scholarly journals Weak convergence towards two independent Gaussian processes from a unique Poisson process

2010 ◽  
Vol 61 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Xavier Bardina ◽  
David Bascompte
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Gaussian Processes

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.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Formation of non-Gaussian random processes, signals and interference, on the basis of stochastic differential equations

2018 ◽  
pp. 45-58
Author(s):  
V. M. Artyushenko ◽  
V. I. Volovach
Keyword(s):  
Differential Equations ◽  
Poisson Process ◽  
Stochastic Differential Equations ◽  
Gaussian Processes ◽  
Random Processes ◽  
Parametric Noise ◽  
Non Gaussian

Reviewed and analyzed issues associated with the formation of naguszewski random processes using stochastic differential equations. Algorithms of formation of scalar, vector and n –connected continuous Markov non-Gaussian sequences are considered. Forming filters with parametric noise and with disturbing influences, which are not Gaussian processes, are analyzed. The analysis of formation of non Gaussian sequences by means of Poisson process and stochastic filters is carried out.

Some properties of the crossings process generated by a stationary χ2 process

1978 ◽  
Vol 10 (2) ◽  
pp. 373-391 ◽  
Author(s):  
Ken Sharpe
Keyword(s):  
Poisson Process ◽  
Gaussian Processes ◽  
Simple Formula ◽  
Sufficient Conditions ◽  
Rayleigh Distribution ◽  
Time Interval ◽  
Expected Number ◽  
Fixed Level ◽  
Image Position

The process generated by the crossings of a fixed level, u, by the process Pn(t) is considered, where and the Xi(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of Pn(t) above u tends to a Rayleigh distribution as u→ ∞.

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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