Generating functional method for spin variables

1973 ◽  
Vol 16 (2) ◽  
pp. 820-829 ◽  
Author(s):  
S. V. Peletminskii ◽  
A. A. Yatsenko
Keyword(s):  
Functional Method ◽  
Generating Functional

Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part

1973 ◽  
Vol 5 (3) ◽  
pp. 391-416 ◽  
Author(s):  
H. Hering
Keyword(s):  
Probability Distributions ◽  
Branching Processes ◽  
Weak Form ◽  
Transition Function ◽  
Functional Method ◽  
Distribution Functions ◽  
Generating Functional ◽  
Second Moments ◽  
Vector Valued ◽  
Population Probability

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part

1973 ◽  
Vol 5 (03) ◽  
pp. 391-416 ◽  
Author(s):  
H. Hering
Keyword(s):  
Probability Distributions ◽  
Branching Processes ◽  
Weak Form ◽  
Transition Function ◽  
Functional Method ◽  
Distribution Functions ◽  
Generating Functional ◽  
Second Moments ◽  
Vector Valued ◽  
Population Probability

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

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