I have read several questions on the difference between weak and strong laws in terms of convergence in probability vs convergence a.s. and I think I grasp them.
However, I have uncertainty on the hypotheses. Is the following claim true ?
For i.i.d. samples from a random variable with finite mean, both strong and weak laws apply ?
My trouble comes from the Wikipedia explanation where a small line at the end of the "weak law" section states:
There are also examples of the weak law applying even though the expected value does not exist.
and indeed, in the section "difference between weak and strong law", the examples are all mentioning "no expected value".
What troubles me is that the weak law, as far as I know, is stated as:
$$\forall \epsilon>0, \quad \lim\limits_{n \to +\infty} P(|\bar{X}-\mu| > \epsilon) =0$$
right ?
Well, then, what does this law become if, as examples mentioned above claim, expected value does not exist ? What's $\mu$ then ? The Wikipedia article refers to "conditional convergence", but what does that mean ?
So, to sum up and facilitate the answers, I have two questions:
(i): Is the claim "For i.i.d. samples from a random variable with finite mean, both strong and weak laws apply." true ? (If not, precise which additional hypothesis for the strong law to hold in some cases)
(ii): What does the weak law look like when "the expected value does not exist", which does not seem very easily "inferable" for me when looking at the standard definition despite the fact that Wikipedia mentions several "examples" showing that strong law does not hold in these cases, but only the weak law ?