Sums of realizations of dependent variables become independent

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I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance:



$X sim mathcalN(0,1)$



$Y = cos(X) +Z$,$~~~~$ where $Zsim mathcalN(0,0.5)$



Here you can see some samples generated for these variables. I use this implementation of distance correlation as a measure for dependence:





Now consider two additional random variables $X_sum$ and $Y_sum$ generated as the summation of $n$ realizations of the original variables respectively. The thing I have noticed is that $X_sum$ and $Y_sum$ become more and more independent as $n$ grows (click on the image to enlarge):



enter image description here



I know that by the Central Limit Theorem $X_sum$ and $Y_sum$ will be approximately normal, but why are they becoming independent? Is there a general explanation for this?. Also, I have noticed that this phenomenon only occurs if $X$ and $Y$ are not correlated.



Thanks in advance.




random-variables independence correlation central-limit-theorem




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    up vote
    2
    down vote

    favorite












    I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance:



    $X sim mathcalN(0,1)$



    $Y = cos(X) +Z$,$~~~~$ where $Zsim mathcalN(0,0.5)$



    Here you can see some samples generated for these variables. I use this implementation of distance correlation as a measure for dependence:





    Now consider two additional random variables $X_sum$ and $Y_sum$ generated as the summation of $n$ realizations of the original variables respectively. The thing I have noticed is that $X_sum$ and $Y_sum$ become more and more independent as $n$ grows (click on the image to enlarge):



    enter image description here



    I know that by the Central Limit Theorem $X_sum$ and $Y_sum$ will be approximately normal, but why are they becoming independent? Is there a general explanation for this?. Also, I have noticed that this phenomenon only occurs if $X$ and $Y$ are not correlated.



    Thanks in advance.




    random-variables independence correlation central-limit-theorem




    share|cite|improve this question





















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance:



      $X sim mathcalN(0,1)$



      $Y = cos(X) +Z$,$~~~~$ where $Zsim mathcalN(0,0.5)$



      Here you can see some samples generated for these variables. I use this implementation of distance correlation as a measure for dependence:





      Now consider two additional random variables $X_sum$ and $Y_sum$ generated as the summation of $n$ realizations of the original variables respectively. The thing I have noticed is that $X_sum$ and $Y_sum$ become more and more independent as $n$ grows (click on the image to enlarge):



      enter image description here



      I know that by the Central Limit Theorem $X_sum$ and $Y_sum$ will be approximately normal, but why are they becoming independent? Is there a general explanation for this?. Also, I have noticed that this phenomenon only occurs if $X$ and $Y$ are not correlated.



      Thanks in advance.




      random-variables independence correlation central-limit-theorem




      share|cite|improve this question











      I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance:



      $X sim mathcalN(0,1)$



      $Y = cos(X) +Z$,$~~~~$ where $Zsim mathcalN(0,0.5)$



      Here you can see some samples generated for these variables. I use this implementation of distance correlation as a measure for dependence:





      Now consider two additional random variables $X_sum$ and $Y_sum$ generated as the summation of $n$ realizations of the original variables respectively. The thing I have noticed is that $X_sum$ and $Y_sum$ become more and more independent as $n$ grows (click on the image to enlarge):



      enter image description here



      I know that by the Central Limit Theorem $X_sum$ and $Y_sum$ will be approximately normal, but why are they becoming independent? Is there a general explanation for this?. Also, I have noticed that this phenomenon only occurs if $X$ and $Y$ are not correlated.



      Thanks in advance.





      random-variables independence correlation central-limit-theorem





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      asked Jul 25 at 22:10









      Daniel López

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          The multidimensional CLT implies that
          $$sqrtn beginbmatrix frac1n X_sum - mathbbE X \ frac1n Y_sum - mathbbE Y endbmatrix$$
          is approximately bivariate normal with mean zero and covariance matrix
          $$beginbmatrixtextVar(X) & textCov(X,Y) \ textCov(X,Y) & textVar(Y)endbmatrix.$$
          Since your $X$ and $Y$ are uncorrelated, we have $textCov(X_sum/sqrtn, Y_sum/sqrtn) approx 0$.



