A central limit theorem for dependent random variable.












1












$begingroup$


Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e.
$$
u_jstackrel{d}{=}text{N}(0,1).
$$
Call $mu_r=mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution of the random variable
$$
sqrt{n},left(frac{1}{n}sum_{j=1}^{n}u_j^2-frac{mu_1^{-2}}{n}sum_{j=1}^{n}|u_j|,|u_{j+1}|right)=n^{-1/2},sum_{j=1}^n|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)=n^{-1/2},sum_{j=1}^nxi_j,
$$
where $xi_j=|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)$. However, since the $xi_j$ are dependent, I do not know how to proceed.






probability-distributions central-limit-theorem







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$endgroup$

















    1












    $begingroup$


    Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e.
    $$
    u_jstackrel{d}{=}text{N}(0,1).
    $$
    Call $mu_r=mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution of the random variable
    $$
    sqrt{n},left(frac{1}{n}sum_{j=1}^{n}u_j^2-frac{mu_1^{-2}}{n}sum_{j=1}^{n}|u_j|,|u_{j+1}|right)=n^{-1/2},sum_{j=1}^n|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)=n^{-1/2},sum_{j=1}^nxi_j,
    $$
    where $xi_j=|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)$. However, since the $xi_j$ are dependent, I do not know how to proceed.






    probability-distributions central-limit-theorem







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e.
      $$
      u_jstackrel{d}{=}text{N}(0,1).
      $$
      Call $mu_r=mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution of the random variable
      $$
      sqrt{n},left(frac{1}{n}sum_{j=1}^{n}u_j^2-frac{mu_1^{-2}}{n}sum_{j=1}^{n}|u_j|,|u_{j+1}|right)=n^{-1/2},sum_{j=1}^n|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)=n^{-1/2},sum_{j=1}^nxi_j,
      $$
      where $xi_j=|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)$. However, since the $xi_j$ are dependent, I do not know how to proceed.






      probability-distributions central-limit-theorem







      share|cite|improve this question











      $endgroup$




      Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e.
      $$
      u_jstackrel{d}{=}text{N}(0,1).
      $$
      Call $mu_r=mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution of the random variable
      $$
      sqrt{n},left(frac{1}{n}sum_{j=1}^{n}u_j^2-frac{mu_1^{-2}}{n}sum_{j=1}^{n}|u_j|,|u_{j+1}|right)=n^{-1/2},sum_{j=1}^n|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)=n^{-1/2},sum_{j=1}^nxi_j,
      $$
      where $xi_j=|u_j|left(|u_j|-mu_1^{-2},|u_{j+1}|right)$. However, since the $xi_j$ are dependent, I do not know how to proceed.






      probability-distributions central-limit-theorem




      probability-distributions central-limit-theorem






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Jan 2 at 16:24







      AlmostSureUser

















      asked Jan 2 at 15:05









      AlmostSureUserAlmostSureUser

      326418




      326418






















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          $begingroup$

          Since $(xi_j)_{jle N}$ and $(xi_j)_{jge N+k}$ are independent for $kge 2$, the stochastic process $(xi_j)_{jin mathbb{N}}$ is strongly($alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $alpha(k)$ vanishes for $kgeq 2$. Note that $Exi_j =0$ and every $p$-th moment $E|xi_j|^p$ is finite. By $alpha$-mixing central limit theorem, we have
          $$
          frac{1}{sqrt{n}}sum_{j=1}^n xi_j to_d mathcal{N}(0,sigma^2)
          $$
          where $sigma^2 = E[xi_j^2] +2 E[xi_j xi_{j+1}]ge 0$.






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            $begingroup$

            Since $(xi_j)_{jle N}$ and $(xi_j)_{jge N+k}$ are independent for $kge 2$, the stochastic process $(xi_j)_{jin mathbb{N}}$ is strongly($alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $alpha(k)$ vanishes for $kgeq 2$. Note that $Exi_j =0$ and every $p$-th moment $E|xi_j|^p$ is finite. By $alpha$-mixing central limit theorem, we have
            $$
            frac{1}{sqrt{n}}sum_{j=1}^n xi_j to_d mathcal{N}(0,sigma^2)
            $$
            where $sigma^2 = E[xi_j^2] +2 E[xi_j xi_{j+1}]ge 0$.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Since $(xi_j)_{jle N}$ and $(xi_j)_{jge N+k}$ are independent for $kge 2$, the stochastic process $(xi_j)_{jin mathbb{N}}$ is strongly($alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $alpha(k)$ vanishes for $kgeq 2$. Note that $Exi_j =0$ and every $p$-th moment $E|xi_j|^p$ is finite. By $alpha$-mixing central limit theorem, we have
              $$
              frac{1}{sqrt{n}}sum_{j=1}^n xi_j to_d mathcal{N}(0,sigma^2)
              $$
              where $sigma^2 = E[xi_j^2] +2 E[xi_j xi_{j+1}]ge 0$.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Since $(xi_j)_{jle N}$ and $(xi_j)_{jge N+k}$ are independent for $kge 2$, the stochastic process $(xi_j)_{jin mathbb{N}}$ is strongly($alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $alpha(k)$ vanishes for $kgeq 2$. Note that $Exi_j =0$ and every $p$-th moment $E|xi_j|^p$ is finite. By $alpha$-mixing central limit theorem, we have
                $$
                frac{1}{sqrt{n}}sum_{j=1}^n xi_j to_d mathcal{N}(0,sigma^2)
                $$
                where $sigma^2 = E[xi_j^2] +2 E[xi_j xi_{j+1}]ge 0$.






                share|cite|improve this answer









                $endgroup$



                Since $(xi_j)_{jle N}$ and $(xi_j)_{jge N+k}$ are independent for $kge 2$, the stochastic process $(xi_j)_{jin mathbb{N}}$ is strongly($alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $alpha(k)$ vanishes for $kgeq 2$. Note that $Exi_j =0$ and every $p$-th moment $E|xi_j|^p$ is finite. By $alpha$-mixing central limit theorem, we have
                $$
                frac{1}{sqrt{n}}sum_{j=1}^n xi_j to_d mathcal{N}(0,sigma^2)
                $$
                where $sigma^2 = E[xi_j^2] +2 E[xi_j xi_{j+1}]ge 0$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 2 at 17:39









                SongSong

                18.5k21651




                18.5k21651






























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