Local growth of the sample paths of random fields with independent increments. II

1980 ◽  
Vol 19 (2) ◽  
pp. 229-239
Author(s):  
N. Kalinauskaite
Keyword(s):  
Random Fields ◽  
Local Growth ◽  
Sample Paths ◽  
Independent Increments

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).

A note on stochastic processes with independent increments taking values in an abelian group

1969 ◽  
Vol 6 (02) ◽  
pp. 449-452
Author(s):  
M.S. Bingham
Keyword(s):  
Abelian Group ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Time Points ◽  
Process With Independent Increments ◽  
Sample Paths ◽  
Finite Dimensional ◽  
Independent Increments ◽  
Processes With Independent Increments ◽  
Basic Ideas

It is well known that any stochastically continuous real valued stochastic process with independent increments defined on a compact time interval can be decomposed into a sum of independent processes, one of which is Gaussian with continuous sample paths, and the remainder of which have sample paths which are continuous except at a finite number of points with the discontinuities occurring at Poisson time points. The purpose of this note is to announce a proof of the above theorem in the case where the process takes values in an abelian group G. The detailed proof will appear elsewhere. The basic ideas of the proof in the case when G is finite dimensional Euclidean space are contained in Chapter VI of Gikhman and Skorohod (1965).

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 FRACTIONAL RANDOM FIELDS ON DOMAINS WITH FRACTAL BOUNDARY

2004 ◽  
Vol 07 (03) ◽  
pp. 395-417 ◽  
Author(s):  
M. D. RUIZ-MEDINA ◽  
J. M. ANGULO ◽  
V. V. ANH
Keyword(s):  
Fractional Order ◽  
Random Fields ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Weak Sense ◽  
Orthogonal Decomposition ◽  
Order Moment ◽  
Fractal Boundary ◽  
Markov Condition ◽  
Sample Paths

For random fields with fractional regularity order (respectively, fractional singularity order), an orthogonal decomposition of the associated reproducing kernel Hilbert space with respect to domains with fractal boundary is derived. The approach presented is based on the theory of generalized random fields on fractional Sobolev spaces. The orthogonal decomposition derived is equivalent to the weak-sense Markov condition, in the second-order moment sense, studied in Ref. 50, and based on the concept of splitting Hilbert spaces. A mean-square fractional order differential representation on bounded domains with fractal boundary is also obtained. In the Gaussian case, the random fields studied have fractal sample paths (see Ref. 1). Examples of fractional-order differential models in the class considered are provided.

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