Central limit theorem

Téorema central limit mangrupa hasil susunan konvergen-lemah dina teori probabilitas. Ieu téorema keur ngagambarkeun yén jumlah unggal variabel acak nu kasebar bébas sacara normal (independent identically distributed atawa i.i.d) ngadeukeutan sebaran normal. Hasil ieu nerangkeun kaunikan sebaran normal.

Hasil nu penting jeung kawentar éta disebutna Teorema Central Limit Theorem; nu museurkeun kana variabel bébas nu pasipatan sebaranna bébas tur nilai eksptasi jeung varianna terhingga. Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others. Two such conditions are the Lindeberg condition and the Lyapunov condition. Other generalizations even allow some "weak" dependence of the random variables.

The réader may find it helpful to consider this illustration of the central limit theorem.

Teorema central limit

Tempo X₁,X₂,X₃,... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the nilai ekspektasi μ and the simpangan baku σ of D exist and are finite.

Consider the sum :S_n=X₁+...+X_n. Then the expected value of S_n is nμ and its simpangan baku is σ n^½. Furthermore, informally spéaking, the distribution of S_n approaches the normal distribution N(nμ,σ²n) as n approaches ∞.

In order to clarify the word "approaches" in the last sentence, we standardize S_n by setting

$Z_{n}={\frac {S_{n}-n\mu }{\sigma {\sqrt {n}}}}$

Then the distribution of Z_n converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution). This méans: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have

$\lim _{n\to \infty }{\mbox{Pr}}(Z_{n}\leq z)=\Phi (z),$

or, equivalently,

$\lim _{n\rightarrow \infty }{\mbox{Pr}}\left({\frac {{\overline {X}}_{n}-\mu }{\sigma /{\sqrt {n}}}}\leq z\right)=\Phi (z)$

where

${\overline {X}}_{n}=S_{n}/n=(X_{1}+\cdots +X_{n})/n$

is the "sample mean".

Proof of the central limit theorem

For a théorem of such fundamental importance to statistics and applied probability, the central limit théorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (wéak) law of large numbers. For any variabel acak, Y, with zero méan and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,

$\varphi _{Y}(t)=1-{t^{2} \over 2}+o(t^{2}),\quad t\rightarrow 0.$

Letting Y_i be (X_i − μ)/σ, the standardised value of X_i, it is éasy to see that the standardised méan of the observations X₁, X₂, ..., Xn is just

$Z_{n}={\frac {{\overline {X}}_{n}-\mu }{\sigma /{\sqrt {n}}}}=\sum _{i=1}^{n}{Y_{i} \over {\sqrt {n}}}.$

By simple properties of characteristic functions, the characteristic function of Z_n is

$\left[\varphi _{Y}\left({t \over {\sqrt {n}}}\right)\right]^{n}=\left[1-{t^{2} \over 2n}+o\left({t^{2} \over n}\right)\right]^{n}\,\rightarrow \,e^{-t^{2}/2},\quad n\rightarrow \infty .$

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit théorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.

Convergence to the limit

If the third central moment E((X₁ − μ)³) exists and is finite, then the above convergence is uniform and the speed of convergence is at léast on the order of 1/n^½ (see Berry-Esséen theorem).

Pictures of a distribution being "smoothed out" by summation (showing original distribution and three subseqent convolutions):

(See Illustration of the central limit theorem for further details on these images.)

An equivalent formulation of this limit théorem starts with A_n = (X₁ + ... + X_n) / n which can be interpreted as the méan of a random sample of size n. The expected value of A_n is μ and the standard deviation is σ / n^½. If we normalize A_n by setting Z_n = (A_n - μ) / (σ / n^½), we obtain the same variable Z_n as above, and it approaches a standard normal distribution.

Note the following apparent "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives? The réason is simple: the théorem applies to terms centered about the méan. Without that standardization, the distribution would, as intuition suggests, escape away to infinity.

Alternative statements of the theorem

The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit théorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions incréases without bound, under the conditions stated above.

Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit théorem has yet another restatement: the product of the characteristic functions of a number of density functions tends to the characteristic function of the normal density as the number of density functions incréases without bound, under the conditions stated above.

An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.

Lyapunov condition

Let X_n be a sequence of independent random variables defined on the same probability space. Assume that X_n has finite expected value μ_n and finite standard deviation σ_n. We define

$s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}$

Assume that the third central moments

$r_{n}^{3}={\mbox{E}}\left({\left|X_{n}-\mu _{n}\right|}^{3}\right)$

are finite for every n, and that

$\lim _{n\to \infty }{\frac {r_{n}}{s_{n}}}=0$

(This is the Lyapunov condition). We again consider the sum S_n=X₁+...+X_n. The expected value of S_n is m_n = ∑_i=1..nμ_i and its standard deviation is s_n. If we normalize S_n by setting

$Z_{n}={\frac {S_{n}-m_{n}}{s_{n}}}$

then the distribution of Z_n converges towards the standard normal distribution N(0,1) as above.

Lindeberg condition

In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following wéaker one: for every ε > 0

$\lim _{n\to \infty }\sum _{i=1}^{n}{\mbox{E}}\left({\frac {(X_{i}-\mu _{i})^{2}}{s_{n}^{2}}}:\left|X_{i}-\mu _{i}\right|>\epsilon s_{n}\right)=0$

(where E( U : V > c) denotes the conditional expected value: the expected value of U given that V > c.) Then the distribution of the normalized sum Z_n converges towards the standard normal distribution N(0,1).

Non-independent case

There are some théorems which tréat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.

track these down

Tumbu kaluar

• Central Limit Theorem Java Archived 2007-09-17 di Wayback Machine

Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantuanna didagoan pikeun narjamahkeun.