Statistics of the Boolean model: from the estimation of means to the estimation of distributions

1995 ◽  
Vol 27 (1) ◽  
pp. 63-86 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Boolean Model ◽  
Higher Order ◽  
Central Limit Theorems ◽  
The Mean ◽  
Estimation Of Distributions ◽  
Non Parametric

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.

Statistics of the Boolean model: from the estimation of means to the estimation of distributions

1995 ◽  
Vol 27 (01) ◽  
pp. 63-86 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Boolean Model ◽  
Higher Order ◽  
Central Limit Theorems ◽  
The Mean ◽  
Estimation Of Distributions ◽  
Non Parametric

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.

scholarly journals On central limit theorems for branching processes with dependent immigration

2020 ◽  
pp. 7-15
Author(s):  
V. Golomoziy ◽  
S. Sharipov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Branching Processes ◽  
Central Limit Theorems ◽  
Standard Normal Distribution ◽  
Immigration Process ◽  
Standard Normal ◽  
Mean And Variance ◽  
The Mean ◽  
Regularly Varying

In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

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