Yes. My PhD advisor had a research interest in axiom systems for probability that are weaker than the familiar Kolmogorov axioms, which are sometimes abbreviated "CMP" for "conventional mathematical probability".
I'll try to remember the setup. The CMP axioms imply that, in a shift-invariant system X(t) (which is a different class than "stationary" -- not necessarily implying existence of second moments), if the mean of X(t) exists finite, then a long-term average of X(t) must converge.
However, you can observe time-invariant physical systems (such as a noisy resistor in a static environment) with spectra that obey the 1/f law down to very low frequencies (i.e., over very long time baselines) -- the time average does not converge. My advisor had a stack of magnetic tapes on his bookshelf with such samples.
These systems would seem to be disobeying the axioms of CMP, thereby motivating searches for alternative formulations that are more general.
I'll try to remember the setup. The CMP axioms imply that, in a shift-invariant system X(t) (which is a different class than "stationary" -- not necessarily implying existence of second moments), if the mean of X(t) exists finite, then a long-term average of X(t) must converge.
However, you can observe time-invariant physical systems (such as a noisy resistor in a static environment) with spectra that obey the 1/f law down to very low frequencies (i.e., over very long time baselines) -- the time average does not converge. My advisor had a stack of magnetic tapes on his bookshelf with such samples.
These systems would seem to be disobeying the axioms of CMP, thereby motivating searches for alternative formulations that are more general.