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Characteristic Functions

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces characteristic functions.

You have already encountered the Moment Generating Function of a pdf in the Part IB probability course. This function was closely related to the Laplace Transform of the pdf.

Now we introduce the Characteristic Function for a random variable, which is closely related to the Fourier Transform of the pdf.

In the same way that Fourier Transforms allow easy manipulation of signals when they are convolved with linear system impulse responses, Characteristic Functions allow easy manipulation of convolved pdfs when they represent sums of random processes.

The Characteristic Function of a pdf is defined as:

ΦX u=Eeiux=eiux fX xdx=u ΦX u u x x u x fX x u
(1)
where u u is the Fourier Transform of the pdf.

Note that whenever fXfX is a valid pdf, Φ0= fX xdx=1 Φ 0 x fX x 1

Properties of Fourier Transforms apply with uu substituted for ωω. In particular:

• Convolution - (sums of independent rv's)
(Y=i=1N Xi )( fY = f X 1 * f X 2 ** f X N )( ΦY u=i=1N Φ X i u) Y i 1 N Xi fY f X 1 f X 2 f X N ΦY u i 1 N Φ X i u
(2)
• Inversion
fX x=12πe(iux) ΦX udu fX x 1 2 u u x ΦX u
(3)
• Moments
(dndun ΦX u=ixneiux fX xdx)(EXn=xn fX xdx=1indndun ΦX u|u=0) un ΦX u x x n u x fX x X n x x n fX x 1 n u 0 u n ΦX u
(4)
• Scaling If Y=aX Y a X , fY y= fX xa fY y fX x a from this equation in our previous discussion of functions of random variables, then
ΦY u=eiuy fY ydy=eiuax fX xdx= ΦX au ΦY u y u y fY y x u a x fX x ΦX a u
(5)

Characteristic Function of a Gaussian pdf

The Gaussian or normal distribution is very important, largely because of the Central Limit Theorem which we shall prove below. Because of this (and as part of the proof of this theorem) we shall show here that a Gaussian pdf has a Gaussian characteristic function too.

A Gaussian distribution with mean μμ and variance σ2 σ 2 has pdf:

fx=12πσ2exμ22σ2 f x 1 2 σ 2 x μ 2 2 σ 2
(6)
Its characteristic function is obtained as follows, using a trick known as completing the square of the exponent:
ΦX u=Eeiux=eiux fX xdx=12πσ2ex22μx+μ22σ2iux2σ2dx=(12πσ2exμ+iuσ222σ2dx)e2iuσ2μu2σ42σ2=eiuμeu2σ22 ΦX u u x x u x fX x 1 2 σ 2 x x 2 2 μ x μ 2 2 σ 2 u x 2 σ 2 1 2 σ 2 x x μ u σ 2 2 2 σ 2 2 u σ 2 μ u 2 σ 4 2 σ 2 u μ u 2 σ 2 2
(7)
since the integral in brackets is similar to a Gaussian pdf and integrates to unity.

Thus the characteristic function of a Gaussian pdf is also Gaussian in magnitude, eu2σ22 u 2 σ 2 2 , with standard deviation 1σ 1 σ , and with a linear phase rotation term, eiuμ u μ , whose rate of rotation equals the mean μμ of the pdf. This coincides with standard results from Fourier analysis of Gaussian waveforms and their spectra (e.g. Fourier transform of a Gaussian waveform with time shift).

Summation of two or more Gaussian random variables

If two variables, X 1 X 1 and X 2 X 2 , with Gaussian pdfs are summed to produce XX, their characteristic functions will be multiplied together (equivalent to convolving their pdfs) to give

ΦX u= Φ X1 u Φ X2 u=eiu( μ1 + μ2 )eu2( σ1 2+ σ2 2)2 ΦX u Φ X1 u Φ X2 u u μ1 μ2 u 2 σ1 2 σ2 2 2
(8)
This is the characteristic function of a Gaussian pdf with mean ( μ1 + μ2 μ1 μ2 ) and variance ( σ1 2+ σ2 2 σ1 2 σ2 2 ).

Further Gaussian variables can be added and the pdf will remain Gaussian with further terms added to the above expressions for the combined mean and variance.

