# Practical behaviors of infinite and undefined expectations

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?

• 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.
– RLC
Apr 27, 2020 at 16:05
• 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. Apr 27, 2020 at 16:20