Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

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 (and 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 multiplicity of exponents, whereas the same is not true for time domain integral representations.

[1]  C. C. Macduffee,et al.  The Theory of Matrices , 1933 .

[2]  W. E. Roth,et al.  On $k$-commutative matrices , 1936 .

[3]  O. Taussky.  Commutativity in Finite Matrices , 1957 .

[4]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[5]  Patrick Billingsley.  Convergence of types in k-space , 1966 .

[6]  Jacob T. Schwartz,et al.  Lie Groups; Lie Algebras , 1968 .

[7]  M. Sharpe.  Operator-stable probability distributions on vector groups , 1969 .

[8]  Scaling limits of Gaussian vector fields , 1978 .

[9]  Operator-stable laws , 1981 .

[10]  V. K. Rohatgi,et al.  Operator self similar stochastic processes in Rd , 1981 .

[11]  J. Mason,et al.  Operator-self-similar processes in a finite-dimensional space , 1982 .

[12]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[13]  J. Glover.  Symmetry groups of Markov processes and the diagonal principle , 1991 .

[14]  Ken-iti Sato,et al.  Self-similar processes with independent increments , 1991 .

[15]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[16]  J. Mason,et al.  Operator-limit distributions in probability theory , 1993 .

[17]  Mark M. Meerschaert,et al.  The structure of the exponents and symmetries of an operator stable measure , 1993 .

[18]  M. Maejima,et al.  Operator-self-similar stable processes , 1994 .

[19]  Symmetry groups in d-space , 1995 .

[20]  15. Operator‐Limit Distributions in Probability Theory , 1995 .

[21]  Limit theorems related to a class of operator-self-similar processes , 1996 .

[22]  Norming Operators for Operator-Self-Similar Processes , 1998 .

[23]  Karl H. Hofmann,et al.  The Structure of Compact Groups , 2020 .

[24]  Spectral decomposition for operator self-similar processes and their generalized domains of attraction , 1999 .

[25]  M. Meerschaert,et al.  Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice , 2001 .

[26]  J. Mason,et al.  Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields , 2002 .

[27]  Dinh Van Huynh,et al.  Algebra and Its Applications , 2006 .

[28]  G. Didier.  Studies in stochastic processes: Adaptive wavelet decompositions and operator fractional Brownian motions , 2007 .

[29]  G. Pap,et al.  Parameter estimation of selfsimilarity exponents , 2008 .

[30]  Donatas Surgailis,et al.  Covariance function of vector self-similar processes , 2009, 0906.4541.

[31]  G. Didier,et al.  Integral representations and properties of operator fractional Brownian motions , 2011, 1102.1822.