Earlier quoted context omitted.
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...)
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?
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.