In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the law of large numbers (LLN) is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
that describes the result of performing the same experiment a large number of times. According to the law, the
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
of the results obtained from a large number of trials should be close to the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
and tends to become closer to the expected value as more trials are performed.
The LLN is important because it guarantees stable long-term results for the averages of some random events.
For example, while a
casino
A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known for hosting live enterta ...
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. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the
gambler's fallacy).
It is also important to note that the LLN only applies to the average. Therefore, while
:
other formulas that look similar are not verified, such as the raw deviation from "theoretical results":
:
not only does it not converge toward zero as ''n'' increases, but it tends to increase in absolute value as ''n'' increases.
Examples
For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
. Therefore, the expected value of the average of the rolls is:
:
According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the
sample mean) will approach 3.5, with the precision increasing as more dice are rolled.
It follows from the law of large numbers that the
empirical probability The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experi ...
of success in a series of
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
s will converge to the theoretical probability. For a
Bernoulli random variable, the expected value is the theoretical probability of success, and the average of ''n'' such variables (assuming they are
independent and identically distributed (i.i.d.)) is precisely the relative frequency.
For example, a
fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to . Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly . In particular, the proportion of heads after ''n'' flips will
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
converge to as ''n'' approaches infinity.
Although the proportion of heads (and tails) approaches , 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 the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips.
Another good example of the LLN is the
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
. These methods are a broad class of
computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. An esp ...
al
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s that rely on repeated
random sampling to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.
Limitation
The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of ''n'' results taken from the
Cauchy distribution or some
Pareto distributions (α<1) will not converge as ''n'' becomes larger; the reason is
heavy tails. The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation, whereas the expectation of the Pareto distribution (α<1) is infinite. One way to generate the Cauchy-distributed example is where the random numbers equal the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
of an angle uniformly distributed between −90° and +90°. The
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
is zero, but the expected value does not exist, and indeed the average of ''n'' such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as ''n'' goes to infinity.
And if the trials embed a selection bias, typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains.
History
The Italian mathematician
Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
(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. A special form of the LLN (for a binary random variable) was first proved by
Jacob Bernoulli. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his (''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
Bernoulli's principle, named after Jacob Bernoulli's nephew
Daniel Bernoulli. In 1837,
S. D. Poisson further described it under the name ("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
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
,
Markov,
Borel,
Cantelli,
Kolmogorov and
Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
exists for the weak law of large numbers to be true.
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, in reference to two different modes of
convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.
Forms
There are two different versions of the law of large numbers that are described below. They are called the'' strong law of large numbers'' and the ''weak law of large numbers''.
Stated for the case where ''X''
1, ''X''
2, ... is an infinite sequence of
independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E(''X''
1) = E(''X''
2) = ...= ''µ'', both versions of the law state that the sample average
:
converges to the expected value:
(Lebesgue integrability of ''X
j'' means that the expected value E(''X
j'') exists according to Lebesgue integration and is finite. It does ''not'' mean that the associated probability measure is
absolutely continuous with respect to
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
.)
Introductory probability texts often additionally assume identical finite
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
(for all
) and no correlation between random variables. In that case, the variance of the average of n random variables is
:
which can be used to shorten and simplify the proofs. This assumption of finite
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
is ''not necessary''. Large or infinite variance will make the convergence slower, but the LLN holds anyway.
Mutual independence of the random variables can be replaced by
pairwise independence or
exchangeability in both versions of the law.
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 (also called
Khinchin's law) states that the sample average
converges in probability towards the expected value
That is, for any positive number ''ε'',
:
Interpreting this result, the weak law states that for any nonzero margin specified (''ε''), 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.
As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first ''n'' values goes to zero as ''n'' goes to infinity.
[ As an example, assume that each random variable in the series follows a ]Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with mean zero, but with variance equal to , which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
to . The variance of the average is therefore asymptotic to and goes to zero.
There are also examples of the weak law applying even though the expected value does not exist.
Strong law
The strong law of large numbers (also called Kolmogorov's law) states that the sample average converges almost surely to the expected value
That is,
:
What this means is that the probability that, as the number of trials ''n'' goes to infinity, 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 subsequence.
The strong law of large numbers can itself be seen as a special case of the 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.
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).
If the summands are independent but not identically distributed, then
provided that each ''X''''k'' has a finite second moment and
:
This statement is known as ''Kolmogorov's strong law'', see e.g. .
Differences between the weak law and the strong law
The ''weak law'' states that for a specified large ''n'', the average is likely to be near ''μ''. Thus, it leaves open the possibility that happens an infinite number of times, although at infrequent intervals. (Not necessarily for all ''n'').
