Random function covariance I was studying signal processing and I was frequently asked to verify if a certain covariance is possible for a given random function. I tried to check by verifying the property to prove it:
${γ}_{xx}(τ) = {γ}_{xx}(t+τ,t)$
but didn't quite got anything.
For exemple, if we suppose $τ_0$ a given constant, how would I prove that I can have the following covariance (or not)?
${γ}_{xx}(τ) = exp(-(τ- τ_0)^2)$
You have two different meanings for ${γ}_{xx}$ when you write stuff such as
$${γ}_{xx}(τ) = {γ}_{xx}(t+τ,t)$$
because on the LHS, the function ${γ}_{xx}$ has one argument while on the RHS it has two arguments. Be that as it may, a very important property of what you call the covariance but what most everyone else on dsp.SE would call the autocovariance function is that
$$\text{For all values of }\tau, |{γ}_{xx}(τ)| \leq {γ}_{xx}(0).\tag{1}$$
It is easy to verify that $\exp\left(-(\tau-t_0)^2\right)$ has value $\exp(-t_0^2)&lt;1$ at $\tau=0$, and larger value $1$ at $\tau=t_0$, in contradiction of $(1)$, and so $\exp\left(-(\tau-t_0)^2\right)$ cannot possibly be a valid autocovariance (except, of course, for the special case of $t_0=0$).