arXiv subjects
Publications and source records attributed to Wojciech Cygan.
We consider a random walk $S_τ$ 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 $S_τ$ converges in the Skorohod space to the symmetric $α$-stable process $B^α$. We also prove asymptotic formula for the transition function of $S_τ$ similar to the Pólya's asymptotic formula for $B^α$.
Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\{\varepsilon (B)\}$ of i.i.d. random variables we define the perturbed function $C(B,ω)$ and the perturbed hierarchical Laplacian $L^{ω}=L_{C(ω)}.$ We study the arithmetic means $\overline{λ}(ω)$ of the $L^{ω}$-eigenvalues. Under some mild assumptions the normalized arithmetic means $\big( \overline{λ}-\mathbb{E}\overline{λ}\big) /σ\big( \overline{λ}\big) $ converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H (t) = \int_Ω\mathbb{P}_{x} (X_t\in Ω^c) dx$ which is called the heat content. We study its asymptotic behaviour as $t$ goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.
We study massive (reccurent) sets with respect to a certain random walk $S_α$ defined on the integer lattice $\mathbb{Z} ^d$, $d=1,2$. Our random walk $S_α$ is obtained from the simple random walk $S$ on $\mathbb{Z} ^d$ by the procedure of discrete subordination. $S_α$ can be regarded as a discrete space and time counterpart of the symmetric $α$-stable Lévy 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.
We introduce and study a class of random walks defined on the integer lattice $ \mathbb{Z} ^d$ -- a discrete space and time counterpart of the symmetric $α$-stable process in $\mathbb{R} ^d$. When $0< α<2$ any coordinate axis in $\mathbb{Z} ^d$, $d\geq 3$, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for the thorn to be a massive set.
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^d$ under the assumption that its Lévy--Khintchine exponent is regularly varying of index between $0$ and $2$.