scholarly journals The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism

Bernoulli ◽  
2022 ◽  
Vol 28 (1) ◽  
Author(s):  
Cheuk Yin Lee
Keyword(s):  
Hausdorff Measure ◽  
Random Fields ◽  
Level Sets ◽  
Gaussian Random Fields ◽  
Local Nondeterminism

A CLASS OF FRACTIONAL BROWNIAN FIELDS FROM BRANCHING SYSTEMS AND THEIR REGULARITY PROPERTIES

2013 ◽  
Vol 16 (03) ◽  
pp. 1350023 ◽  
Author(s):  
YUQIANG LI ◽  
YIMIN XIAO
Keyword(s):  
Modulus Of Continuity ◽  
Random Fields ◽  
Sample Path ◽  
Gaussian Random Fields ◽  
Particle Systems ◽  
Sample Path Properties ◽  
Regularity Properties ◽  
Fractal Properties ◽  
Local Nondeterminism ◽  
Path Properties

In this paper, the smoothness and exact modulus of continuity of a class of fractional Brownian fields are studied. These Gaussian random fields satisfy a kind of operator-scaling property and, depending on the choice of their parameters, may share similar fractal properties as those of fractional Brownian sheets or may be smooth in some (or all) directions. It is proved that these Gaussian random fields satisfy the property of sectorial local nondeterminism which is useful for further studying their sample path properties. In addition, the link between these Gaussian random fields and the functional fluctuation limits of branching particle systems is studied.

Conditional Local Nondeterminism and Hausdorff Measure of Level Sets

1991 ◽  
Vol 34 (1) ◽  
pp. 123-127
Author(s):  
Narn-Rueih Shieh
Keyword(s):  
Stochastic Process ◽  
Level Set ◽  
Hausdorff Measure ◽  
Level Sets ◽  
Positive Probability ◽  
Large Classes ◽  
Measure Function ◽  
Fixed Level ◽  
Local Nondeterminism

AbstractLet X be a real stochastic process. We localize S. M. Berman's formulation on the local nondeterminism of X to a fixed level. With this localized idea, we prove that, for large classes of Gaussian and Markov X, at each x the level set X(t, w) = x has infinite Hausdorff ϕ - measure (ϕ is certain measure function) for w in a set of positive probability.

Comparison of Nonlinear Spatial Correlation Models by the Influence of the Data Augmentation to the Classification Risk

2002 ◽  
Vol 7 (1) ◽  
pp. 31-42
Author(s):  
J. Šaltytė ◽  
K. Dučinskas
Keyword(s):  
Spatial Correlation ◽  
Random Fields ◽  
Data Augmentation ◽  
Gaussian Random Fields ◽  
Classification Rule ◽  
Numerical Comparison ◽  
First Order ◽  
Bayesian Risk ◽  
Correlation Models

The Bayesian classification rule used for the classification of the observations of the (second-order) stationary Gaussian random fields with different means and common factorised covariance matrices is investigated. The influence of the observed data augmentation to the Bayesian risk is examined for three different nonlinear widely applicable spatial correlation models. The explicit expression of the Bayesian risk for the classification of augmented data is derived. Numerical comparison of these models by the variability of Bayesian risk in case of the first-order neighbourhood scheme is performed.

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