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13.8. Central limit theorem for equally distributed terms

Lecture



Different forms of the central limit theorem differ among themselves by the conditions imposed on the distributions forming the sum of random terms. Here we formulate and prove one of the simplest forms of the central limit theorem, which refers to the case of identically distributed terms.

Theorem. If a   13.8.  Central limit theorem for equally distributed terms - independent random variables having the same distribution law with the expectation   13.8.  Central limit theorem for equally distributed terms and variance   13.8.  Central limit theorem for equally distributed terms , then with an unlimited increase   13.8.  Central limit theorem for equally distributed terms amount distribution law

  13.8.  Central limit theorem for equally distributed terms (13.8.1)

unlimited approaches to normal.

Evidence.

We will carry out the proof for the case of continuous random variables.   13.8.  Central limit theorem for equally distributed terms (for discontinuous it will be similar).

According to the second property of the characteristic functions, proved in the previous   13.8.  Central limit theorem for equally distributed terms , characteristic function of magnitude   13.8.  Central limit theorem for equally distributed terms is a product of the characteristic functions of the terms. Random variables   13.8.  Central limit theorem for equally distributed terms have the same distribution law with density   13.8.  Central limit theorem for equally distributed terms and therefore the same characteristic function

  13.8.  Central limit theorem for equally distributed terms . (13.8.2)

Therefore, the characteristic function of a random variable   13.8.  Central limit theorem for equally distributed terms will be

  13.8.  Central limit theorem for equally distributed terms . (13.8.3)

We investigate in more detail the function   13.8.  Central limit theorem for equally distributed terms . Imagine it in a neighborhood of a point.   13.8.  Central limit theorem for equally distributed terms according to the Maclaurin formula with three members:

  13.8.  Central limit theorem for equally distributed terms , (13.8.4)

Where   13.8.  Central limit theorem for equally distributed terms at   13.8.  Central limit theorem for equally distributed terms .

Find the values   13.8.  Central limit theorem for equally distributed terms ,   13.8.  Central limit theorem for equally distributed terms ,   13.8.  Central limit theorem for equally distributed terms . Assuming in the formula (13.8.2)   13.8.  Central limit theorem for equally distributed terms we have:

  13.8.  Central limit theorem for equally distributed terms . (13.8.5)

Differentiate (13.8.2) by   13.8.  Central limit theorem for equally distributed terms :

  13.8.  Central limit theorem for equally distributed terms . (13.8.6)

Putting in (13.8.6)   13.8.  Central limit theorem for equally distributed terms , we get:

  13.8.  Central limit theorem for equally distributed terms . (13.8.7)

Obviously, without limiting the generality, you can put   13.8.  Central limit theorem for equally distributed terms (for this it is enough to move the origin to the point   13.8.  Central limit theorem for equally distributed terms ). Then

  13.8.  Central limit theorem for equally distributed terms .

Let's differentiate (13.8.6) again:

  13.8.  Central limit theorem for equally distributed terms ,

from here

  13.8.  Central limit theorem for equally distributed terms . (13.8.8)

With   13.8.  Central limit theorem for equally distributed terms the integral in expression (13.8.8) is nothing but the dispersion of the quantity   13.8.  Central limit theorem for equally distributed terms with density   13.8.  Central limit theorem for equally distributed terms , Consequently

  13.8.  Central limit theorem for equally distributed terms . (13.8.9)

Substituting in (13.8.4)   13.8.  Central limit theorem for equally distributed terms ,   13.8.  Central limit theorem for equally distributed terms and   13.8.  Central limit theorem for equally distributed terms , we get:

  13.8.  Central limit theorem for equally distributed terms . (13.8.10)

Let's turn to random   13.8.  Central limit theorem for equally distributed terms . We want to prove that its distribution law with increasing   13.8.  Central limit theorem for equally distributed terms approaching normal. To do this, move on from   13.8.  Central limit theorem for equally distributed terms to another ("normalized") random value

  13.8.  Central limit theorem for equally distributed terms . (13.8.11)

This value is convenient because its dispersion does not depend on   13.8.  Central limit theorem for equally distributed terms and is equal to one for any   13.8.  Central limit theorem for equally distributed terms . It is easy to verify this, considering the value   13.8.  Central limit theorem for equally distributed terms as a linear function of independent random variables   13.8.  Central limit theorem for equally distributed terms each of which has a variance   13.8.  Central limit theorem for equally distributed terms . If we prove that the distribution law   13.8.  Central limit theorem for equally distributed terms approaching normal, then obviously it will be true for the magnitude   13.8.  Central limit theorem for equally distributed terms associated with   13.8.  Central limit theorem for equally distributed terms linear dependence (13.8.11).