Central Limit Theorem

The central limit theorem states broadly that if a large number NN of independent random variables of arbitrary pdf, but with equal variance σ2 σ 2 and zero mean, are summed together and scaled by 1N 1 N to keep the total energy independent of NN, then the pdf of the resulting variable will tend to a zero-mean Gaussian with variance σ2 σ 2 as NN tends to infinity.

This result is obvious from the previous result if the input pdfs are also Gaussian, but it is the fact that it applies for arbitrary input pdfs that is remarkable, and is the reason for the importance of the Gaussian (or normal) pdf. Noise generated in nature is nearly always the result of summing many tiny random processes (e.g. noise from electron energy transitions in a resistor or transistor, or from distant worldwide thunder storms at a radio antenna) and hence tends to a Gaussian pdf.

Although for simplicity, we shall prove the result only for the case when all the summed processes have the same variance and pdfs, the central limit result is more general than this and applies in many cases even when the variance and pdfs are not all the same.

Proof:

Let X i X i ( i=1 i 1 to NN) be the NN independent random processes, each will zero mean and variance σ2 σ 2 , which are combined to give

X=1Ni=1N Xi X 1 N i 1 N Xi
(9)
Then, if the characteristic function of each input process before scaling is Φu Φ u and we use Equation 5 to include the scaling by 1N 1 N , the characteristic function of XX is
ΦX u=i=1N Φ X i uN=ΦNuN ΦX u i 1 N Φ X i u N Φ u N N
(10)
Taking logs:
log ΦX u=NlogΦuN ΦX u N Φ u N
(11)
Using Taylor's theorem to expand ΦuN Φ u N in terms of its derivatives at u=0 u 0 (and hence its moments) gives
ΦuN=Φ0+uNdΦ0d+12uN2dΦ0d+16uN3dΦ0d+124uN4dΦ0d+ Φ u N Φ 0 u N Φ 0 1 2 u N 2 2 Φ 0 1 6 u N 3 3 Φ 0 1 24 u N 4 4 Φ 0
(12)
From the Moments property of characteristic functions with zero mean:
• valid pdf Φ0=E Xi 0=1 Φ 0 Xi 0 1
• zero mean Φ0=iE Xi =0 Φ 0 Xi 0
• variance Φ0=i2E Xi 2=σ2 2 Φ 0 2 Xi 2 σ 2
• scaled skewness Φ0=i3E Xi 3=(iγσ3) 3 Φ 0 3 Xi 3 γ σ 3
• scaled kurtosis Φ04=i4E Xi 4=(κ+3)σ4 4 Φ 0 4 Xi 4 κ 3 σ 4
These are all constants, independent of NN, and dependent only on the shape of the pdfs f Xi f Xi .

Substituting these moments into Equation 11 and Equation 12 and using the series expansion, log(1+x)=x 1 x x + (terms of order x2 x 2 or smaller), gives

log ΦX u=NlogΦuN=Nlog(1u22Nσ2+**)=N(u2σ22N+**)=u2σ22+## ΦX u N Φ u N N 1 u 2 2 N σ 2 ** N u 2 σ 2 2 N ** u 2 σ 2 2 ##
(13)
where ** represents the terms of order N32 N 3 2 or smaller and ## represents the terms of order N12 N 1 2 or smaller. As N N , log ΦX uu2σ22 ΦX u u 2 σ 2 2 Therefore, as N N
ΦX ueu2σ22 ΦX u u 2 σ 2 2
(14)
Note that, if the input pdfs are symmetric, the skewness will be zero and the error terms will decay as N-1 N -1 rather than N12 N 1 2 ; and so convergence to a Gaussian characteristic function will be more rapid.

Hence we may now infer from Equation 6, Equation 7 and Equation 14 that the pdf of XX as N N will be given by

fX x=12πσ2ex22σ2 fX x 1 2 σ 2 x 2 2 σ 2
(15)
Thus we have proved the required central limit result.

Figure 1(a) shows an example of convergence when the input pdfs are uniform, and NN is gradually increased from 11 to 5050. By N=12 N 12 , convergence is good, and this is how some 'Gaussian' random generator functions operate - by summing typically 1212 uncorrelated random numbers with uniform pdfs.

For some less smooth or more skewed pdfs, convergence can be slower, as shown for a highly skewed triangular pdf in Figure 1(b); and pdfs of discrete processes are particularly problematic in this respect, as illustrated in Figure 1(c).

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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