Give an example of independent integrable random variables $X_n$ such that $E[X_n] = 0$ for all n, but $\overline{S}_n = (\sum_{i=0}^n X_i)/n$ does not converge to 0 in probability.
As far as I know, $E[X_n] = 0$ is essentially the same as $\int_{-\infty}^\infty X_n(t)dt = 0$ when the $X_n$'s use the real numbers as their domain, so this question is asking for a sequence of functions which are 0 almost everywhere, but whose sum, divided by n, is not 0 almost everywhere. It seems that I need a sequence of partial sums that go to infinity "faster" than $1/n$ goes to 0, but anything I can think of that might do that seems to be made up of non-integrable functions (functions for which $\int_{-\infty}^\infty X_n(t)dt$ diverge), much less functions that are 0 almost everywhere.
The other idea I had was to perhaps have $X_1$ be $x^2$ (or some other "big enough" function) on the rationals, 0 elsewhere, and each successive $X_i$ be that same function on the rationals shifted by some tiny irrational number, 0 elsewhere, but this doesn't seem like it can work, since the resulting sum would still only be nonzero on a countable union of countable sets, which should itself then be countable.
I would really appreciate just a hint on how to get started with finding the right kind of sequence of functions.
As you ask, this is a hint:
Define $a_0 = 0, a_1 = 1$ and $a_{n} = (n+1)\sum_{k=1}^{n-1}a_k$.
Define $X_n$ take values $-a_n$ and $a_n$ with equal probability, i.e. $\frac{1}{2}$. All $X_n$'s are independent.
This idea is that $X_n$ take a value so large that it ignores what happened before. And the value is so large that even divided by $n$ doesn't eliminate its influence.