Some remarks on the Hausdorff dimension of a multiple point set of random fields with continuous sample paths

1992 ◽  
Vol 32 (3) ◽  
pp. 285-290
Author(s):  
N. Kalinauskaitė
Keyword(s):  
Hausdorff Dimension ◽  
Random Fields ◽  
Multiple Point ◽  
Sample Paths ◽  
Point Set

scholarly journals Extremes of Homogeneous Gaussian Random Fields

2015 ◽  
Vol 52 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Natalia Soja-Kukieła
Keyword(s):  
Correlation Function ◽  
Asymptotic Expansion ◽  
Random Fields ◽  
Survival Function ◽  
Random Variable ◽  
Gaussian Random Fields ◽  
Extremal Index ◽  
Gaussian Field ◽  
Gaussian Fields ◽  
Sample Paths

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn1(u),tn2(u)) ∈[0,x]∙[0,y]X(s, t) ≤ u), where n1(u)n2(u) = u2/α1+2/α2Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

Hausdorff Dimension of Operator Stable Sample Paths

2003 ◽  
Vol 140 (2) ◽  
pp. 91-101 ◽  
Author(s):  
Peter Becker-Kern ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Hausdorff Dimension ◽  
Sample Paths ◽  
Operator Stable

scholarly journals Random fields of bounded variation and computation of their variation intensity

2016 ◽  
Vol 48 (4) ◽  
pp. 947-971
Author(s):  
Bruno Galerne
Keyword(s):  
Random Field ◽  
Bounded Variation ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Random Sets ◽  
Functions Of Bounded Variation ◽  
Boolean Models ◽  
Sample Paths ◽  
Directional Variation ◽  
The Mean

AbstractThe main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

scholarly journals Extremes of Homogeneous Gaussian Random Fields

2015 ◽  
Vol 52 (01) ◽  
pp. 55-67 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Enkelejd Hashorva ◽  
Natalia Soja-Kukieła
Keyword(s):  
Correlation Function ◽  
Asymptotic Expansion ◽  
Random Fields ◽  
Survival Function ◽  
Random Variable ◽  
Gaussian Random Fields ◽  
Extremal Index ◽  
Gaussian Field ◽  
Gaussian Fields ◽  
Sample Paths

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2 ), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) &lt; 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn 1(u),tn 2(u)) ∈[0,x]∙[0,y] X(s, t) ≤ u), where n 1(u)n 2(u) = u 2/α1+2/α2 Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

scholarly journals On the carrying dimension of occupation measures for self-affine random fields

2019 ◽  
Vol 39 (2) ◽  
pp. 459-479
Author(s):  
Peter Kern ◽  
Ercan Sönmez
Keyword(s):  
Hausdorff Dimension ◽  
Large Class ◽  
General Formula ◽  
Random Fields ◽  
Value Function ◽  
Singular Value ◽  
Occupation Measure ◽  
Occupation Measures ◽  
Alternative Approach ◽  
Path Properties

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle 1984, 1988, 1990, 1991. Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.

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