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As in St. Petersburg paradox, let $X:\mathbb Z_{>0}\to\mathbb R$ be a discrete random variable with $\operatorname{Pr}[X = 2^k] = \frac{1}{2^k}$ for all $k\ge 0$ (and $\operatorname{Pr}[X = n] = 0$ if $n$ is not a power of two). Then $\mathbb E[X] = +\infty$ or is undefined, depending on the definition.

Closely related, let $Y:\mathbb Z_{>0}\to\mathbb R$ be a discrete random variable with $\operatorname{Pr}[X = (-1)^k 2^k] = \frac{1}{2^k}$ for all $k\ge 0$ (and $\operatorname{Pr}[X = n] = 0$ if $n$ is not a power of two). Then $\mathbb E[Y]$ is undefined.

What does it mean for repeated experiments?

  1. Does $\mathbb E[X]=+\infty$ implies that one should expect that the more one repeats the experiment, the larger the mean of the observed values is? Of course, defining $N$ copies $X_1$, ..., $X_N$ of $X$, one has $\mathbb E[\frac{1}{N}(X_1+\dotsb+X_N)] = +\infty$ so it seems to have no sense to say that the observed values grow in expectation...

  2. What does $\mathbb E[Y]$ being undefined implies? The observed values seems distributed around $0$: Can we quantify this or at least make some sense of it?

The general question is: The expectation of these variables is either infinite or undefined; Yet one can perform the experiment and compute the means that are obtained. What do they look like?

In a different direction: Is there some mathematical quantity that can describe the practical behavior of these experiments, if expectation is not relevant?

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  • $\begingroup$ Maybe your trouble is with the difference between "theoretical mean" and "sample mean". For example, if $X$ is a Cauchy r.v., it is known that $E|X|=\infty$ (theoretical value), yet one can simulate values from $|X|$ and calculate a sample mean. $\endgroup$
    – RLC
    Apr 27, 2020 at 16:05
  • $\begingroup$ I didn't know the terms. My question is maybe: What can an infinite or undefined theoretical mean (expectation) say about the expected¹ sample mean? ¹ Used in non-technical sense. $\endgroup$
    – Bruno
    Apr 27, 2020 at 16:20

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