Stable Distribution of Random Variables

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If X and Y are identical independent Random Variables. I have shown that X+Y and X have same distribution also i used the definition of stable distribution but i am enable to prove X=0 almost surely? Can someone give me hint how to proceed further?

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Hint: What is the variance of the sum, $\mathsf {Var}(X+Y)$, equal to when $X,Y$ are independent and identically distributed?   What is it equal to when $X+Y$ is identically distributed to $X$?   How is it possible to be equal to both?   If that is the case, what distribution must $X+Y$ have?   (Which is also the distribution of $X$ if they are identically distributed.)