Extreme values and crossings for the X2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

1980 ◽  
Vol 12 (3) ◽  
pp. 746-774 ◽  
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Processes ◽  
Survival Probability ◽  
Extreme Values ◽  
Extreme Points ◽  
Linear Functions ◽  
Degree Polynomial ◽  
Engineering Sciences ◽  
Multivariate Gaussian ◽  
Safe Region ◽  
Image Position

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks the survival probability where X(t) = (X1(t), …, Xn(t)) is a stationary, multivariate Gaussian load process, and S is a safe region. In general, the asymptotic survival probability for large T-values is the most interesting quantity.By considering the point process formed by the extreme points of the vector process X(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x1, …, xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf>x∉S ||x||, i.e. the smallest distance from the origin to an unsafe point.

Extreme values and crossings for the X 2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

1980 ◽  
Vol 12 (03) ◽  
pp. 746-774
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Processes ◽  
Survival Probability ◽  
Extreme Values ◽  
Extreme Points ◽  
Linear Functions ◽  
Degree Polynomial ◽  
Engineering Sciences ◽  
Multivariate Gaussian ◽  
Safe Region ◽  
Image Position

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks the survival probability where X(t) = (X 1(t), …, X n (t)) is a stationary, multivariate Gaussian load process, and S is a safe region. In general, the asymptotic survival probability for large T-values is the most interesting quantity. By considering the point process formed by the extreme points of the vector process X(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x 1, …, x n ). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf>x∉S ||x||, i.e. the smallest distance from the origin to an unsafe point.

Extreme values of the cyclostationary Gaussian random process

1993 ◽  
Vol 30 (01) ◽  
pp. 82-97 ◽  
Author(s):  
D. G. Konstant ◽  
V.I. Piterbarg
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Gaussian Processes ◽  
Extreme Values ◽  
Random Processes ◽  
Mean Square ◽  
Gaussian Random Process ◽  
Covariance Functions ◽  
Limit Formula ◽  
Image Position

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

Extreme values of the cyclostationary Gaussian random process

1993 ◽  
Vol 30 (1) ◽  
pp. 82-97 ◽  
Author(s):  
D. G. Konstant ◽  
V.I. Piterbarg
Keyword(s):  
Gaussian Process ◽  
Random Process ◽  
Gaussian Processes ◽  
Extreme Values ◽  
Random Processes ◽  
Mean Square ◽  
Gaussian Random Process ◽  
Covariance Functions ◽  
Limit Formula ◽  
Image Position

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

scholarly journals Extreme values of a portfolio of Gaussian processes and a trend

Extremes ◽  
2005 ◽  
Vol 8 (3) ◽  
pp. 171-189 ◽  
Author(s):  
Jürg Hüsler ◽  
Christoph M. Schmid
Keyword(s):  
Gaussian Processes ◽  
Extreme Values

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

scholarly journals On Kadison's condition for extreme points of the unit ball in a B*-algebra

1969 ◽  
Vol 16 (3) ◽  
pp. 245-250 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Unit Ball ◽  
Banach Algebra ◽  
Extreme Points ◽  
Complex Banach Algebra ◽  
Image Position

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which

scholarly journals I.—Elimination in the case of Equality of Fractions whose Numerators and Denominators are Linear Functions of the Variables.

1906 ◽  
Vol 45 (1) ◽  
pp. 1-7
Author(s):  
Thomas Muir
Keyword(s):  
Linear Functions ◽  
Short Paper ◽  
Left Hand ◽  
Image Position ◽  
Factor Α

(1) It is well known that if equations of the type referred to in the title be dealt with like ordinary quadrics, the eliminant obtained is marred by association with an irrelevant factor. Thus, to take the simplest case, viz.the left-hand member of which contains the irrelevant factor |α2β2|, being readily shown to be equal toThe object of the present short paper is to draw attention to other modes of procedure, and to formulate the results for n variables.

scholarly journals Joint segmentation of multivariate Gaussian processes using mixed linear models

2011 ◽  
Vol 55 (2) ◽  
pp. 1160-1170 ◽  
Author(s):  
F. Picard ◽  
E. Lebarbier ◽  
E. Budinskà ◽  
S. Robin
Keyword(s):  
Gaussian Processes ◽  
Linear Models ◽  
Mixed Linear Models ◽  
Multivariate Gaussian

The extreme points of some classes of polynomials

1985 ◽  
Vol 101 (1-2) ◽  
pp. 99-110 ◽  
Author(s):  
D. A. Brannan ◽  
J. G. Clunie
Keyword(s):  
Extreme Points ◽  
The Other ◽  
Other Hand ◽  
Image Position

SynopsisWe study the extreme points of two classes of polynomials of degree at most n:It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.

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