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

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φ ( t ) = E [ e i t X ] and φ n ( t ) = E [ e i t S n * ] = φ n ( t / σ n )

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

φ ( t ) = 1 - σ 2 t 2 2 + β ( t ) where β ( t ) = o ( t 2 ) as t → 0

This implies

| φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | = | β ( t / σ n ) | = o ( t 2 / σ 2 n )

so that

n | φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | → 0 as n → ∞

A standard lemma of analysis ensures

| φ n ( t / σ n ) - ( 1 - t 2 / 2 n ) n | ≤ n | φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | → 0 as n → ∞

It is a well known property of the exponential that

1 - t 2 2 n n → e - t 2 / 2 as n → ∞

so that

φ ( t / σ n ) → e - t 2 / 2 as n → ∞ for all t

By the convergence theorem on characteristic functions, above, F n ( t ) → Φ ( t ) .

— □

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] )
Figure one is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from -20 in increments of 10 to 40. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.4. There are two captions inside the graph. The first reads, X = [-3.2 -1.05 2.1 4.6 5.3 7.2]. The second reads, PX = 0.1*[2 2 1 3 1 1]. There are two graphs, one, a solid blue line, listed as a sum and the other a dashed and dotted line, listed as gaussian, but they both follow the same path and are nearly indistinguishable as they lay on top of one another. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (10, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (35, 1), it has again reduced in slope enough to be a horizontal line. Figure one is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from -20 in increments of 10 to 40. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.4. There are two captions inside the graph. The first reads, X = [-3.2 -1.05 2.1 4.6 5.3 7.2]. The second reads, PX = 0.1*[2 2 1 3 1 1]. There are two graphs, one, a solid blue line, listed as a sum and the other a dashed and dotted line, listed as gaussian, but they both follow the same path and are nearly indistinguishable as they lay on top of one another. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (10, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (35, 1), it has again reduced in slope enough to be a horizontal line.
Distribution for the sum of five iid random variables.
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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] )
Figure two is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from 5 in increments of 1 to 15. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.2. There are two captions inside the graph. The first reads, X = [1 2 3]. The second reads, PX = [0.3 0.5 0.2]. There are two graphs, one, a solid blue line, labeled step dbn fn, and the other a dashed and dotted line, labeled gaussian, but they both follow the same path. The step dbn fn is a series of horizontal line segments followed by vertical line segments in varying sizes that fit the shape of the smoother curve, the gaussian curve. A third labeled item is a series of small blue circles that sit at the upper corners of the steps of the solid lined curve, labeled, shifted gaussian. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (5, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (14, 1), it has again reduced in slope enough to be a horizontal line. Figure two is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from 5 in increments of 1 to 15. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.2. There are two captions inside the graph. The first reads, X = [1 2 3]. The second reads, PX = [0.3 0.5 0.2]. There are two graphs, one, a solid blue line, labeled step dbn fn, and the other a dashed and dotted line, labeled gaussian, but they both follow the same path. The step dbn fn is a series of horizontal line segments followed by vertical line segments in varying sizes that fit the shape of the smoother curve, the gaussian curve. A third labeled item is a series of small blue circles that sit at the upper corners of the steps of the solid lined curve, labeled, shifted gaussian. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (5, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (14, 1), it has again reduced in slope enough to be a horizontal line.
Distribution for the sum of five iid random variables.
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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] )
Figure three is a distribution graph. It is titled, partial distribution for sum of 21 iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range in value from 40 to 90 at increments of 5, and the vertical axis ranges from 0 to 1 in increments of .1. There are two labeled equations. The first reads, X = [0 1 3 5 6]. The second reads, PX = 0.1*[1 2 3 2 2]. There are two graphs, one a smooth curve, labeled gaussian dbn, and the other a series of steps closely following the smooth curve, labeled Dbn for sum. Both graphs begin at the point (40, 0) at the bottom-left of the graph. The slope of the smooth curve is flat, and increases until approximately (70, 0.5). At this point, the graph continues increasing, but its slope begins decreasing until at approximately (90, 0.99), the path is again nearly flat. The steps follow the smooth curve along the same path. Figure three is a distribution graph. It is titled, partial distribution for sum of 21 iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range in value from 40 to 90 at increments of 5, and the vertical axis ranges from 0 to 1 in increments of .1. There are two labeled equations. The first reads, X = [0 1 3 5 6]. The second reads, PX = 0.1*[1 2 3 2 2]. There are two graphs, one a smooth curve, labeled gaussian dbn, and the other a series of steps closely following the smooth curve, labeled Dbn for sum. Both graphs begin at the point (40, 0) at the bottom-left of the graph. The slope of the smooth curve is flat, and increases until approximately (70, 0.5). At this point, the graph continues increasing, but its slope begins decreasing until at approximately (90, 0.99), the path is again nearly flat. The steps follow the smooth curve along the same path.
Distribution for the sum of twenty one iid random variables.
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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