3 ms·

Yup, it has the fun property that you'll get the same error in your estimate of the mean regardless of averaging time (though since it's not stationary this isn

by rcxdude 2mo ago

Yup, it has the fun property that you'll get the same error in your estimate of the mean regardless of averaging time (though since it's not stationary this isn't even really that well defined). (though of course, a random walk means you'll get even worse as you measure for longer...)

plus 2mo ago
This makes me think of the Cauchy distribution, a probability distribution whose average follows the distribution itself rather than converging (hence, the distribution has no "mean" despite being symmetric). Is there any connection here, or is that just a coincidental similarity?
- spider-mario 2mo ago
  At the same time, though, it doesn’t mean you can’t infer the location parameter of a Cauchy distribution with more and more precision as you receive more samples. It just means that averaging the samples is not the way to do it.
mturmon 2mo ago
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.

3 ms·

Yup, it has the fun property that you'll get the same error in your estimate of the mean regardless of averaging time (though since it's not stationary this isn

by rcxdude 2mo ago

plus 2mo ago
This makes me think of the Cauchy distribution, a probability distribution whose average follows the distribution itself rather than converging (hence, the distribution has no "mean" despite being symmetric). Is there any connection here, or is that just a coincidental similarity?
- spider-mario 2mo ago
  At the same time, though, it doesn’t mean you can’t infer the location parameter of a Cauchy distribution with more and more precision as you receive more samples. It just means that averaging the samples is not the way to do it.
mturmon 2mo ago
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.