), \(d=1,2\). Our random walk \(S_\alpha \) is obtained from the simple random walk \(S\) on \(\mathbb{Z} ^d\) by the procedure of discrete subordination. \(S_\alpha \) can be regarded as a discrete space and time counterpart of the symmetric \(\alpha \)-stable L\'{e}vy process in \(\mathbb{R}^d\). In the case \(d=1\) we show that some remarkable proper subsets of \(\mathbb{Z}\) , e.g. the set \(\mathcal{P}\) of primes, are massive whereas some proper subsets of \(\mathcal{P}\) such as Leitmann primes \(\mathcal{P}_h\) are massive/non-massive depending on the function \(h\). Our results can be regarded as an extension of the results of McKean (1961) about massiveness of the set of primes for the simple random walk in \(\mathbb{Z}^3\). In the case \(d=2\) we study massiveness of thorns and their proper subsets.