According to the Wikipedia page on the law of large numbers:

The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).[11]

I was under the impression that the conditions of the weak law of large numbers are not enough to imply an application of the strong law. But it appears to me this Wikipedia snippet is saying otherwise?

Question: So, just to clarify, do the following conditions imply that one can safely apply the strong law of large numbers?

  1. $X_1, \ldots, X_n$ are IID random variables
  2. $E[X_1] = \ldots = E[X_n] = \mu \in \mathbb{R}$

That is, from these assumptions, can we then conclude both that

  1. $X_n \xrightarrow{a.s.} X$ (weak law)
  2. $X_n \xrightarrow{a.s.} X$ (strong law)?

or only just (1)?

  • 1
    $\begingroup$ When the strong law applies, it implies the weak law, hence the name (since the weak law only implies convergence in probability which is strictly weaker than convergence a.s.). This probably doesn't answer your question though because I was too lazy to read it, hence why I am just posting a comment. I apologize for not being more helpful. $\endgroup$ – Chill2Macht Jun 19 '16 at 2:28

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