The maximum expected deviation from the sample average matrix?












1












$begingroup$


I have reached to $mathbb{E}[|x_tx_t^T - G_t|^2_F]$, and I need an upperbound for it in terms of probabilistic characteristics of $x_t$ where:



$x_t$ is a random vector in $mathbb{R}^n$ drawn from an i.i.d with mean of $mu$ and variance $sigma^2$



$G_t$ is a symmetric matrix which I want it to be the sample average of $x_tx_t^T$ (I am not sure whether it can be simply the average of all $x_tx_t^T$ matrices according to the following or not?).



To handle the stated problem, I am going to generalize vector problem to matrix one as the following:



Assume we want to find an upperbound for $mathbb{E}[|x_t - z_t|^2]$



$$
|x_t - z_t|^2=|x_t - mu + mu -z|^2 = |x_t - mu |^2 + |mu -z_t|^2 +
2langle x_t - mu , mu -z_trangle
$$

where $z_t = frac{1}{t-1}sum_{s=1}^{t-1}x_s$ is the average of $t-1$ number of data. Taking the expected value vanishes the last part and gives $sigma^2$ as the variance of data.



$$
mathbb{E}[|x_t - z_t|^2]=mathbb{E}[|x_t - mu |^2] + mathbb{E}[|z_t- mu |^2]=sigma^2 + mathbb{E}[|mu -z_t|^2]
$$



Also, $mathbb{E}[|z_t- mu |^2] = mathbb{E}[|frac{1}{t-1}x_1 -frac{1}{t-1}mu +cdots + frac{1}{t-1}x_{t-1} -frac{1}{t-1}mu|^2] leq frac{1}{(t-1)^2}sigma^2$
Therefore,
$$
mathbb{E}[|x_t - z_t|^2] leq sigma^2 + frac{1}{(t-1)^2}sigma^2
$$



Now, my questions are:



1- Is the sample average of $x_tx_t^T$ can be defined simply as $G_t=frac{1}{t-1}sum_{s=1}^{t-1}x_sx_s^T$ because $z_t$ had beautiful property that helped me to simplify the expression.



2- Is there any $G_t$ in the matrix form that I can use it to get some good result?



My guess is $G_t$ has to be something which gives us an upperbound in terms of covariance of the data:



$$|x_tx_t^T - G_t|^2_F$$










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I have reached to $mathbb{E}[|x_tx_t^T - G_t|^2_F]$, and I need an upperbound for it in terms of probabilistic characteristics of $x_t$ where:



    $x_t$ is a random vector in $mathbb{R}^n$ drawn from an i.i.d with mean of $mu$ and variance $sigma^2$



    $G_t$ is a symmetric matrix which I want it to be the sample average of $x_tx_t^T$ (I am not sure whether it can be simply the average of all $x_tx_t^T$ matrices according to the following or not?).



    To handle the stated problem, I am going to generalize vector problem to matrix one as the following:



    Assume we want to find an upperbound for $mathbb{E}[|x_t - z_t|^2]$



    $$
    |x_t - z_t|^2=|x_t - mu + mu -z|^2 = |x_t - mu |^2 + |mu -z_t|^2 +
    2langle x_t - mu , mu -z_trangle
    $$

    where $z_t = frac{1}{t-1}sum_{s=1}^{t-1}x_s$ is the average of $t-1$ number of data. Taking the expected value vanishes the last part and gives $sigma^2$ as the variance of data.



    $$
    mathbb{E}[|x_t - z_t|^2]=mathbb{E}[|x_t - mu |^2] + mathbb{E}[|z_t- mu |^2]=sigma^2 + mathbb{E}[|mu -z_t|^2]
    $$



    Also, $mathbb{E}[|z_t- mu |^2] = mathbb{E}[|frac{1}{t-1}x_1 -frac{1}{t-1}mu +cdots + frac{1}{t-1}x_{t-1} -frac{1}{t-1}mu|^2] leq frac{1}{(t-1)^2}sigma^2$
    Therefore,
    $$
    mathbb{E}[|x_t - z_t|^2] leq sigma^2 + frac{1}{(t-1)^2}sigma^2
    $$



    Now, my questions are:



    1- Is the sample average of $x_tx_t^T$ can be defined simply as $G_t=frac{1}{t-1}sum_{s=1}^{t-1}x_sx_s^T$ because $z_t$ had beautiful property that helped me to simplify the expression.



    2- Is there any $G_t$ in the matrix form that I can use it to get some good result?



