law of large numbers

An illustration of the Law of Large Numbers using die rolls. As the number of die rolls increases, the average of the values of all the rolls approaches 3.5.

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

For example, a single roll of a dice produces one of the numbers 1, 2, 3, 4, 5, 6, each with equal probability. Therefore, the expected value of a single die roll is

According to the law of large numbers, if a large number of dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled.

Similarly, when a fair coin is flipped once, the expected value of the number of heads is equal to one half. Therefore, according to the law of large numbers, the proportion of heads in a large number of coin flips should be roughly one half. In particular, the proportion of heads after n flips will almost surely converge to one half as n approaches infinity.

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

Already the Indian mathematician Brahmagupta (598–668) and modernly the Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. The LLN was first proved by Jacob Bernoulli.^[1] 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").^[2] 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 -- with virtual certainty -- the sample average

converges to the expected value

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. For interpretation of these modes, see Convergence of random variables.

Weak law

The weak law of large numbers states that the sample average converges in probability towards the expected value^[3]

That is to say that for any positive number ε,

(see Proof of the law of large numbers for a proof of the above statement).

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.

Strong law

The strong law of large numbers states that the sample average converges almost surely to the expected value^[4]

That is,

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.

Moreover, if the summands are independent but not identically distributed, then

provided that each X_k has finite second moment and

This statement is known as Kolmogorov's strong law, see e.g. Theorem 2.3.10 in Sen and Singer (1993) ^[5]

Differences between the weak law and the strong law

The Weak Law states that, for a specified large n, is likely to be near μ. Thus, it leaves open the possibility that 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 , the inequality is true only a finite number of times (as opposed to an infinite, but infrequent, number of times). ^[6]

See also

• Central limit theorem
• Law of averages
• Infinite Monkey Theorem

Notes

1. ^ Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4, (Translated into English by Oscar Sheynin)
2. ^ Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism"
3. ^ Michel Loève, Probability Theory 1, 1977, 4th edition, Chapter 1.4, page 14, Springer Verlag
4. ^ Michel Loève, Probability Theory 1, 1977, 4th edition, Chapter 17.3, page 251, Springer Verlag
5. ^ Sen, P. K. and Singer J. M. (1993). Large Sample Methods In Statistics, New York, London: Chapman \& Hall, Inc
6. ^ Sheldon Ross, A First Course in Probability, Fifth edition, Prentice Hall press

References

• Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford. ISBN 0-19-853665-8. 
• Richard Durrett (1995). Probability: Theory and Examples, 2nd Edition. Duxbury Press. 
• Martin Jacobsen (1992). Videregående Sandsynlighedsregning (Advanced Probability Theory) 3rd Edition. HCØ-tryk, Copenhagen. ISBN 87-91180-71-6. 

External links

• Weisstein, Eric W., "Weak Law of Large Numbers" from MathWorld.
• Weisstein, Eric W., "Strong Law of Large Numbers" from MathWorld.
• Animations for the Law of Large Numbers by Yihui Xie using the R package animation