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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 firs
by mturmon 12y ago
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.
- pash 12y agoTheory 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.
- dnautics 12y agolevy-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.
- mturmon 12y agoI'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.