30 research outputs found
On the positive eigenvalues and eigenvectors of a non-negative matrix
The paper develops the general theory for the items in the title, assuming
that the matrix is countable and cofinal.Comment: Version 2 allows the matrix to have zero row(s) and rows with
infinitely many non-zero entries. In addition the introduction has been
rewritte
Hausdorff dimension of operator semistable L\'evy processes
Let be an operator semistable L\'evy process in \rd
with exponent , where is an invertible linear operator on \rd and
is semi-selfsimilar with respect to . By refining arguments given in
Meerschaert and Xiao \cite{MX} for the special case of an operator stable
(selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we
determine the Hausdorff dimension of the partial range in terms of the
real parts of the eigenvalues of and the Hausdorff dimension of .Comment: 23 page
Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime
In this paper we derive a technique of obtaining limit theorems for suprema
of L\'evy processes from their random walk counterparts. For each , let
be a sequence of independent and identically distributed
random variables and be a L\'evy processes such that
, and as . Let .
Then, under some mild assumptions, , for some random variable and some function
. We utilize this result to present a number of limit theorems
for suprema of L\'evy processes in the heavy-traffic regime
On infinite-volume mixing
In the context of the long-standing issue of mixing in infinite ergodic
theory, we introduce the idea of mixing for observables possessing an
infinite-volume average. The idea is borrowed from statistical mechanics and
appears to be relevant, at least for extended systems with a direct physical
interpretation. We discuss the pros and cons of a few mathematical definitions
that can be devised, testing them on a prototypical class of infinite
measure-preserving dynamical systems, namely, the random walks.Comment: 34 pages, final version accepted by Communications in Mathematical
Physics (some changes in Sect. 3 -- Prop. 3.1 in previous version was
partially incorrect