Abstract
We propose some matrix generalizations of square Bessel processes and we indicate their first properties: hitting time of 0 of the smallest eigenvalue, additivity property, associated Martingales, distributions, which mainly extend the real-valued classical results. We explain why these processes are indecomposable and therefore differ from the real-valued ones. We conclude with some formulae concerning matrix quadratic functionals analogous to the Cameron Martin formula.
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References
Atkinson, F. V. (1964).Discrete and Continuous Boundary Problems, Academic Press, New York.
Barra, J. R. (1981).Mathematical Basis of Statistics, Academic Press, New York.
Bru, M. F. (1987). Thèse de 3ième cycle, Université Paris Nord.
Bru, M. F. (1989). Diffusions of perturbed principal component analysis.J. Multivariate Anal. 29, 127–136.
Bru, M. F. (1989). Processus de Wishart,C. R. Acad. Sci. Paris 308, 29–32.
Dieudonné, J. (1965).Fondements de l'analyse moderne, Gauthier Villars, Paris.
Doss, H., and Lenglart, E. (1978). Sur l'existence, l'unicité et le comportement asymptotique des solutions d'équations différentielles stochastiques,Ann. Inst. Henri Poincaré XIV, 189–214.
Faraut, J. (1988). Personal communication.
Farrell, R. H. (1985).Multivariate Calculation. Use of Continuous Groups, Springer-Verlag, New York.
Fourgeaud, C., and Fuchs, A. (1972).Statistique, Dunod, Paris.
Gindikin, S. G. (1975). Invariant generalized functions in homogeneous domains.Funct. Anal. Appl. 9, 50–52.
Ikeda, N., and Watanabe, S. (1981).Stochastic Differential Equations and Diffusion Processes, Amsterdam, North-Holland.
Johnson, N. L., and Kotz, S. (1972).Distributions in Statistics: Continuous Multivariate Distributions, J. Wiley and Sons, New York.
Kendall, D. G. (1984). Shape manifolds, procustean matrices, and complex projective spaces.Bull. London Math. Soc. 16, 81–121.
Kendall, W. (1989). The diffusion of Euclidean shape. Disorder in Physical Systems. InFestschift for Hammersley. To appear.
Letac, G. (1989). A characterization of the Wishart exponential families by an invariance property.J. Theor. Prob. 2, 71–86.
Lévy, P. (1948). The arithmetical character of the Wishart distribution.Proc. Cambridge Phil. Soc. 44, 295–297.
Marshall, A. W., and Olkin, I. (1979).Inequalities: Theory of Majorization and Its Applications, Academic Press, New York.
Magnus, I. R., and Neudecker, H. (1988).Matrix Differential Calculus with Applications in Statistics and Economics, J. Wiley and Sons, New York.
McKean, H. P. (1969).Stochastic Integrals, Academic Press, New York.
Norris, J. R., Rogers, L. C. G., and Williams, D. (1986). Brownian motions of ellipsoids.Trans. Amer. Math. Soc. 294, 757–765.
Pitman, J., and Yor, M. (1982). A decomposition of Bessel bridges.Z. Wahrsch. verw. Gebiete 59, 425–457.
Revuz, D., and Yor, M. (1990).Continuous Martingales and Brownian Motion. Springer-Verlag, New York.
Rogers, L. C. G., and Williams, D. (1987).Diffusions, Markov Processes, and Martingales: Vol. 2. Itô Calculus. J. Wiley and Sons, New York.
Shiga, T., and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes.Z. Wahrsch. verw. Gebiete 27, 37–46.
Siegel, C. L. (1936). The volume of the fundamental domain for some infinite groups.Trans. Amer. Math. Soc. 39, 209–218.
Williams, D. (1979).Diffusions, Markov Processes, and Martingales: Vol. 1. Foundations. J. Wiley and Sons, New York.
Williams, D. (1985).A Note on Brownian Motions and Symmetric Matrices. Unpublished manuscript.
Wishart, J. (1928). The generalized product moment distribution in samples from a normal multivariate population.Biometrika 20A, 32–52.
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Bru, MF. Wishart processes. J Theor Probab 4, 725–751 (1991). https://doi.org/10.1007/BF01259552
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DOI: https://doi.org/10.1007/BF01259552