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1
Content available remote Urbanik type subclasses of free-infinitely~divisible~transforms
Jurek Zbigniew J.
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 1
23--39
EN
For the class of free-infinitely divisible transforms we introduce three families of increasing Urbanik type subclasses. They begin with the class of free-normal transforms and end up with the whole class of free- infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which random integral repre- sentations are known. Special functions like Hurwitz–Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.
2
Content available remote On a Relation Between Classical and Free Infinitely Divisible Transforms
Jurek Zbigniew J.
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
349--367
EN
We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.
3
Content available remote Advantages of the Use of Characteristic Functions for the Statistical Analysis of Technological Processes
Bondariev A., Nestor N.
Machine Dynamics Research
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2017
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Vol. 41, No. 3
5--12
EN
This paper proposed method of statistical analysis of technological processes with the use of characteristic functions makes it possible to construct algorithms that operate with multidimensional probability distributions of relative deviations of parameters of individual technological operations, to calculate the distribution in sections and at the output of the technological process.
4
Content available remote Variance of random signal mean square value digital estimator
Lal-Jadziak J., Sienkowski S.
Metrology and Measurement Systems
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2009
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Vol. 16, nr 2
267-277
EN
In the article, original relations enabling the estimation of the variance of a random signal mean square value digital estimator are derived. Three cases are considered: first when the estimator is determined from quantized samples; second, when it is additionally assumed that the conditions of Widrow's theorem are satisfied; and third, when the samples have not been quantized. The obtained relations can be used e.g. to determine uncertainty in precision measurements and to evaluate signal degradation in radio astronomy. As an example, the variance of the mean square value estimator of a random Gaussian signal for the three above-mentioned situations is analyzed. It has been observed that in the first and second cases, an increase in variance as well as in type A standard uncertainty takes place in comparison with the estimation based on unquantized samples. This increase diminishes along with an increase in the ratio of the signal rms value to the quantization step size.
5
Content available remote Models of bias of mean square value digital estimator for selected deterministic and random signals
Lal-Jadziak J., Sienkowski S.
Metrology and Measurement Systems
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2008
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Vol. 15, nr 1
55-67
EN
The article presents the probability density functions as well as characteristic functions of selected periodic and random signals. On their basis, original models of the bias of the mean square value digital estimator have been designed. These models were employed in investigating estimation errors caused by analog to digital conversion and analog to digital conversion with a dither signal. Selected graphical model representations and their analyses are demonstrated. It has been shown that for a triangular probability density function random signal with the amplitude At = kq, k∈N \ {0}, mean square value reconstruction occurs on the basis of a signal quantized with the accuracy of Sheppard's correction. Whereas for periodic signals as well as for the sum of periodic and random signals, the δ component of bias due to the nonsatisfaction of the reconstruction condition is a suppressed oscillating function of the quotient of the amplitude A and the quantization step size q. It has been proved that by adding, prior to quantization, a triangular distribution random signal with zero mean and the amplitude At = kq (k = 1, 2, ...) in the mean square value measurement of any periodic signal, this bias component can be brought to zero.
6
Content available Modelowanie obciążenia estymatora wartości średniokwadratowej wybranych klas sygnałów
Lal-Jadziak J., Sienkowski S.
Pomiary Automatyka Kontrola
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2007
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R. 53, nr 9 bis
93--96
PL
Modele obciążenia znajdują zastosowanie w badaniach estymatorów parametrów i charakterystyk sygnałów, a także w określaniu ich niepewności. W publikacji przedstawiono modele obciążenia estymatora wartości średniokwadratowej powodowanego kwantowaniem. Specjalne miejsce poświęcono sygnałom poliharmonicznym oraz sygnałom poliharmonicznym z sygnałami losowymi o rozkładzie równomiernym, gaussowskim oraz trójkątnym.
EN
Models of bias are used in research of parameters and characteristics of signal estimators and in determination their uncertainties. In this article are presented models of mean square value estimator bias caused by quantization. Special attention is paid to the poliharmonic signals and poliharmonic signals with uniform, Gaussian and triangular PDF signal.
7
Content available remote Langevin equation as a model of COP sway
Bosek M., Grzegorzewski B., Kowalczyk A.
Biocybernetics and Biomedical Engineering
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2005
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Vol. 25, no. 2
53-59
EN
Static posturography is a method of analysis of the human postural control system, in which the center of pressure (COP) sway at quiet standing is investigated. In this study, the Langevin equation is used as a model of the COP sway. The solution of this equation is the probability density function P(r,t/r(t0),t0). For experimental data, it is possible to represent this function in the form of a histogram. In the wave vector space, the probability density function is represented by the characteristic function, containing full information about the distribution of the variable r - r(t0). For young, elderly and parkinsonian populations, for the excluded as well as for the included visual input, for short wave vector k, the experimental functions are shown to be wen approximated by the Gaussian theoretical functions. However, in these cases, for greater wave vectors, non-Gaussian properties of this variable are observed.
8
Content available remote Alternative conditions for attraction to stable vectors
de Haan L., Li D., Peng L., Iglesias-Pereira H.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 2
303--317
EN
Relying on Geluk and de Haan [3] we derive alternative necessary and sufficient conditions for the domain of attraction of a stable distribution in Rd which are phrased entirely in terms of (joint distributions of) linear combinations of the marginals. The conditions in terms of characteristic functions should be useful for determining rates of convergence, as in de Haan and Peng [4].
9
Content available remote Stable probability distributions and their domains of attraction : a direct approach
Geluk J. L., de Haan L.
Probability and Mathematical Statistics
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2000
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Vol. 20, Fasc. 1
169--188
EN
The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.
10
Content available remote The generalization of the Kac-Bernstein Theorem
Quine M. P., Seneta E.
Probability and Mathematical Statistics
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1999
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Vol. 19, Fasc. 2
441--452
EN
The Skitovich-Darmois Theorem of the early 1950's establishes the normality of independent X1, X2,…, Xn from the independence of two linear forms in these random variables. Existing proofs generally rely on the theorems of Marcinkiewicz and Cramér, which are based on analytic function theory. We present a self-contained real-variable proof of the essence of this theorem viewed as a generalization of the case n = 2, which is generally called Bernstein's Theorem, and also adapt an early little known argument of Kac to provide a direct simple proof when n = 2. A large bibliography is provided.
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