Operator self-similar processes on Banach spaces

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0, ∞), for some s0 ≥ 1, have a unique scaling family of operators of the form {sH: s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Original languageEnglish (US)
Article number82838
JournalJournal of Applied Mathematics and Stochastic Analysis
Volume2006
DOIs
StatePublished - Jun 29 2006

Fingerprint

Self-similar Processes
Banach spaces
Mathematical operators
Banach space
Operator
Scaling
Hilbert spaces
Random processes
Positive Operator
Time Constant
Linear Space
Differentiable
Stochastic Processes
Multiplicative
Closure
Hilbert space
Non-negative
Exponent
Subspace
Family

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

Cite this

@article{b210ff4e49aa4bb481c57a3d234a428b,
title = "Operator self-similar processes on Banach spaces",
abstract = "Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0, ∞), for some s0 ≥ 1, have a unique scaling family of operators of the form {sH: s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.",
author = "Matache, {Mihaela T.} and Valentin Matache",
year = "2006",
month = "6",
day = "29",
doi = "10.1155/JAMSA/2006/82838",
language = "English (US)",
volume = "2006",
journal = "International Journal of Stochastic Analysis",
issn = "2090-3332",
publisher = "Hindawi Publishing Corporation",

}

TY - JOUR

T1 - Operator self-similar processes on Banach spaces

AU - Matache, Mihaela T.

AU - Matache, Valentin

PY - 2006/6/29

Y1 - 2006/6/29

N2 - Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0, ∞), for some s0 ≥ 1, have a unique scaling family of operators of the form {sH: s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

AB - Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such apower-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0, ∞), for some s0 ≥ 1, have a unique scaling family of operators of the form {sH: s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

UR - http://www.scopus.com/inward/record.url?scp=33745321910&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745321910&partnerID=8YFLogxK

U2 - 10.1155/JAMSA/2006/82838

DO - 10.1155/JAMSA/2006/82838

M3 - Article

AN - SCOPUS:33745321910

VL - 2006

JO - International Journal of Stochastic Analysis

JF - International Journal of Stochastic Analysis

SN - 2090-3332

M1 - 82838

ER -