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Extra resources for Multivariate Analysis–III. Proceedings of the Third International Symposium on Multivariate Analysis Held at Wright State University, Dayton, Ohio, June 19–24, 1972

Example text

E. e. convergence of the martingale is equivalent to that of the sequence (1). Proof. The simple proof will be given here to illustrate the problem. First note that if vh(A) = jA h άμ, Λ e sé, then vh is semibounded. Jn fact, the variation measure | ν Λ |(·) is σ-fìnite, which trivially implies the asserted prop­ erty of vh. To see the former property, /? (and hn) being clearly measurable, let Sk = {k} x Ω G sé, and S = U*°°=i Sk. Then | vh\(Sk) = JSJA| άμ = ^\Tkf\ dP ^ \\T\\k JQI/I dP < oo, so that | vh| is σ-fìnite.

C. and Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. (English translation. ) 4. Hida, T. (1960). Canonical representations of Gaussian processes and their applications. Mem. Coll. , Univ. Kyoto Ser. A 33 109-155. 5. Hitsuda, M. (1968). Representation of Gaussian processes equivalent to Wiener process. Osaka J. Math. 5 299-312. 44 G. KALLIANPUR 6. Kailath, T. and Duttweiler, D. (1972). An RKHS approach to detection and estimation problems, Part III: Generalized innovations, representations and a likelihood-ratio formula.

Math. 9 699-708. 2. Girsanov, I. V. (1960). On transforming a certain class of stochastic processes by abso­ lutely continuous substitution of measures. Theor. Probability Appi. 5 285-301. 3. Gohberg, I. C. and Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. (English translation. ) 4. Hida, T. (1960). Canonical representations of Gaussian processes and their applications. Mem. Coll. , Univ. Kyoto Ser. A 33 109-155. 5. Hitsuda, M. (1968). Representation of Gaussian processes equivalent to Wiener process.

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