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I'm in a stochastics class this semester, and during our last class, a classmate casually said that "correlation coefficients of random processes can be greater than 1 in the case of exponential growth series."

I thought this was wrong, but the professor confirmed it, although he didn't elaborate further.

Our definition of the correlation coefficient for a random process Y is:

$$\rho_Y(t_1,t_2) = \frac{Cov_{YY}(t_1,t_2)}{\sigma_Y(t_1),\sigma_Y(t_2)}$$

Could someone please explain how $\rho_Y(t_1,t_2)$ isn't bounded from -1 to 1?

Camellia99
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    The Cauchy Schwartz theorem says that it must be bounded between -1 and 1. – user317176 Apr 19 '22 at 17:51
  • See this question – lulu Apr 19 '22 at 17:52
  • This question is useful https://math.stackexchange.com/questions/4035086/pearson-coefficient-may-not-be-bounded-by-1 – oliverjones Apr 19 '22 at 17:52
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    Reminds me of a joke that a drill sergeant says that if needed, value of $\pi$ can be as high as $4$. Outside of army though it's usually around $3.14$ – SBF Apr 19 '22 at 17:58
  • I see the comments are all saying that the correlation coefficient is bounded by [-1,1] (sometimes a tighter bound). I didn't think this was important when I wrote the question but my professor said "the correlation coefficient can be greater than 1 for an exponential growth series when we remove the condition of stationarity." Does the lack of stationarity affect anything? – Camellia99 Apr 19 '22 at 18:12
  • @Camellia99 In what context did you hear that? –  Apr 19 '22 at 18:52
  • @d.k.o. My class was discussing first order autoregressive processes- and how the coefficient for an AR(p=1) is equivalent to the correlation coefficient (WSS process, zero mean, with additive White noise). My teacher remarked how this coefficient was bounded from [-1,1]. A student corrected him and said that was only true for stationary processes and then the teacher said yes, that the correlation coefficient was only bounded from [-1,1] for stationary processes and that an example of a correlation coefficient greater than 1 could be seen for exponential time series. – Camellia99 Apr 19 '22 at 19:31
  • @Camellia99 The discussion about the equivalence of the autoregressive coefficient and the autocorrelation of a time-series makes sense only if the series is stationary, i.e., when the coefficient lies in $(-1,1)$. Regarding "exponential time series" you should probably ask your prof. –  Apr 19 '22 at 20:27

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