Candidate distributions whose linear combination is power law

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This question is a further exploration of my initial question here. I am essentially trying to solve the problem: if $Z = rho X + (1-rho) Y$ and $rho in (0,1)$ follows a power law distribution beyond a certain minimum value $z_min$, so that $f_Z(z) = (alpha - 1) z_min^alpha-1 z^-alpha$, is there any criterion we can establish on the distribution of $X$ and $Y$? Should they follow any known distribution? A starting point I explored here was to see if power law distributions on $X$ and $Y$ would yield a power law distribution for $Z$ but this proof is elusive to me.



Any suggestions?




probability probability-theory probability-distributions




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    up vote
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    This question is a further exploration of my initial question here. I am essentially trying to solve the problem: if $Z = rho X + (1-rho) Y$ and $rho in (0,1)$ follows a power law distribution beyond a certain minimum value $z_min$, so that $f_Z(z) = (alpha - 1) z_min^alpha-1 z^-alpha$, is there any criterion we can establish on the distribution of $X$ and $Y$? Should they follow any known distribution? A starting point I explored here was to see if power law distributions on $X$ and $Y$ would yield a power law distribution for $Z$ but this proof is elusive to me.



    Any suggestions?




    probability probability-theory probability-distributions




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      This question is a further exploration of my initial question here. I am essentially trying to solve the problem: if $Z = rho X + (1-rho) Y$ and $rho in (0,1)$ follows a power law distribution beyond a certain minimum value $z_min$, so that $f_Z(z) = (alpha - 1) z_min^alpha-1 z^-alpha$, is there any criterion we can establish on the distribution of $X$ and $Y$? Should they follow any known distribution? A starting point I explored here was to see if power law distributions on $X$ and $Y$ would yield a power law distribution for $Z$ but this proof is elusive to me.



      Any suggestions?




      probability probability-theory probability-distributions




      share|cite|improve this question











      This question is a further exploration of my initial question here. I am essentially trying to solve the problem: if $Z = rho X + (1-rho) Y$ and $rho in (0,1)$ follows a power law distribution beyond a certain minimum value $z_min$, so that $f_Z(z) = (alpha - 1) z_min^alpha-1 z^-alpha$, is there any criterion we can establish on the distribution of $X$ and $Y$? Should they follow any known distribution? A starting point I explored here was to see if power law distributions on $X$ and $Y$ would yield a power law distribution for $Z$ but this proof is elusive to me.



      Any suggestions?





      probability probability-theory probability-distributions





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      asked Jul 26 at 10:22









      buzaku

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