scholarly journals Approximation of free convolutions by free infinitely divisible laws

2021 ◽  
Vol 66 (4) ◽  
pp. 806-838
Author(s):  
Геннадий Петрович Чистяков ◽  
Gennadii Petrovich Chistyakov ◽  
Фридрих Гeтце ◽  
Friedrich Gotze
Keyword(s):  
Infinitely Divisible ◽  
Infinitely Divisible Laws ◽  
Free Convolutions

Основываясь на методе сyбoрдинационных функций, мы получаем оценки минимальных ошибок аппроксимации $n$-кратных свободных сверток вероятностных мер безгранично делимыми свободными вероятностными мерами.

scholarly journals A martingale central limit theorem without negligibility conditions

1978 ◽  
Vol 18 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Martingale Central Limit Theorem ◽  
Triangular Arrays ◽  
Infinitely Divisible Laws ◽  
Martingale Case

We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

scholarly journals Structural properties of generalised Planck distributions

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Structural Properties ◽  
Genetic Mutation ◽  
Balance Model ◽  
Density Functions ◽  
Infinitely Divisible ◽  
Weighted Version ◽  
Infinitely Divisible Laws ◽  
Gamma Law

AbstractA family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.

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