nLab law of large numbers

Idea

In probability theory, the law or large numbers states that the empirical mean of a process where the random variables are independent and identically distributed tends to its expectation value.

It can be seen as a mathematical formalization of the idea that, even in presence of randomness, the average behavior of a system tends to be predictable, and the larger the sample size is, the better the prediction is.

For example, when rolling a fair die repeatedly, the average of the number rolled tends to

$$\frac{1+2+3+4+5+6}{6} \;=\; 3.5 .$$

Main statements

The term law of large numbers refers to a few different statements, depending on the particular convergence of random variables? considered.

The weak law of large numbers

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The strong law of large numbers

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Category-theoretic approaches

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See also

• probability theory, categorical probability
• random variable, expectation value
• empirical mean, empirical distribution
• iid random variables, de Finetti's theorem, zero-one law
• central limit theorem, Glivenko-Cantelli theorem?
• ergodicity, ergodic theorem?
• martingale convergence theorem?
• asymptotic equipartition property?

References

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category: probability