Joint measures and cross-covariance operators

Author:
Charles R. Baker

Journal:
Trans. Amer. Math. Soc. **186** (1973), 273-289

MSC:
Primary 60G15; Secondary 28A40

DOI:
https://doi.org/10.1090/S0002-9947-1973-0336795-3

MathSciNet review:
0336795

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let ${H_1}$ (resp., ${H_2}$) be a real and separable Hilbert space with Borel $\sigma$-field ${\Gamma _1}$ (resp., ${\Gamma _2}$), and let $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$ be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$, i.e., joint measures, and the projections of such measures on $({H_1},{\Gamma _1})$ and $({H_2},{\Gamma _2})$. In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are obtained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized.

- Charles R. Baker,
*Mutual information for Gaussian processes*, SIAM J. Appl. Math.**19**(1970), 451–458. MR**266672**, DOI https://doi.org/10.1137/0119044 - R. G. Douglas,
*On majorization, factorization, and range inclusion of operators on Hilbert space*, Proc. Amer. Math. Soc.**17**(1966), 413–415. MR**203464**, DOI https://doi.org/10.1090/S0002-9939-1966-0203464-1 - Jacob Feldman,
*Equivalence and perpendicularity of Gaussian processes*, Pacific J. Math.**8**(1958), 699–708. MR**102760** - I. M. Gel’fand and N. Ya. Vilenkin,
*Generalized functions. Vol. 4: Applications of harmonic analysis*, Academic Press, New York - London, 1964, 1964. Translated by Amiel Feinstein. MR**0173945** - I. M. Gel′fand and A. M. Yaglom,
*Calculation of the amount of information about a random function contained in another such function*, Amer. Math. Soc. Transl. (2)**12**(1959), 199–246. MR**0113741** - Yaroslav Gaek,
*On a property of normal distribution of any stochastic process*, Czechoslovak Math. J.**8(83)**(1958), 610–618 (Russian, with English summary). MR**104290** - Jaroslav Hájek,
*On linear statistical problems in stochastic processes*, Czechoslovak Math. J.**12(87)**(1962), 404–444 (English, with Russian summary). MR**152090** - Kiyoshi Itô,
*The topological support of Gauss measure on Hilbert space*, Nagoya Math. J.**38**(1970), 181–183. MR**261332** - Edith Mourier,
*Eléments aléatoires dans un espace de Banach*, Ann. Inst. H. Poincaré**13**(1953), 161–244 (French). MR**64339** - K. R. Parthasarathy,
*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684** - C. Radhakrishna Rao and V. S. Varadarajan,
*Discrimination of Gaussian processes*, Sankhyā Ser. A**25**(1963), 303–330. MR**183090** - Frédéric Riesz and Béla Sz.-Nagy,
*Leçons d’analyse fonctionnelle*, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR**0056821** *Proceedings of the Symposium on Time Series Analysis*, John Wiley and Sons, Inc., New York-London, 1963. Held at Brown University, June 11-14. MR**0145634**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60G15,
28A40

Retrieve articles in all journals with MSC: 60G15, 28A40

Additional Information

Keywords:
Joint measures,
Gaussian measures,
absolute continuity of measures,
covariance operators,
mutual information

Article copyright:
© Copyright 1973
American Mathematical Society