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Comment by mturmon | Hacker News Reader

Thanks for this link.

Many investigators have been dissatisfied with the strong assumptions that conventional measure theoretic probability requires you to make. The community that I'm most familiar with that is stretching these bounds is the imprecise probability people, http://www.sipta.org/index.php?id=cs

They have a conference every other year. The simplest generalization is to interval valued probabilities, but there are several other interesting theories.

There are some phenomena, such as a failure of long term averages of some actual physical-world stochastic processes to converge, that seem to indicate that conventional mathematical probability does not predict their behavior correctly. I.e., it is a theorem that long term averages of stationary processes should converge, and these don't.

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There are some phenomena, such as a failure of long term averages of some actual physical-world stochastic processes to converge

That's a general property of levy-stable probability distributions, which occur all over the place, and you don't have to resort to unconventional probability to explain most of these processes.

No, this is different. There is a theorem, which you can find in Loeve (and other places too I'm sure), that any stationary stochastic process with finite first moment will have long-term averages that converge. IIRC, you don't even need finite second moments. In any case, it has great generality, and basically all you need is stationarity.

Presumably a physical system that has stable components in a stable environment will be stationary. You can build a simple electrical circuit of a certain type and compute its long term averages and they do not, in this case, converge. Not slow convergence, like you would see in a process with long range dependence or heavy tails, but lack of convergence, despite averaging over very long times.

I have not done is my experiment myself, but my PhD advisor, who did some of the early work on unconventional axiomatizations of probability, did.

Theory is unfortunately silent on how many observations it takes to see convergence. If you mean that there are processes that "should" converge that do not seem to empirically—yes, there are, but that is not evidence for failure of the theory.

Is this unsatisfying? Yes, it is.

levy-stable distributions with an alpha less than or equal to 1, (e.g. cauchy distribution) don't have finite first moments.

The cauchy distribution is most simply explained with the blind archer analogy. An archer is blindfolded and placed in a random orientation. (Let's restrict it to [0-pi] radians for simplicity). She then shoots the arrow in the direction she's pointing until it hits an infinitely long, straight wall. What is the position along the wall where the arrow strikes?

The CDF is the normalized arctan(x) - should be easy to visualize, but the integral of x arctan x (the first moment) is undefined.

I'm aware that there are distributions without finite first moments; that's why I specified it. But a real electrical signal cannot have infinite expected value, so that's not an explanation.

Note that your archer analogy implicitly requires infinite expected energy input.

> I.e., it is a theorem that long term averages of stationary processes should converge, and these don't.

Maybe they're non-stationary then? Doesn't sound like a failure of probability, just a failure of a particular probabilistic model.

The real answer is, we don't know yet.

There are applications of this notion to financial modeling.

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