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.Comment: 16 pages, 1 figur

Asymptotic behavior of densities of unimodal convolution semigroups

“…Then using equicontinouity of the distributions we conclude the convergence. The above technique was previously used in [3] where subordinated random walks on Z d were considered. Beside the transition density p(t, x) we also investigate the Lévy measure ν of the process X.…”

Stationary entrance Markov chains, inducing, and level-crossings of random walks

“…It appears (based on the estimates of Green's function in Uchiyama [34, Section 8]) that such results should be fully analogous to the lattice ones for walks with EX 1 = 0 and E X 1 2 < ∞ if the distribution of X 1 has density with respect to the Lebesgue measure. The case of heavy-tailed random walks on Z d , including transient walks in dimensions d = 1, 2, is considered by Bendikov and Cygan [4,5].…”

Transition probability estimates for subordinate random walks

“…It implies that the corresponding estimates hold only in that region whereas (1.2) is true for all ∈ ℤ and ∈ ℕ. There are more papers where subordinate random walks were studied from potential-theoretic point of view, see [5,6,11,16,17]. Note that discrete subordination allows us to efficiently construct examples of random walks with the controlled tail behaviour.…”