scholarly journals Martingale-difference Gibbs random fields and central limit theorem

1992 ◽  
Vol 17 ◽  
pp. 105-110 ◽  
Author(s):  
B.S. Nahapetian ◽  
A.N. Petrosian
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Martingale Difference ◽  
Gibbs Random Fields

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1996 ◽  
Vol 28 (02) ◽  
pp. 333-334
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
Individual Grains

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

A central limit theorem for super-Brownian motion with super-Brownian immigration

1999 ◽  
Vol 36 (4) ◽  
pp. 1218-1224 ◽  
Author(s):  
Wen-Ming Hong ◽  
Zeng-Hu Li
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Potential Operator ◽  
Super Brownian Motion

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d ≥ 3. For d = 3 the field is spatially uniform; for d ≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.

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