# Law of large numbers

The law of large numbers (LLN) is a theorem in probability that describes the long-term stability of the mean of a random variable. Given a random variable with a finite expected value, if its values are repeatedly sampled, as the number of these observations increases, their mean will tend to approach and stay close to the expected value (the mean (average) for the population).

The LLN can easily be illustrated using the rolls of a die. That is, outcomes of a multinomial distribution in which the numbers 1, 2, 3, 4, 5, and 6 are equally likely to be chosen. The population mean (or "expected value") of the outcomes is:

(1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.

The graph to the right plots the results of an experiment of rolls of a die. In this experiment we see that the average of die rolls deviates wildly at first. As predicted by LLN the average stabilizes around the expected value of 3.5 as the number of observations becomes large.

Another example is the flip of a coin. Given repeated flips of a fair coin, the frequency of heads (or tails) will increasingly approach 50% over a large number of trials. Almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as number of flips becomes large. Also, almost surely the ratio of the absolute difference to number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.

For example, we may see 520 heads after 1000 flips and 4904 heads after 10000 flips. While the average has moved from 0.52 to 0.4904, closer to the expected 50%, the absolute difference from the expected mean has increased from 20 to 96.

The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be "balanced" by the others. See the Gambler's fallacy.

## History

The LLN was first described by Jacob Bernoulli. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's Theorem". This should not be confused with the principle in physics with the same name, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1835, S.D. Poisson further described it under the name "La loi des grands nombres" ("The law of large numbers"). Thereafter, it was known under both names, but the "Law of large numbers" is most frequently used.

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law. These forms do not describe different laws but instead refer to different ways of describing the mode of convergence of the cumulative sample means to the expected value, and the strong form implies the weak.

## Forms

Both versions of the law state that the sample average $\overline{X}_n=\frac1n(X_1+\cdots+X_n)$

converges to the expected value $\overline{X}_n \, \to \, \mu \qquad\textrm{for}\qquad n \to \infty$

where X1, X2, ... is an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞.

An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.

The difference between the strong and the weak version is concerned with the mode of convergence being asserted.

### The weak law

The weak law of large numbers states that the sample average converges in probability towards the expected value $\overline{X}_n \, \xrightarrow{P} \, \mu \qquad\textrm{for}\qquad n \to \infty.$

That is to say that for any positive number ε, $\lim_{n\to\infty}\operatorname{P}\left(\left|\overline{X}_n-\mu\right|<\varepsilon\right)=1.$

(Proof)

Interpreting this result, the weak law essentially states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value, that is, within the margin.

Convergence in probability is also called weak convergence of random variables. This version is called the weak law because random variables may converge weakly (in probability) as above without converging strongly (almost surely) as below.

A consequence of the weak LLN is the asymptotic equipartition property.

### The strong law

The strong law of large numbers states that the sample average converges almost surely to the expected value $\overline{X}_n \, \xrightarrow{\mathrm{a.s.}} \, \mu \qquad\textrm{for}\qquad n \to \infty .$

That is, $\operatorname{P}\left(\lim_{n\rightarrow\infty}\overline{X}_n=\mu\right)=1,$

The proof is more complex than that of the weak law. This law justifies the intuitive interpretation of the expected value of a random variable as the "long-term average when sampling repeatedly."

Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law.

The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem.

### Differences between the weak law and the strong law

The Weak Law states that, for a specified large n, (X1 + ... + Xn) / n is likely to be near μ. Thus, it leaves open the possibility that |(X1 + ... + Xn) / n − μ| > ε happens an infinite number of times, although it happens at infrequent intervals.

The strong law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have for any positive value ε, the inequality |(X1 + ... + Xn) / n − μ| > ε is true only a finite number of times (as opposed to an infinite, but infrequent, number of times). 

## Activities and demonstrations

There are varieties of ways to illustrate the theory and applications of the laws of large numbers using interactive aids. The SOCR resource provides a hands-on learning activity paired with a Java applet (select the Coin Toss LLN Experiment) that demonstrate the power and usability of the law of large numbers.