          I am not sure how to quantify what happens when you scale up by $sqrtn$ to consider $textCov(X_sum, Y_sum)$; you may need a Berry-Esseen type of non-asymptotic result.






          share|cite|improve this answer





















          • Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
            – Daniel López
            Jul 25 at 23:08






          • 1




            But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
            – Daniel López
            Jul 25 at 23:20










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          up vote
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          The multidimensional CLT implies that
          $$sqrtn beginbmatrix frac1n X_sum - mathbbE X \ frac1n Y_sum - mathbbE Y endbmatrix$$
          is approximately bivariate normal with mean zero and covariance matrix
          $$beginbmatrixtextVar(X) & textCov(X,Y) \ textCov(X,Y) & textVar(Y)endbmatrix.$$
          Since your $X$ and $Y$ are uncorrelated, we have $textCov(X_sum/sqrtn, Y_sum/sqrtn) approx 0$.



          I am not sure how to quantify what happens when you scale up by $sqrtn$ to consider $textCov(X_sum, Y_sum)$; you may need a Berry-Esseen type of non-asymptotic result.






          share|cite|improve this answer





















          • Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
            – Daniel López
            Jul 25 at 23:08






          • 1




            But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
            – Daniel López
            Jul 25 at 23:20














          up vote
          1
          down vote













          The multidimensional CLT implies that
          $$sqrtn beginbmatrix frac1n X_sum - mathbbE X \ frac1n Y_sum - mathbbE Y endbmatrix$$
          is approximately bivariate normal with mean zero and covariance matrix
          $$beginbmatrixtextVar(X) & textCov(X,Y) \ textCov(X,Y) & textVar(Y)endbmatrix.$$
          Since your $X$ and $Y$ are uncorrelated, we have $textCov(X_sum/sqrtn, Y_sum/sqrtn) approx 0$.



          I am not sure how to quantify what happens when you scale up by $sqrtn$ to consider $textCov(X_sum, Y_sum)$; you may need a Berry-Esseen type of non-asymptotic result.






          share|cite|improve this answer





















          • Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
            – Daniel López
            Jul 25 at 23:08






          • 1




            But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
            – Daniel López
            Jul 25 at 23:20












          up vote
          1
          down vote










          up vote
          1
          down vote









          The multidimensional CLT implies that
          $$sqrtn beginbmatrix frac1n X_sum - mathbbE X \ frac1n Y_sum - mathbbE Y endbmatrix$$
          is approximately bivariate normal with mean zero and covariance matrix
          $$beginbmatrixtextVar(X) & textCov(X,Y) \ textCov(X,Y) & textVar(Y)endbmatrix.$$
          Since your $X$ and $Y$ are uncorrelated, we have $textCov(X_sum/sqrtn, Y_sum/sqrtn) approx 0$.



          I am not sure how to quantify what happens when you scale up by $sqrtn$ to consider $textCov(X_sum, Y_sum)$; you may need a Berry-Esseen type of non-asymptotic result.






          share|cite|improve this answer













          The multidimensional CLT implies that
          $$sqrtn beginbmatrix frac1n X_sum - mathbbE X \ frac1n Y_sum - mathbbE Y endbmatrix$$
          is approximately bivariate normal with mean zero and covariance matrix
          $$beginbmatrixtextVar(X) & textCov(X,Y) \ textCov(X,Y) & textVar(Y)endbmatrix.$$
          Since your $X$ and $Y$ are uncorrelated, we have $textCov(X_sum/sqrtn, Y_sum/sqrtn) approx 0$.



          I am not sure how to quantify what happens when you scale up by $sqrtn$ to consider $textCov(X_sum, Y_sum)$; you may need a Berry-Esseen type of non-asymptotic result.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 25 at 22:48









          angryavian

          34.5k12874




          34.5k12874











          • Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
            – Daniel López
            Jul 25 at 23:08






          • 1




            But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
            – Daniel López
            Jul 25 at 23:20
















          • Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
            – Daniel López
            Jul 25 at 23:08






          • 1




            But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
            – Daniel López
            Jul 25 at 23:20















          Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
          – Daniel López
          Jul 25 at 23:08




          Thanks for your answer. That might explain the near cero correlation, but I am more interested in the increasingly lower dependence.
          – Daniel López
          Jul 25 at 23:08




          1




          1




          But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
          – Daniel López
          Jul 25 at 23:20




          But now that I think about it, if two variables follow a bivariate normal with cero co-variance, they are by definition independent right?
          – Daniel López
          Jul 25 at 23:20












           

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