The structure of Gaussian fields near a level crossing

1982 ◽  
Vol 14 (3) ◽  
pp. 543-565 ◽  
Author(s):  
Richard J. Wilson ◽  
Robert J. Adler
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Final Section ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Gaussian Fields ◽  
Level Crossing ◽  
Model Field ◽  
Definition Of

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

The structure of Gaussian fields near a level crossing

1982 ◽  
Vol 14 (03) ◽  
pp. 543-565 ◽  
Author(s):  
Richard J. Wilson ◽  
Robert J. Adler
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Final Section ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Gaussian Fields ◽  
Level Crossing ◽  
Model Field ◽  
Definition Of

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

Model fields in crossing theory: a weak convergence perspective

1988 ◽  
Vol 20 (4) ◽  
pp. 756-774
Author(s):  
Richard J. Wilson
Keyword(s):  
Weak Convergence ◽  
Random Field ◽  
Lebesgue Measure ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Asymptotic Results ◽  
Model Field ◽  
Fixed Level ◽  
Excursion Set ◽  
Crossing Theory

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

Model fields in crossing theory: a weak convergence perspective

1988 ◽  
Vol 20 (04) ◽  
pp. 756-774
Author(s):  
Richard J. Wilson
Keyword(s):  
Weak Convergence ◽  
Random Field ◽  
Lebesgue Measure ◽  
Gaussian Random Field ◽  
Asymptotic Distributions ◽  
Asymptotic Results ◽  
Model Field ◽  
Fixed Level ◽  
Excursion Set ◽  
Crossing Theory

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

The horizon problem for random surfaces

1991 ◽  
Vol 109 (1) ◽  
pp. 211-219 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Computer Simulation ◽  
Stochastic Processes ◽  
Random Field ◽  
Random Fields ◽  
Sample Function ◽  
Gaussian Fields ◽  
Random Surfaces ◽  
Horizon Problem ◽  
Typical Sample

Computer simulation of landscapes and skylines has recently attracted a great deal of interest: see [6, 7]. Specification of a ‘landscape’ requires a function f: D → ℝ on a subset D of ℝ2, selected so that the apparent irregularity and randomness of the surface {(t,f(t)): t ∈ D} corresponds to what might be observed in nature. It is natural to look to random fields (that is, stochastic processes in two variables), and in particular to Gaussian fields, for functions with such properties. Even when an appropriate random field has been selected, determination of a typical sample function is far from easy [7].

scholarly journals Independence of the increments of Gaussian random fields

1982 ◽  
Vol 85 ◽  
pp. 251-268 ◽  
Author(s):  
Kazuyuki Inoue ◽  
Akio Noda
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Euclidean Distance ◽  
Gaussian Random Field ◽  
Gaussian Random Fields

Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.

A VARIATIONAL FORMULA FOR SOME RANDOM FIELDS: AN ANALOGUE OF ITO'S FORMULA

1999 ◽  
Vol 02 (02) ◽  
pp. 305-313
Author(s):  
SI SI
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Gaussian Random Field ◽  
Nonlinear Function ◽  
Variational Formula ◽  
Canonical Representation ◽  
Two Dimensional ◽  
Dimensional Parameter

We shall first establish a canonical representation of a Gaussian random field X(C) indexed by a smooth contour C in terms of two-dimensional parameter white noise. Then, we take a nonlinear function F(X(C)) of the X(C) and obtain its variation when C deforms slightly. The variational formula is analogous to the Ito formula for a stochastic process X(t), but somewhat simpler.

On the asymptotic form of convex hulls of Gaussian random fields

2014 ◽  
Vol 12 (5) ◽  
Author(s):  
Youri Davydov ◽  
Vygantas Paulauskas
Keyword(s):  
Banach Space ◽  
Asymptotic Behavior ◽  
Random Field ◽  
Hausdorff Distance ◽  
Random Fields ◽  
Asymptotic Form ◽  
Gaussian Random Field ◽  
Gaussian Random Fields ◽  
Convex Hulls ◽  
Limit Set

AbstractWe consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $$\mathbb{B}$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n), where f is some function, is also studied.

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