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I'm having trouble grasping the autocorrelation function for stationary signals, both strict stationary and WSS.

First for strict sense, we have $$\forall(\tau,t_1, \ldots, t_n) \in \mathbb{R} \land \forall n \in \mathbb{N} \ \ \ F_{X}(X_{t_1+\tau} ,\ldots, X_{t_n+\tau}) = F_{X}(X_{t_1},\ldots, X_{t_n}) \ \ $$ For unconditional cumulative distribution function $F_X$

This implies that all time instants are identically distributed $$\forall(t_1,t_2) \ \ F_X(X_{t_1})=F_X(X_{t_2})$$

My question then is, how is it possible for the autocorrelation to be anything other than a constant? The only thing I can think of is each time instant random variable $X_t$ is not independent, so the product distribution function would be different for different time instances and $$E[X_{t_1}X_{t_2}]\ne E[X_{t_1}X_{t_3}]$$ if they were independent then the product distribution would be the same for all time instants and therefore the expected value as well giving a constant autocorrelation.

Additionally, for WSS, I can't conceptualize how autocorrelation could be independent of time, but still possibly dependent on shift. In this case however, each time instant is not necessarily identically distributed, only they have the same mean, but then it would seem the autocorrelation would depend on time.

Altogether, it seems that if the autocorrelation is time independent, then each time instant random variable is identically distributed (is this not the case?) and if they are identically distributed, the only way shift can affect the value is if

each time instant random variable has a different dependence on the other time instant random variables, but the dependence is consistent for all the random variables with respect to their "relative neighbors"

Is this understanding ok, or am I completely off the mark? Can someone provide an example where the autocorrelation of the stationary process is time independent, but does vary with shift?

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You need the joint distribution of $X_{t_1}$ and $X_{t_2}$ in order to calculate $E[X_{t_1}X_{t_2}]$ and nobody has claimed that the joint distribution of $X_{t_1}$ and $X_{t_2}$ is the same as the joint distribution of $X_{t_1}$ and $X_{t_3}$. Stationarity requires that the joint distribution of $X_{t_1}$ and $X_{t_2}$ be the same as the joint distribution of $X_{t_1+\tau}$ and $X_{t_2+\tau}$ which is a quite different requirement than the one you need for claiming that the autocorrelation function of the process is a constant. See this answer of mine for more than you might ever have wanted to know about autocorrelation functions.

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  • $\begingroup$ If the random variables are independent, the product distribution can be computed from the marginal probability distributions the joint distribution would not be needed. So I guess they aren’t independent? $\endgroup$ – Colin Hicks Nov 14 at 6:24
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    $\begingroup$ If all the random variables comprising the process are independent of each other, then the autocorrelation function has value $$E[X_{t_1}X_{t_2}] = E[X_{t_1}]E[X_{t_2}] = \mu^2$$ for all $t_1 \neq t_2$. This is not quite the same as saying that the autocorrelation function $R_X(\tau)=E[X(t)X(t+\tau)]$ is a constant (meaning it has the same value for all arguments) since the autocorrelation value at $0$ ($R_X(0)=E[(X(t))^2]$) is not specified in the above equation. In general, in typical processes, the random variables are not independent or uncorrelated. $\endgroup$ – Dilip Sarwate Nov 14 at 15:13
  • $\begingroup$ Thank you! Great response! $\endgroup$ – Colin Hicks Nov 14 at 15:15

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