Autocorrelation of a constant function

Question

I have to compute the autocorrelation function (see here) of a signal $X_t$

$${\displaystyle {\hat {R}}(k)={\frac {1}{(n-k)\sigma ^{2}}}\sum _{t=1}^{n-k}(X_{t}-\mu )(X_{t+k}-\mu )}.$$

My problem is that the function always takes the same value for all times, i.e. it is constant. This would lead to $\sigma^2 = 0$ and $X_t=\mu$ for all $t$, i.e. a "$\frac{0}{0}$ situation".

How is the autocorrelation defined in such a situation?

My attempt at resolving this would be $\hat R(k)=0$ since the numerator is always $0$. However, the signal is always perfectly correlated with itself so should we have $\hat R(k) = 1$?

Answer 1

Same as with regular correlation, it is undefined. Same as with correlation with a constant variable, if a function is constant, there is no point in calculating autocorrelation.