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I know that a stationary Gaussian process $X(t)$ that has an exponential autocorrelation is called a Gauss-Markov process.

$$R(t)=\sigma^2 e^{-t/τ}$$

In a book, it says that the autocorrelation function approaches zero as the time constant τ is close to infinity, and thus the mean value of the process must be zero. I know the stationary means the expected value of $X(t)$ is constant. But I don't understand why the mean value of the process must be zero?

Samuel Adrian Antz
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Hengwei
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  • Hi: The DSP definition of autocorrelation is very different from the statistical definition. The DSP definition for autocorrelation i the definition of autocorrelation in statistics ( but it doesn't de-mean ). The lack of de-meaning is probably related to why they say what they say in your question. I sort of gave up trying to understand the DSP version. I just stick to the statistical version . You are correct that the autocorrelation as defined in statistics says nothing about the mean of the process. In DSP, it does because the mean is not removed. – mark leeds Jul 08 '22 at 14:21
  • Thank you for your helping. In the digital signal process (DSP), you say it does because the mean is not removed. I want to know why the mean of Gauss-Markov process is zero? In other word, τ is close to infinity and the autocorrelation function R(t)=0, but how to obtain mean is zero from the R(t)=0? – Hengwei Jul 09 '22 at 01:26
  • It's definitely confusing because, in statistics, there is no way of inferring the mean from the autocorrelation of the process. This is because the autocorrelation is calculated using the de-meaned series. In DSP, they don't de-mean so I think you can just look at the function and see that the mean is zero because it decays to zero. But again, I am not an expert here because of the difference in definitions. I just happen to be able to tell that you're looking at it from a DSP viewpoint because I've seen that expression in DSP land. It's very popular there and not popular in statistics land. – mark leeds Jul 10 '22 at 02:23
  • It might be interesting to post your question on dsp.stackexchange and see what they say. I've had a few arguments with experts over there so I would just read the replies and not jump in. There are definitely DSP experts over there but I'm not sure about their knowledge when it comes to autocorrelation issues. It might be helpful for you to read some old threads regarding the topic over there. I'm in some of them. – mark leeds Jul 10 '22 at 02:25

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