    My guess is $G_t$ has to be something which gives us an upperbound in terms of covariance of the data:



    $$|x_tx_t^T - G_t|^2_F$$










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I have reached to $mathbb{E}[|x_tx_t^T - G_t|^2_F]$, and I need an upperbound for it in terms of probabilistic characteristics of $x_t$ where:



      $x_t$ is a random vector in $mathbb{R}^n$ drawn from an i.i.d with mean of $mu$ and variance $sigma^2$



      $G_t$ is a symmetric matrix which I want it to be the sample average of $x_tx_t^T$ (I am not sure whether it can be simply the average of all $x_tx_t^T$ matrices according to the following or not?).



      To handle the stated problem, I am going to generalize vector problem to matrix one as the following:



      Assume we want to find an upperbound for $mathbb{E}[|x_t - z_t|^2]$



      $$
      |x_t - z_t|^2=|x_t - mu + mu -z|^2 = |x_t - mu |^2 + |mu -z_t|^2 +
      2langle x_t - mu , mu -z_trangle
      $$

      where $z_t = frac{1}{t-1}sum_{s=1}^{t-1}x_s$ is the average of $t-1$ number of data. Taking the expected value vanishes the last part and gives $sigma^2$ as the variance of data.



      $$
      mathbb{E}[|x_t - z_t|^2]=mathbb{E}[|x_t - mu |^2] + mathbb{E}[|z_t- mu |^2]=sigma^2 + mathbb{E}[|mu -z_t|^2]
      $$



      Also, $mathbb{E}[|z_t- mu |^2] = mathbb{E}[|frac{1}{t-1}x_1 -frac{1}{t-1}mu +cdots + frac{1}{t-1}x_{t-1} -frac{1}{t-1}mu|^2] leq frac{1}{(t-1)^2}sigma^2$
      Therefore,
      $$
      mathbb{E}[|x_t - z_t|^2] leq sigma^2 + frac{1}{(t-1)^2}sigma^2
      $$



      Now, my questions are:



      1- Is the sample average of $x_tx_t^T$ can be defined simply as $G_t=frac{1}{t-1}sum_{s=1}^{t-1}x_sx_s^T$ because $z_t$ had beautiful property that helped me to simplify the expression.



      2- Is there any $G_t$ in the matrix form that I can use it to get some good result?



      My guess is $G_t$ has to be something which gives us an upperbound in terms of covariance of the data:



      $$|x_tx_t^T - G_t|^2_F$$










      share|cite|improve this question









      $endgroup$




      I have reached to $mathbb{E}[|x_tx_t^T - G_t|^2_F]$, and I need an upperbound for it in terms of probabilistic characteristics of $x_t$ where:



      $x_t$ is a random vector in $mathbb{R}^n$ drawn from an i.i.d with mean of $mu$ and variance $sigma^2$



      $G_t$ is a symmetric matrix which I want it to be the sample average of $x_tx_t^T$ (I am not sure whether it can be simply the average of all $x_tx_t^T$ matrices according to the following or not?).



      To handle the stated problem, I am going to generalize vector problem to matrix one as the following:



      Assume we want to find an upperbound for $mathbb{E}[|x_t - z_t|^2]$



      $$
      |x_t - z_t|^2=|x_t - mu + mu -z|^2 = |x_t - mu |^2 + |mu -z_t|^2 +
      2langle x_t - mu , mu -z_trangle
      $$

      where $z_t = frac{1}{t-1}sum_{s=1}^{t-1}x_s$ is the average of $t-1$ number of data. Taking the expected value vanishes the last part and gives $sigma^2$ as the variance of data.



      $$
      mathbb{E}[|x_t - z_t|^2]=mathbb{E}[|x_t - mu |^2] + mathbb{E}[|z_t- mu |^2]=sigma^2 + mathbb{E}[|mu -z_t|^2]
      $$



      Also, $mathbb{E}[|z_t- mu |^2] = mathbb{E}[|frac{1}{t-1}x_1 -frac{1}{t-1}mu +cdots + frac{1}{t-1}x_{t-1} -frac{1}{t-1}mu|^2] leq frac{1}{(t-1)^2}sigma^2$
      Therefore,
      $$
      mathbb{E}[|x_t - z_t|^2] leq sigma^2 + frac{1}{(t-1)^2}sigma^2
      $$



      Now, my questions are:



      1- Is the sample average of $x_tx_t^T$ can be defined simply as $G_t=frac{1}{t-1}sum_{s=1}^{t-1}x_sx_s^T$ because $z_t$ had beautiful property that helped me to simplify the expression.



      2- Is there any $G_t$ in the matrix form that I can use it to get some good result?



      My guess is $G_t$ has to be something which gives us an upperbound in terms of covariance of the data:



      $$|x_tx_t^T - G_t|^2_F$$







      probability covariance variance expected-value






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




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      asked Dec 23 '18 at 4:34









      SaeedSaeed

      1,149310




      1,149310






















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