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How are they all related? You can define them as:

  • Orthogonal Processes: $E[XY] = 0$
  • Uncorrelated Processes: $E[XY] = E[(X - \mu_x)(Y - \mu_y)] = 0$
  • Statistically Independent Processes: $E[XY] = E[X] \cdot E[Y]$

If two processes are orthogonal:

  • they are also uncorrelated
  • they are not necessary independent

If two processes are uncorrelated:

  • they are not necessary orthogonal
  • they are not necessary independent

If two processes are independent:

  • they are uncorrelated
  • they are orthogonal

Is that correct? I'm not sure.

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You got some definitions wrong. It's correct that orthogonality means that $E[XY]=0$. Uncorrelated means that $X-\mu_X$ and $Y-\mu_Y$ are orthogonal, i.e., $E[(X-\mu_X)(Y-\mu_Y)]=0$. If you work that out you should arrive at the equivalent condition $E[XY]=\mu_X\mu_Y$ for uncorrelatedness (not for independence!). Consequently, if at least one of the two RVs $X$ and $Y$ have a zero mean, then orthogonality implies uncorrelatedness and vice versa.

Statistical independence means that the joint PDF of two random variables can be written as the product of the individual PDFs:

$$f_{XY}(x,y)=f_X(x)f_Y(y)\tag{1}$$

Independence implies uncorrelatedness, but the opposite is generally not true. If $X$ and $Y$ are jointly Gaussian, then independence and uncorrelatedness are equivalent. Consequently, in the special case that $X$ and $Y$ are jointly Gaussian and at least one of them has a mean of zero, then orthogonality, uncorrelatedness, and independence are all equivalent.

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  • $\begingroup$ So independence does not imply orthogonality? $\endgroup$
    – user674907
    May 18, 2020 at 12:21
  • 1
    $\begingroup$ @user674907: No, unless one of the two RVs has a zero mean. $\endgroup$
    – Matt L.
    May 18, 2020 at 12:51

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