Editing Law of large numbers (section)

===Strong law===
The '''strong law of large numbers''' (also called [[Andrey Kolmogorov|Kolmogorov]]'s law) states that the sample average [[Almost sure convergence|converges almost surely]] to the expected value<ref>{{harvnb|Loève|1977|loc=Chapter 17.3, p. 251}}</ref>
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{\text{a.s.}}{\longrightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|3}}}}

That is,

<math display="block">
    \Pr\!\left( \lim_{n\to\infty}\overline{X}_n = \mu \right) = 1.
</math>

What this means is that, as the number of trials ''n'' goes to infinity, the probability that the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate sub-sequence.<ref name="TaoBlog" /> 

The strong law of large numbers can itself be seen as a special case of the [[Ergodic theory#Ergodic theorems|pointwise ergodic theorem]]. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average".  

Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability). See [[#Differences between the weak law and the strong law|Differences between the weak law and the strong law]].

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).<ref name=EMStrong>{{cite web|author1=Yuri Prokhorov| title=Strong law of large numbers|url=https://www.encyclopediaofmath.org/index.php/Strong_law_of_large_numbers| website=Encyclopedia of Mathematics}}</ref>

If the summands are independent but not identically distributed, then
{{NumBlk||<math display="block">
    \overline{X}_n - \operatorname{E}\big[\overline{X}_n\big]\ \overset{\text{a.s.}}{\longrightarrow}\ 0,
</math>|{{EquationRef|2}}}}

provided that each ''X''<sub>''k''</sub> has a finite second moment and

<math display="block">
    \sum_{k=1}^{\infty} \frac{1}{k^2} \operatorname{Var}[X_k] < \infty.
</math>

This statement is known as ''Kolmogorov's strong law'', see e.g. {{harvtxt|Sen|Singer|1993|loc=Theorem 2.3.10}}.