# 13.1 Convergence and the central limit theorem  (Page 2/4)

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$\phi \left(t\right)=E\left[{e}^{itX}\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\phi }_{n}\left(t\right)=E\left[{e}^{it{S}_{n}^{*}}\right]={\phi }^{n}\left(t/\sigma \sqrt{n}\right)$

Using the power series expansion of φ about the origin noted above, we have

$\phi \left(t\right)=1-\frac{{\sigma }^{2}{t}^{2}}{2}+\beta \left(t\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\beta \left(t\right)=o\left({t}^{2}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\to 0$

This implies

$|\phi \left(t/\sigma \sqrt{n}\right)-\left(1-{t}^{2}/2n\right)|=|\beta \left(t/\sigma \sqrt{n}\right)|=o\left({t}^{2}/{\sigma }^{2}n\right)$

so that

$n|\phi \left(t/\sigma \sqrt{n}\right)-\left(1-{t}^{2}/2n\right)|\to 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty$

A standard lemma of analysis ensures

$|{\phi }^{n}\left(t/\sigma \sqrt{n}\right)-{\left(1-{t}^{2}/2n\right)}^{n}|\le n|\phi \left(t/\sigma \sqrt{n}\right)-\left(1-{t}^{2}/2n\right)|\to 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty$

It is a well known property of the exponential that

${\left(1,-,\frac{{t}^{2}}{2n}\right)}^{n}\to {e}^{-{t}^{2}/2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty$

so that

$\phi \left(t/\sigma \sqrt{n}\right)\to {e}^{-{t}^{2}/2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{0.277778em}{0ex}}t$

By the convergence theorem on characteristic functions, above, ${F}_{n}\left(t\right)\to \Phi \left(t\right)$ .

$\square$

The theorem says that the distribution functions for sums of increasing numbers of the X i converge to the normal distribution function, but it does not tell how fast. It is instructive to consider some examples, which are easily worked out with the aid of our m-functions.

Demonstration of the central limit theorem

Discrete examples

We first examine the gaussian approximation in two cases. We take the sum of five iid simple random variables in each case. The first variable has six distinct values; thesecond has only three. The discrete character of the sum is more evident in the second case. Here we use not only the gaussian approximation, but the gaussian approximationshifted one half unit (the so called continuity correction for integer-values random variables). The fit is remarkably good in either case with only five terms.

A principal tool is the m-function diidsum (sum of discrete iid random variables). It uses a designated number of iterations of mgsum.

## First random variable

X = [-3.2 -1.05 2.1 4.6 5.3 7.2];PX = 0.1*[2 2 1 3 1 1];EX = X*PX' EX = 1.9900VX = dot(X.^2,PX) - EX^2 VX = 13.0904[x,px] = diidsum(X,PX,5); % Distribution for the sum of 5 iid rvF = cumsum(px); % Distribution function for the sum stairs(x,F) % Stair step plothold on plot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function% Plotting details (see [link] ) Distribution for the sum of five iid random variables.

## Second random variable

X = 1:3; PX = [0.3 0.5 0.2]; EX = X*PX'EX = 1.9000 EX2 = X.^2*PX'EX2 = 4.1000 VX = EX2 - EX^2VX = 0.4900 [x,px]= diidsum(X,PX,5); % Distribution for the sum of 5 iid rv F = cumsum(px); % Distribution function for the sumstairs(x,F) % Stair step plot hold onplot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function plot(x,gaussian(5*EX,5*VX,x+0.5),'o') % Plot with continuity correction% Plotting details (see [link] ) Distribution for the sum of five iid random variables.

As another example, we take the sum of twenty one iid simple random variables with integer values. We examine only part of the distribution function where most of theprobability is concentrated. This effectively enlarges the x-scale, so that the nature of the approximation is more readily apparent.

## Sum of twenty-one iid random variables

X = [0 1 3 5 6];PX = 0.1*[1 2 3 2 2];EX = dot(X,PX) EX = 3.3000VX = dot(X.^2,PX) - EX^2 VX = 4.2100[x,px] = diidsum(X,PX,21);F = cumsum(px); FG = gaussian(21*EX,21*VX,x);stairs(40:90,F(40:90)) hold onplot(40:90,FG(40:90)) % Plotting details (see [link] ) Distribution for the sum of twenty one iid random variables.

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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive

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