## Strongly and weakly harmonizable stochastic processes of H-valued random variables.(English)Zbl 0589.60034

Author’s summary: Let H be a Hilbert space and ($$\Omega$$,$${\mathcal F},\mu)$$ be a probability measure space. Consider the Hilbert space $$L^ 2_ 0(\Omega,H)$$ consisting of all H-valued strong random variables on $$\Omega$$ with zero mean which are square integrable with respect to $$\mu$$. The author studies $$L^ 2_ 0(\Omega;H)$$-valued processes. It is shown that, as in the scalar valued case, every weakly harmonizable process is approximated pointwisely on R by a sequence of strong harmonizable processes. To prove this he obtains a series representation of a continuous process.
Reviewer: H.Salehi

### MSC:

 60G12 General second-order stochastic processes 60B11 Probability theory on linear topological spaces 46G10 Vector-valued measures and integration
Full Text:

### References:

 [1] {\scKakihara, Y.} Hilbert A-module-valued measures and a Riesz type theorem. Unpublished. [2] Kakihara, Y., A note on harmonizable and V-bounded processes, J. multivariate anal., 16, 140-156, (1985) · Zbl 0561.60041 [3] Kakihara, Y., Semivariation and operator semivariation of Hilbert space valued measures, (), 456-458 · Zbl 0547.28006 [4] Kallianpur, G.; Mandrekar, V., Spectral theory of stationary $$h$$-valued processes, J. multivariate anal., 1, 1-16, (1971) · Zbl 0248.60028 [5] Loeve, M., () [6] Mandrekar, V.; Salehi, H., The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana univ. math. J, 20, 545-563, (1970) · Zbl 0252.46040 [7] Niemi, H., Stochastic processes as Fourier transforms of stochastic measures, Ann. acad. sci. fenn. ser. A I math., 591, 1-47, (1975) · Zbl 0307.60034 [8] Ozawa, M., Hilbert B(H)-modules and stationary processes, Kodai math. J, 3, 26-39, (1980) · Zbl 0435.60033 [9] Rao, M.M., Representations of weakly harmonizable processes, (), 5288-5289 · Zbl 0529.60032 [10] Rao, M.M., Harmonizable processes: structure theory, Enseign. math., 28, 295-351, (1982) · Zbl 0501.60046 [11] Rosenberg, M., Spectral integrals of operator valued functions. II, from the study of stationary processes, J. multivariate anal., 6, 538-571, (1976) · Zbl 0343.47017 [12] Rozanov, Yu.A., Spectral theory of abstract functions, Theory probab. appl., 4, 271-287, (1959) · Zbl 0089.32602 [13] Salehi, H., The continuation of Wiener’s work on q-variate linear prediction and its extension to infinite-dimensional spaces, (), 307-338, Norbert Wiener [14] Schatten, R., () [15] Umegaki, H.; Bharucha-Reid, A.T., Banach space-valued random variables and tensor products of Banach spaces, J. math. anal. appl., 31, 49-67, (1970) · Zbl 0292.60008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.