A weak convergence theorem for the empirical characteristic function

1975 ◽  
Vol 12 (03) ◽  
pp. 515-523 ◽  
Author(s):  
John T. Kent
Keyword(s):  
Weak Convergence ◽  
Characteristic Function ◽  
Gaussian Process ◽  
Convergence Theorem ◽  
Stationary Gaussian Process ◽  
Empirical Characteristic Function ◽  
Autocovariance Function ◽  
Weak Convergence Theorem

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.

A weak convergence theorem for the empirical characteristic function

1975 ◽  
Vol 12 (3) ◽  
pp. 515-523 ◽  
Author(s):  
John T. Kent
Keyword(s):  
Weak Convergence ◽  
Characteristic Function ◽  
Gaussian Process ◽  
Convergence Theorem ◽  
Stationary Gaussian Process ◽  
Empirical Characteristic Function ◽  
Autocovariance Function ◽  
Weak Convergence Theorem

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.

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.

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 Weak Convergence Theorem for Functionals of Sums of Martingale Differences

1987 ◽  
Vol 130 (1) ◽  
pp. 151-155
Author(s):  
Z. Rychlik ◽  
I. Szyszkowski
Keyword(s):  
Weak Convergence ◽  
Convergence Theorem ◽  
Martingale Differences ◽  
Weak Convergence Theorem

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

1977 ◽  
Vol 14 (1) ◽  
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.

scholarly journals WEAK CONVERGENCE THEOREM FOR NEW NONEXPANSIVE MAPPINGS IN BANACH SPACES AND ITS APPLICATIONS

2007 ◽  
Vol 11 (3) ◽  
pp. 929-944 ◽  
Author(s):  
Takanori Ibaraki ◽  
Wataru Takahashi
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Weak Convergence Theorem
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