A gaussian process with parabolic covariances

1991 ◽  
Vol 28 (04) ◽  
pp. 898-902 ◽  
Author(s):  
Enrique M. Cabaña
Keyword(s):  
Gaussian Process ◽  
Brownian Bridge ◽  
Stationary Gaussian Process ◽  
String Equation ◽  
Distribution Of The Maximum ◽  
Close Relationship ◽  
Vibrating String

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.

A gaussian process with parabolic covariances

1991 ◽  
Vol 28 (4) ◽  
pp. 898-902 ◽  
Author(s):  
Enrique M. Cabaña
Keyword(s):  
Gaussian Process ◽  
Brownian Bridge ◽  
Stationary Gaussian Process ◽  
String Equation ◽  
Distribution Of The Maximum ◽  
Close Relationship ◽  
Vibrating String

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.

On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process

1977 ◽  
Vol 14 (01) ◽  
pp. 114-126 ◽  
Author(s):  
A. M. Liebetrau
Keyword(s):  
Weak Convergence ◽  
Gaussian Process ◽  
Point Process ◽  
Stationary Point ◽  
Stationary Gaussian Process ◽  
Time Curve ◽  
Second Moment ◽  
Stationary Point Process ◽  
Class Of Estimators

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ (V*(tTα ) – V(tTα )), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα ) is replaced by a suitable estimator.

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