### Abstract

The Law of Large Numbers states that, informally, if one performs a random experiment $X$ many times and takes the average of the results, that average will be very close to the expected value $E[X]$.

More formally, let $(X_i)_{i\in\mathbb{N}}$ be a sequence of independently identically distributed random variables whose expected value $E[X_1]$ exists. Denote the running average of $X_1, \ldots, X_n$ as $\overline{X}_n$. Then:

- The Weak Law of Large Numbers states that $\overline{X}_{n} \longrightarrow E[X_1]$ in probability for $n\to\infty$, i.e. $\mathcal{P}(|\overline{X}_{n} - E[X_1]| > \varepsilon) \longrightarrow 0$ as $n\to\infty$ for any $\varepsilon > 0$.
- The Strong Law of Large Numbers states that $\overline{X}_{n} \longrightarrow E[X_1]$ almost surely for $n\to\infty$, i.e. $\mathcal{P}(\overline{X}_{n} \longrightarrow E[X_1]) = 1$.

In this entry, I formally prove the strong law and from it the weak law. The approach used for the proof of the strong law is a particularly quick and slick one based on ergodic theory, which was formalised by Gouëzel in another AFP entry.

### License

BSD License

### Topics

- Mathematics/Probability theory

### Session Laws_of_Large_Numbers

- Laws_of_Large_Numbers
- Laws_of_Large_Numbers_Example

### Depends on

- Ergodic Theory

## Cite

×

Laws_of_Large_Numbers-AFP

    @article{Laws_of_Large_Numbers-AFP,
      author  = {Manuel Eberl},
      title   = {The Laws of Large Numbers},
      journal = {Archive of Formal Proofs},
      month   = {February},
      year    = {2021},
      note    = {\url{https://isa-afp.org/entries/Laws_of_Large_Numbers.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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