A note on concentration functions

1984 ◽  
Vol 96 (3-4) ◽  
pp. 181-184
Author(s):  
J. E. A. Dunnage
Keyword(s):  
Random Variables ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Absolute Moment ◽  
Third Order ◽  
Concentration Functions

SynopsisWe give an inequality for the concentration function of a sum X1 + … + Xn of independent random variables when Xv has a finite absolute moment of order kv (2 < kv ≦ 3). It is an extension of somewhat similar inequalities found earlier by Offord and by the author in the case of finite third-order absolute moments.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (1) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

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