Instead of proving that the distribution law   13.8.  Central limit theorem for equally distributed terms while increasing   13.8.  Central limit theorem for equally distributed terms approaches normal, we show that its characteristic function approaches the characteristic function of a normal law.

Find the characteristic function of   13.8.  Central limit theorem for equally distributed terms . From relation (13.8.11), according to the first property of the characteristic functions (13.7.8), we obtain

  13.8.  Central limit theorem for equally distributed terms , (13.8.12)

Where   13.8.  Central limit theorem for equally distributed terms - characteristic function of a random variable   13.8.  Central limit theorem for equally distributed terms .

From formulas (13.8.12) and (13.8.3) we get

  13.8.  Central limit theorem for equally distributed terms (13.8.13)

or using the formula (13.8.10),

  13.8.  Central limit theorem for equally distributed terms . (13.8.14)

Let us name the expression (13.8.14):

  13.8.  Central limit theorem for equally distributed terms   13.8.  Central limit theorem for equally distributed terms .

We introduce the notation

  13.8.  Central limit theorem for equally distributed terms . (13.8.15)

Then

  13.8.  Central limit theorem for equally distributed terms . (13.8.16)

We will increase without limit   13.8.  Central limit theorem for equally distributed terms . The value of   13.8.  Central limit theorem for equally distributed terms , according to the formula (13.8.15), tends to zero. With significant   13.8.  Central limit theorem for equally distributed terms it can be considered very small. Decompose   13.8.  Central limit theorem for equally distributed terms in a row and confine ourselves to one member of the expansion (the rest   13.8.  Central limit theorem for equally distributed terms become negligible):

  13.8.  Central limit theorem for equally distributed terms .

Then we get

  13.8.  Central limit theorem for equally distributed terms

  13.8.  Central limit theorem for equally distributed terms .

By definition, the function   13.8.  Central limit theorem for equally distributed terms tends to zero at   13.8.  Central limit theorem for equally distributed terms ; Consequently,

  13.8.  Central limit theorem for equally distributed terms

and

  13.8.  Central limit theorem for equally distributed terms ,

from where

  13.8.  Central limit theorem for equally distributed terms . (13.8.17)

This is nothing but the characteristic function of a normal law with parameters   13.8.  Central limit theorem for equally distributed terms ,   13.8.  Central limit theorem for equally distributed terms (see example 2,   13.8.  Central limit theorem for equally distributed terms 13.7).

Thus, it is proved that by increasing   13.8.  Central limit theorem for equally distributed terms random characteristic function   13.8.  Central limit theorem for equally distributed terms unboundedly approaches the characteristic function of the normal law; hence we conclude that the law of distribution of magnitude   13.8.  Central limit theorem for equally distributed terms (and therefore the values   13.8.  Central limit theorem for equally distributed terms a) unlimited approach to normal law. The theorem is proved.

We have proved the central limit theorem for a particular, but important case of equally distributed terms. However, in a fairly wide class of conditions, it is also valid for unequally distributed terms. For example, A. M. Lyapunov proved the central limit theorem for the following conditions:

  13.8.  Central limit theorem for equally distributed terms , (13.8.18)

Where   13.8.  Central limit theorem for equally distributed terms - the third absolute central moment of magnitude   13.8.  Central limit theorem for equally distributed terms :

  13.8.  Central limit theorem for equally distributed terms   13.8.  Central limit theorem for equally distributed terms .

  13.8.  Central limit theorem for equally distributed terms - variance of magnitude   13.8.  Central limit theorem for equally distributed terms .

The most general (necessary and sufficient) condition for the validity of the central limit theorem is the Lindeberg condition: for any   13.8.  Central limit theorem for equally distributed terms

  13.8.  Central limit theorem for equally distributed terms ,

Where   13.8.  Central limit theorem for equally distributed terms - expected value,   13.8.  Central limit theorem for equally distributed terms - distribution density of a random variable   13.8.  Central limit theorem for equally distributed terms ,   13.8.  Central limit theorem for equally distributed terms .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis