By Ashot Vazrikievich Kakosyan, Leo Borisovich Klebanov, Joseph Aleksandrovich Melamed (auth.)

ISBN-10: 3540138579

ISBN-13: 9783540138570

ISBN-10: 3540390502

ISBN-13: 9783540390503

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**Extra resources for Characterization of Distributions by the Method of Intensively Monotone Operators**

**Sample text**

TXj . We shall say that random variables ~-conditionally X4 ~ X ~ . . , X ~ , are ... ,$~, .... 3. Let X0 , X4, ... , X n ,... ,X~,... f. f. ~ ~(t; J ~)-t with probability one and with a positive probability at least two of them do not vanish; CO iii) the series ~ X] converges with probability one. f. o f a n o r m a l law. are identically distri- $2 Proof. 's with regard to the conditio- nal independence of X 1 ,X 2 ~"" has the form of CO _q Using the assumption ii) we may rewrite this condition as the equation which is valid in some neighborhood U 0 of the point ~ = .

Note that the exponential distribution arises also as a limiting one for sums of a random amount of random variables (see Kova- lenko (1965)). Namely, the following result is valid. 10. Let X~ ~ X g ~ , . ~ X ~ , , , be a sequence of inde- pendent identically distributed random variables. Assume that EX4=~>O , EX~ < . Let V be a geometric random variable P{V=Kl=p(~-p) independent of the sequence K-4 ,K=~,~, Then 'l-P. O , 0 , cz

Of the exponential distribu- . 's. 9, making the result somewhat mere general. 11. be a sequence of independent and identically distributed positive random variables. Assume that V tribution is an integer-valued random variable with the geometric disP[~---K]=~(~-~)K, sequence X i tribution ~ K =0,4,,.. , ~(0,~) . Assume that the parameter ~ , independent of the is random and has a dis- whose support is not reduced to the point ~= ~ . The X o is identically distributed with the stun random variable ¢ o~Xo+ p~IX j , #= Ep , iff tion kwe assume here that under Xo has the exponential distribu~_, X i = 0 V= 0 j-1 ).

### Characterization of Distributions by the Method of Intensively Monotone Operators by Ashot Vazrikievich Kakosyan, Leo Borisovich Klebanov, Joseph Aleksandrovich Melamed (auth.)

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