Gaussian and their Subordinated Self-similar Random Generalized Fields
Reads0
Chats0
TLDR
In this paper, a large class of generalized random fields is defined, containing random elements $F$ of $\mathscr{J}'', where the dual of the Schwartz space is defined.Abstract:
A large class of generalized random fields is defined, containing random elements $F$ of $\mathscr{J}'$, where $\mathscr{J}'$ is the dual of the Schwartz space $\mathscr{J} = \mathscr{J}(\mathbb{R}^\nu)$. Such a generalized random field is translation-invariant if $F\phi$ is the same as $F\psi$ for any translate $\psi$ of $\phi$; it is invariant under the renormalization group with index $_\kappa$ (or self-similar with index $_\kappa$) if $F\phi_\lambda = \lambda^{-\alpha}F\phi$ for all $\lambda > 0$ and $\phi \in \mathscr{L}$, where $\phi_\lambda$ is the rescaled test function $\phi_\lambda(x) = \lambda^{-\nu}\phi(x/\lambda)$. Recent work of several authors has shown that self-similar generalized random fields on $\mathbb{R}^\nu$, and self-similar random fields on $\mathbb{Z}^\nu$ which can be constructed from them, arise naturally in problems of statistical mechanics and limit laws of probability theory. They generalize the theory of stable distributions. Here the class of all translation-invariant self-similar Gaussian generalized random fields on $\mathbb{R}^\nu$ is completely described. By the discretization of such fields the class of self-similar Gaussian fields with discrete arguments (found by Sinai) is extended. Finally, a class of generalized random fields subordinated to the self-similar translation-invariant Gaussian ones is constructed. These non-Gaussian generalized random fields are Wick powers (multiple Ito integrals) of the Gaussian ones.read more
Citations
More filters
Book
The Fractal Geometry of Nature
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Book
Random Fields and Geometry
Robert J. Adler,Jonathan Taylor +1 more
TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.
Journal ArticleDOI
Convergence of integrated processes of arbitrary Hermite rank
TL;DR: In this article, the weak limit in C[0, 1] of the integrated process is investigated, and it is shown that it converges for all m≧1 to some process that depends essentially on m.
Journal ArticleDOI
Non-central limit theorems for non-linear functional of Gaussian fields
R. L. Dobrushin,Péter Major +1 more
TL;DR: In this article, the authors studied the limit behavior as N→∞ and showed that the norming constants tend to infinity more rapidly than the usual norming sequence when the correlation function r(n) tends slowly to 0, and generalized the results to the case when the parameter set is multi-dimensional.
Non-central limit theorems of non-linear functions of Gaussian fields
TL;DR: In this paper, a stationary Gaussian sequence Xn, n = −1,0,1,... and a real function H(x) was given, where ANs were appropriate norming constants.
Related Papers (5)
Non-central limit theorems for non-linear functional of Gaussian fields
R. L. Dobrushin,Péter Major +1 more