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Matt_Simpson comments on Anticipating critical transitions - Less Wrong
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Similar weird things happen for the Cauchy distribution (whose probability density function is proportional to 1/(1+x^2)), which is symmetric around 0 but does not have mean 0 because the sum doesn't converge.
Exercise: what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?
Off the cuff: it's probably a random walk.
Edit: It's now pretty clear to me that's false, but plotting the ergodic means of several "chains" seems like a good way to figure it out.
Edit 2: In retrospect, I should have predicted that. If anyone is interested, I can post some R code so you can see what happens.