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National Repository of Grey Literature 2 records found  Search took 0.00 seconds. 
α-symmetric measures
 Ranošová, Hedvika ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
Spherically symmetric measures in Rn are rotationally invariant, indicating that their characteristic functions can be written as a composition of the Euclidean norm with a univariate function. If we replace the Euclidean norm with an ℓα norm, the resulting distributions are known as α-symmetric. This thesis aims to provide a general description of α-symmetric measures and explore various non-trivial examples. The existence of α- symmetric measures for a given α and dimension n ∈ N is discussed, along with the connection between the existence of α-symmetric measures and isometric embedding into Lp spaces through strictly stable distributions. One of the main properties explored in this thesis is the relationship between moments of non-integer order and α-symmetry in distributions. Additionally, several sufficient conditions for the existence and the form of α-symmetric measures are described. In the final chapter, a further generalization of α-symmetric distributions toward quasi-norms is discussed, along with the properties of the resulting concept of pseudo-isotropy. 1
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Spherically symmetric measures
 Ranošová, Hedvika ; Nagy, Stanislav (advisor) ; Dvořák, Jiří (referee)
A probability distribution is called spherically symmetric if it is invariant with respect to rotations about the origin. This class includes the multivariate standard normal distribution, a multivariate extension of the t-distribution and uniform distribu- tions inside the unit ball or the unit sphere surface. The first part of the thesis summarizes the basic properties of spherically symmetric distributions such as the form of their char- acteristic function and provides expressions for their moments and the density function. It turns out that spherically symmetric distributions are fully characterized by the dis- tribution of their Euclidean norm or by any of their univariate marginal distributions. As any marginal distribution of a spherically symmetric distribution is also spherically symmetric, the aim of the second part of this thesis is to study the inverse relationship using fractional calculus. For a given n-dimensional spherically symmetric distribution we solve the problem of deciding whether there is a spherically symmetric distribution in higher dimensions whose n-dimensional marginal is as given. 1
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