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Positive definite norm dependent matrices in stochastic modeling

Kuniewski S. P., Misiewicz J.K.

Demonstratio Mathematica

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2014

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Vol. 47, nr 1

211--231

EN

Positive definite norm dependent matrices are of interest in stochastic modeling of distance/norm dependent phenomena in nature. An example is the application of geostatistics in geographic information systems or mathematical analysis of varied spatial data. Because the positive definiteness is a necessary condition for a matrix to be a valid correlation matrix, it is desirable to give a characterization of the family of the distance/norm dependent functions that form a valid (positive definite) correlation matrix. Thus, the main reason for writing this paper is to give an overview of characterizations of norm dependent real functions and consequently norm dependent matrices, since this information is somehow hidden in the theory of geometry of Banach spaces.

2

Some remarks on S alpha S, beta - substable random vectors

Misiewicz J.K., Takenaka S.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

253--258

EN

An S α S random vector Xis β-substable, α < β ≤ 2, if Xd = Y Θ 1/β for some symmetric β-stable random vector Y Θ ≥ 0 a random variable with the Laplace transform exp{−tα/β}, Y and Θ are independent. We say that an S α S random vector is maximal if it is not β-substable for any β > α. In the paper we show that the ca,nonical spectral measure for every S α S, β-substablerandom vector X, β > α is equivalent to the Lebesgue measure on Sn−1.We show also that every such vector admits the representation X=Y+Z, where Y is an S α S sub-Gaussian random vector, Z is a maximal S α S random vector, Y and Z are independent. The last representation is not unique.

3

On the Dugue problem with a solution in the set of signed measures

Malinowski M. T., Misiewicz J. K.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

319--331

EN

There are two methods of obtaining symmetric probability measure ona base of an arbitrary probability measure μ corresponding to the random variable X. The first relies on considering distribution of Y=X−X′, where X′ is an independent copy of X. In the language of measures we have then L(Y) = μ∗μ−, where μ−(A) = μ(−A). In the second method we consider the mean of two measures μ and μ−. In the paper we want to present some known and new results on characterizing such measures μ for which both methods coincide, i.e. measures for which [formula]. In the literature one can find also the following generalization of this question: for fixed p∈ (0,1 ]what is the characterization of such pairs of distributions μ and ν for which pμ+ (1−p)ν=μ∗ν? This problem was posed by Dugué in 1939 and it was extensively studied since then. However, the full characterization has not been found yet. In the paper we show some constructions of the Dugué question with the properties of simple fractions classes of characteristic functions. We give also a collection of new solutions and an example of three measures μ, ν and η such that pμ+qν+rη=μ∗ν∗η. In the last section we give also some solutions in the set of signed σ-finite measures. The authors would like to express their gratitude to Professor D. Szynal for his interesting questions and discussions.