The ''strong law'' shows that this almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
will not occur. Note that it does not imply that with probability 1, we have that for any the inequality holds for all large enough ''n'', since the convergence is not necessarily uniform on the set where it holds.
The strong law does not hold in the following cases, but the weak law does.
Uniform law of large numbers
Suppose ''f''(''x'',''θ'') is some function defined for ''θ'' ∈ Θ, and continuous in ''θ''. Then for any fixed ''θ'', the sequence will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E 'f''(''X'',''θ'') This is the ''pointwise'' (in ''θ'') convergence.
The uniform law of large numbers states the conditions under which the convergence happens ''uniformly'' in ''θ''. If
# ''Θ'' is compact,
# ''f''(''x'',''θ'') is continuous at each ''θ'' ∈ Θ for almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
''x''s, and measurable function of ''x'' at each ''θ''.
# there exists a dominating function ''d''(''x'') such that E 'd''(''X'')< ∞, and
:::
Then E 'f''(''X'',''θ'')is continuous in ''θ'', and
:
This result is useful to derive consistency of a large class of estimators (see Extremum estimator In statistics and econometrics, extremum estimators are a wide class of estimators for parametric models that are calculated through maximization (or minimization) of a certain objective function, which depends on the data. The general theory of e ...
).
Borel's law of large numbers
Borel's law of large numbers, named after Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel was ...
, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if ''E'' denotes the event in question, ''p'' its probability of occurrence, and ''Nn''(''E'') the number of times ''E'' occurs in the first ''n'' trials, then with probability one,An Analytic Technique to Prove Borel's Strong Law of Large Numbers Wen, L. Am Math Month 1991
/ref>
:
This theorem makes rigorous the intuitive notion of probability as the long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.
Chebyshev's inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
. Let ''X'' be a random variable with finite expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
''μ'' and finite non-zero variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
''σ''2. Then for any real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
,
:
Proof of the weak law
Given ''X''1, ''X''2, ... an infinite sequence of i.i.d. random variables with finite expected value , we are interested in the convergence of the sample average
The weak law of large numbers states:
Proof using Chebyshev's inequality assuming finite variance
This proof uses the assumption of finite variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
(for all ). The independence of the random variables implies no correlation between them, and we have that
The common mean μ of the sequence is the mean of the sample average:
Using Chebyshev's inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
on results in
This may be used to obtain the following:
As ''n'' approaches infinity, the expression approaches 1. And by definition of convergence in probability
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications t ...
, we have obtained
Proof using convergence of characteristic functions
By Taylor's theorem for complex functions, the characteristic function of any random variable, ''X'', with finite mean μ, can be written as
All ''X''1, ''X''2, ... have the same characteristic function, so we will simply denote this ''φ''''X''.
Among the basic properties of characteristic functions there are
if ''X'' and ''Y'' are independent.
These rules can be used to calculate the characteristic function of in terms of ''φ''''X'':
The limit ''e''''it''μ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem
Levy, Lévy or Levies may refer to:
People
* Levy (surname), people with the surname Levy or Lévy
* Levy Adcock (born 1988), American football player
* Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker
* Levy ...
, converges in distribution to μ:
μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see Convergence of random variables.) Therefore,
This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.
Consequences
The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the probability distribution. By applying Borel's law of large numbers, one could easily obtain the probability mass function. For each event in the objective probability mass function, one could approximate the probability of the event's occurrence with the proportion of times that any specified event occurs. The larger the number of repetitions, the better the approximation. As for the continuous case: , for small positive h. Thus, for large n:
With this method, one can cover the whole x-axis with a grid (with grid size 2h) and obtain a bar graph which is called a histogram.
See also
* Asymptotic equipartition property
* Central limit theorem
* Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...
* Law of averages
* Law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by ...
* Law of truly large numbers
* Lindy effect
* Regression toward the mean
* Sortition
In governance, sortition (also known as selection by lottery, selection by lot, allotment, demarchy, stochocracy, aleatoric democracy, democratic lottery, and lottocracy) is the selection of political officials as a random sample from a large ...
* Strong law of small numbers
Notes
References
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External links
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Animations for the Law of Large Numbers
by Yihui Xie using the R packag
animation
Apple CEO Tim Cook said something that would make statisticians cringe
"We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody . said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform.'' explained Business Insider
''Insider'', previously named ''Business Insider'' (''BI''), is an American financial and business news website founded in 2007. Since 2015, a majority stake in ''Business Insider''s parent company Insider Inc. has been owned by the German publ ...
''
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Probability theorems
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Asymptotic theory (statistics)
Theorems in statistics