Operator-Self-Similar Processes in a Finite-Dimensional Space

Hudson, William N.; Mason, J. David

doi:10.2307/1999205

Cited by 34 publications

(91 citation statements)

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“…For R n -valued OSS processes, a connection between this class of processes and operator-stable probability measures has been observed by the authors of [4]. Their idea extends to Banach space valued processes as follows.…”

Section: Definition 13mentioning

confidence: 98%

“…4) In order that the inequalities above hold we also require that the OSS processes have a norm-continuous expectation-function t → E(X(t)), t > 0. The main ingredient in the proof of the theorem is the fact that OSS processes whose expectations span a dense linear subspace of the whole space have a unique scaling family of operators which is necessarily a multiplicative group of operators.…”

Section: Introductionmentioning

confidence: 99%

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Operator self‐similar processes on Banach spaces

Matache

2006

International Journal of Stochastic Analysis

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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 a power-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.

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Section: Definition 13mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Operator self‐similar processes on Banach spaces

Matache

2006

International Journal of Stochastic Analysis

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“…Remark 4.2 Theorem 6 in Hudson and Mason (1982) shows that every maximal symmetry o.s.s. process has an exponent of the form h I, h ∈ R. For the case of OFBMs, the proof of Theorem 4.1 retrieves this result (see expression (4.5)).…”

Section: Theorem 41 Consider An Ofbm Given By the Spectral Representmentioning

confidence: 99%

“…Such issues are strongly connected. Since the fundamental work of Hudson and Mason (1982), it is well known that one given o.s.s. process X may have multiple exponents.…”

Section: Introductionmentioning

confidence: 99%

Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

Didier

Pipiras

2011

J Theor Probab

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Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group.In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

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“…Let X = {X(t)} t≥0 be a Lévy process in R d starting from 0 such that the distribution of X(t) is full for every t > 0. Hudson and Mason [10,Theorem 7] proved that X is operator self-similar if and only if the distribution of X(1), ν := P • (X(1)) −1 , is strictly operator stable. In this case, every stability exponent B of ν is also a self-similarity exponent of X.…”

Section: Now We Derive the Lower Bound For Dim P X([0 1]) It Follomentioning

confidence: 99%

Packing dimension of the range of a Lévy process

Khoshnevisan

Xiao

2008

Proc. Amer. Math. Soc.

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Abstract. Let {X(t)} t≥0 denote a Lévy process in R d with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X ([0 , 1]) is given by the indexWe provide an alternative formulation of γ in terms of the Lévy exponent Ψ. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.

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Operator-Self-Similar Processes in a Finite-Dimensional Space

Cited by 34 publications

References 5 publications

Operator self‐similar processes on Banach spaces

Operator self‐similar processes on Banach spaces

Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

Packing dimension of the range of a Lévy process

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