I want to know that the addition of a uncorrelated time series x with a correlated time series, say y generates a time series but is that correlated or uncorrelated output? For example: x is White Gaussian noise and y is pink noise, then z = x+y Looking at the plot which is the output of the noisy time series, it is hard to say whether z is correlated or uncorrelated. Is there a rule about the nature of the output?

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  • $\begingroup$ Can you re-consider your question a little? You begin with x being a correlated time series and y an uncorrelated series, but in the very next sentence, x is white Gaussian noise (which most people would regard as an uncorrelated series) and y is pink noise (which most people would regard as a correlated series). So, which is which? and what, if any, is the cross-correlation between x and y? $\endgroup$ – Dilip Sarwate Sep 1 '20 at 2:03
  • $\begingroup$ @DilipSarwate: sorry for the typo. I have fixed it. $\endgroup$ – Sm1 Sep 1 '20 at 2:17

Are $x$ and $y$ uncorrelated? In this case the autocorrelation function of $x+y$ is just the sum of the autocorrelation functions of $x$ and $y$, which if one of them is not white will lead to its sum also not being white.

In general, for stationary processes you have $\phi_{zz}(\tau) = \phi_{xx}(\tau) + \phi_{yy}(\tau) + \phi_{xy}(\tau) + \phi_{yx}(\tau)$ where $\phi_{xy}(\tau) = \mathbb{E}\{x(t)y(t+\tau)\}$. Moreover, since the process is stationary you also have $\phi_{xy}(\tau) = \phi_{xy}(-\tau) = \phi_{yx}(\tau)$. So in general, the autocorrelation of $z$ depends not only on the autocorrelations of $x$ and $y$ but also their cross correlations.

  • $\begingroup$ thank you for your answer, but it does not address completely my problems. One variable has a flat power spectrum and so is white (x r.v ) and the other has a non flat spectrum so is colored, consider say brown color (y r.v). Since x is white it is uncorrelated to itself and y is correlated to itself (each random number for the red noise is positively correlated to the one preceding it). In general, does the addition of an uncorrelated r.v with a correlated r.v give an output that is correlated or uncorrlelated? i.e., z=x+y is z correlated/uncorrelated? $\endgroup$ – Sm1 Sep 1 '20 at 12:19
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    $\begingroup$ It does answer your question. If your variables are independent, the correlation of the sum is equal to the sum of the correlations. Therefore, if one is not white (your $y$), the sum $x+y$ is also not white (i.e., samples are correlated to each other correlated), as I wrote in my second sentence. $\endgroup$ – Florian Sep 1 '20 at 13:59
  • $\begingroup$ Thank you, now it is clear. $\endgroup$ – Sm1 Sep 1 '20 at 15:58
  • $\begingroup$ You're welcome :) Glad it helped. $\endgroup$ – Florian Sep 1 '20 at 16:45

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