### 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 (s_{0}, ∞), for some s_{0} ≥ 1, have a unique scaling family of operators of the form {s^{H}: 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 {s^{H}: s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Original language | English (US) |
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Article number | 82838 |

Journal | Journal of Applied Mathematics and Stochastic Analysis |

Volume | 2006 |

DOIs | |

State | Published - Jun 29 2006 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

**Operator self-similar processes on Banach spaces.** / Matache, Mihaela T.; Matache, Valentin.

Research output: Contribution to journal › Article

}

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 -