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1

J-regularity of multivariate stationary sequences for some families J

Klotz L., Mädler C.

Probability and Mathematical Statistics

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2015

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Vol. 35, Fasc. 1

31--40

EN

Let n ϵ N, S be a nonempty finite subset of the set of integers, Sc be its complement, and J be the family of translations of Sc by ln, l ϵ Z. For such a family, J-regularity of a q-variate stationary sequence over Z is studied. If S contains exactly n elements, a description of a J-regular sequence in terms of its spectral density is obtained. Some examples are given for the case where S contains more than n elements.

2

Periodic observations of harmonizable symmetric stable sequences

Klotz L., Riedel M.

Probability and Mathematical Statistics

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2005

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Vol. 25, Fasc. 2

289--306

EN

For harmonizable symmetric stable sequences we solve the following prediction problem: Assume that the values of the sequence are known at all odd integers. Compute the metric projection of an unknown value onto the space spanned by the known values as well as the corresponding approximation error. We study several questions related to this prediction problem such as regularity and singularity, Wold type decomposition, interrelations between the spaces spanned by the values at the even and odd integers, respectively.

3

The maximal J-regular part of a q-variate weakly stationary process

Klotz L.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

155--165

EN

Let x be a q-variate (weakly) stationary process over a locally compact Abelian group G, and J a family of subsets of G invariant under translation. We show that the set of all regular non-negative Hermitian matrix-valued measures M not exceeding the (non-stochastic) spectral measure of x and such that the Hilbert space L2(M) is J-regular contains a unique maximal element. Moreover, this maximal element coincides with the spectral measure of the J-regular part of the Wold decomposition of x.

4

Spectral representation and extrapolation of stationary random processes on linear spaces

Klotz L., Riedel M.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

179--197

EN

The paper deals with continuous Banach-space-valued stationary random processes on linear spaces. From von Waldenfels’ [13] integral representation of positive definite functions on a linear space L we derive an analogue of Stone’s theorem for a group of unitary operators over L. It is used to obtain spectral representations of a general Banach-space-valued stationary random process over L and its covariance function. For the special class of Hilbert-Schmidt operator-valued stationary processes the explicit form of Kolmogorov’s isomorphism theorem between temporal space and spectral space is established and with its aid there are studied some prediction problems. Our prediction results are similar to those proved in [5] for multivariate stationary processes on groups.

5

On the completeness of some Lp-spaces of operator-valued functions

Klotz L.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

463--468

EN

In [3] there were studied Banach spaces of (equivalence classes of) functions Φ whose values are unbounded operators, in general, and which are p-integrable with respect to operator-valued measures having an operator density N with respect to some non-negative scalar measure μ. In the present short note it is shown that the values of all functions Φ are even bounded linear operators if and only if there is not any set A of positive finite measure μ such that the values of N on A have non-closed ranges. The result is used to answer a question raised by Górniak et al. [2].

6

Some remarks on Jo-regularity and Jo-singularity of q-variate stationary processes

Klotz L., Schmidt F.

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 2

351--357

EN

We give a new proof of Makagon’s and Weron’s criterion for Jo-regularity (see [4], Theorem 5.3), and discuss some conditions of Jo-singularity of q-variate stationary processes.