An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

1985 ◽  
Vol 22 (2) ◽  
pp. 454-460 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Asymptotic Formula ◽  
Gaussian Process ◽  
Variance Function ◽  
Stationary Increments ◽  
Distribution Of The Maximum

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ2(t) is convex and σ2(t) = o(t) for t → 0, then P(max[o,t]X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

1985 ◽  
Vol 22 (02) ◽  
pp. 454-460 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Asymptotic Formula ◽  
Gaussian Process ◽  
Variance Function ◽  
Stationary Increments ◽  
Distribution Of The Maximum

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ 2(t) is convex and σ 2(t) = o(t) for t → 0, then P(max[o,t] X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process

1978 ◽  
Vol 15 (02) ◽  
pp. 433-439 ◽  
Author(s):  
A. M. Liebetrau
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Gaussian Process ◽  
Higher Dimensions ◽  
Variance Function ◽  
Stationary Gaussian Process ◽  
Two Dimensional ◽  
Second Moment ◽  
Class Of Estimators

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t 1, t 2) of the variance function V(t 1, t 2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

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.

scholarly journals Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

2017 ◽  
Vol 46 (3-4) ◽  
pp. 67-78 ◽  
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Gaussian Process ◽  
Unknown Parameter ◽  
Strong Consistency ◽  
Likelihood Function ◽  
Likelihood Estimator ◽  
Stationary Increments ◽  
Drift Estimation ◽  
Mixed Fractional Brownian Motion

The paper deals with the regression model X_t = \theta t + B_t , t\in[0, T ],where B=\{B_t, t\geq 0\} is a centered Gaussian process with stationary increments.We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation.Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process

1978 ◽  
Vol 15 (2) ◽  
pp. 433-439 ◽  
Author(s):  
A. M. Liebetrau
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Gaussian Process ◽  
Higher Dimensions ◽  
Variance Function ◽  
Stationary Gaussian Process ◽  
Two Dimensional ◽  
Second Moment ◽  
Class Of Estimators

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

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