By Ashot Vazrikievich Kakosyan, Leo Borisovich Klebanov, Joseph Aleksandrovich Melamed (auth.)
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Additional info for Characterization of Distributions by the Method of Intensively Monotone Operators
2. Charaqterizations of the normal distribution b2~ro~erties of random linear forms Characterization of the normal distribution by the property of identical distribution of a moncmial and a linear form may be strengthened at the expense of consideration of linear forms with random coefficients. Zinger and Yu,V,Linnik nik (1970)). Later on characterizations random coeffi- (see Zinger, Lin- of the normal and stable distributions by the property of identical distribution of a monomial and a random linear form of independent and identically distributed random variables were obtained (see Shimizu (1968), zu and Davies (1979), Avksentiev (1978),Shimi- (1982)).
X0 . Assume ~hat the set of functions is equicontinuous in some neighborhood V of and d,. Ct) =t for all t . Ass e that the series X; J eonver es w i t h probability one. ,X~, ~ Xj j=1 is equivalent to normality of all variables . . 2. Charaqterizations of the normal distribution b2~ro~erties of random linear forms Characterization of the normal distribution by the property of identical distribution of a moncmial and a linear form may be strengthened at the expense of consideration of linear forms with random coefficients.
A~=~ for all ~ >0 is a solution of the equation and utilize Theorem 1. I. O Note that conditionally independent random variables occur in the problems of mathematical statistics often enough. In particuTar, if X4 9 X ~ , • • • , X ~ ~,.. are symmetrically dependent random variables, then they are conditionally independent with respect to the subalgebra generated by some random variable (see de ~inetti (193~ - 33 - 1933~), (1937)). And if X ~ , . , . , X ~ , (~4, ~ , ''''~ ~ °'') ... are independent random variables and is a random sequence, then the linear form CO z:, jx i with random coefficients ~i may be considered as a sum of conditioc~O nally independent variables ~ ~i J=4 , where characterization of the normal distribution ~ = ~j Xj - Thus, by the property of iden- tical distribution of a monomial and a linear form with random coefficients is a special case of the problem on identical distribution of a monomial and the sum of conditionally independent random variables.
Characterization of Distributions by the Method of Intensively Monotone Operators by Ashot Vazrikievich Kakosyan, Leo Borisovich Klebanov, Joseph Aleksandrovich Melamed (auth.)