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Operator-stable laws

Author

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  • Hudson, William N.
  • Mason, J. David

Abstract

Sharpe investigated the structure of full operator-stable measures [mu] on a vector group V and obtained decompositions, [mu] = [mu]1 * [mu]2 and V = V1 [circle plus operator] V2, in terms of the Gaussian component [mu]1 and the Poisson component [mu]2. The subspaces V1 and V2 are here identified in terms of an exponent B for [mu]. Sharpe also pointed out that the Lévy measure M of [mu] is a mixture of Lévy measures concentrated on single orbits of tB. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Lévy measure be the Lévy measure of a full operator-stable measure. The final result deals with full Gaussian measures [mu] and establishes the connection between its covariance operator and the class of all exponents of [mu].

Suggested Citation

  • Hudson, William N. & Mason, J. David, 1981. "Operator-stable laws," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 434-447, September.
  • Handle: RePEc:eee:jmvana:v:11:y:1981:i:3:p:434-447
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    Citations

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    Cited by:

    1. Hudson, William N. & Veeh, Jerry Alan, 2001. "Complex Stable Sums of Complex Stable Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 77(2), pages 229-238, May.
    2. Gustavo Didier & Vladas Pipiras, 2012. "Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 25(2), pages 353-395, June.
    3. Andrzej Łuczak, 2010. "Centering Problems for Probability Measures on Finite-Dimensional Vector Spaces," Journal of Theoretical Probability, Springer, vol. 23(3), pages 770-791, September.
    4. Steven N. Evans & Ilya Molchanov, 2018. "Polar Decomposition of Scale-Homogeneous Measures with Application to Lévy Measures of Strictly Stable Laws," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1303-1321, September.

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