Operator-self-similar processes in a finite-dimensional space

Abstract

A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar. 1. Introduction. In 1962 Lamperti [8] introduced the notion of a self-similar process, {X(t): t s* 0), taking values in a real finite-dimensional inner product space T. A stochastic process {X(t)} is called self-similar (s.s.) if it is continuous in probability at each t > 0 and if for every a > 0 there exist a positive number B(a) and a vector b(a) in Tsuch that the process {X(at)} and {B(a)X(t) + b(a)} have the same finite-dimensional distributions. Lamperti actually called such processes " semi-stable" but currently this term is being used in another sense. These processes arise in diverse areas (see Taqqu [14]). Self-similar processes are also obtained by various limiting procedures (e.g., see Gorodetskii [4], Kesten and Spitzer [6], Sinai

Dimension results for sample paths of operator stable Lévy processes

“…Here the linear operator B is called a self-similarity exponent of X. Hudson and Mason [11] proved that if X is a Le´vy process in R d such that the distribution of X ð1Þ is full, then X is operator self-similar if and only if X ð1Þ is strictly operator stable. In this case, every stability exponent B of X is also a self-similarity exponent of X.…”

A note on operator self-similar Gaussian vector fields

“…For the operator self-similar R n valued processes with d = 1, Laha and Rohatgi [8] proved that if E = I and A(c) is positive definite, A(c) can be expressed as A(c) = c Q , where Q is a square matrix. Hudson and Mason [5] have shown that if d = 1 and E = I, this is again true, i.e., A(c) = c Q . For the Gaussian vector fields, Pitt [6] showed that if E = I, A(c) = c Q .…”

“…We refer the reader to Biermé et al [4], and Hudson and Mason [5] and references therein. In our work, we use a more general definition of operator self-similar processes.…”

Operator semi-self-similar processes and their space-scaling matrices

“…When we replace b in (1.1) by a d  d matrix B, we call such a process fX ðtÞ; tX0g operator self-similar. Wide sense operator self-similar processes have been studied by Laha and Rohatgi (1982), Hudson and Mason (1982), Sato (1991) and Maejima and Mason (1994). In these cases, B is not necessarily determined uniquely for each given a40.…”

“…In these cases, B is not necessarily determined uniquely for each given a40. As to the exponent, Hudson and Mason (1982) proved that if an operator self-similar process fX ðtÞg is proper in the sense that the distribution of X ðtÞ for some t40 is full (which is defined later), then there exists an invertible d  d matrix D such that B ¼ a D . Without the assumption that fX ðtÞg is proper, the existence of an invertible d  d matrix D was also proved by Sato (1991).…”