Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Probability Theory and Stochastic Processes
Steven R. Dunbar
Analytic Proof of the DeMoivre-Laplace Central Limit Theorem
Mathematicians Only: prolonged scenes of intense rigor.
Derive the Fourier transform of the rectangle function , where
Deﬁnition. Let be a random variable with probability density function . The Fourier transform, also known as the characteristic function of (and also of ) is
Example. The uniform random variable with density function on has Fourier transform
The Fourier transform has many properties, but the subsequent proofs only need one, the Fourier inversion formula:
(valid if ).
Remark. The proof is easily generalized to cumulative distribution functions and with associated probability measures and . See .
Rearranging the constants and using the evenness of
so that is the probability density that is the convolution of with the scaled normal density, .
Remark. This proof is adapted from Feller [3, pages 507-511]. This is one of several proofs possible for the inversion theorem. For alternative proofs see [1, pages 177-178], [2, page 155], or [5, pages 20-21].
Proof. Left as an exercise. □
Using the analytic expression for the probability of the sum of Bernoulli trials
Changing the exponential into a product of cosines comes from equations (2) and (3) in Analytic Model of Coin Flipping.
Interchange the order of integration, justiﬁed since assumes only a ﬁnite set of values.
Proof. This follows from
Alternatively, taking logarithms and using L’Hospital’s Rule□
Using this limit
The interchange of the operations of integration and taking the limit as needs justiﬁcation. However, the limits of integration are and and the function
is not absolutely integrable. Recall that a function is integrable on if . A function is absolutely integrable on if . A theorem from Lebesgue integration theory says that a function is integrable if and only if is absolutely integrable. The interchange of the operations of integration and taking the limit as is justiﬁed if the integrand is integrable, hence absolutely integrable. However, the function
is not absolutely integrable on , see the exercises. Therefore, although this proof of the Central Limit Theorem is fairly direct with Fourier transforms, it is not rigorously justiﬁed.
The equation (4) is a special case of Fourier’s general formula
applied to the indicator function (3). To make the proof above rigorous, introduce two auxiliary functions and , both graphed in Figure 1.
By construction, and then
are absolutely integrable as functions of on . (See the exercises for a proof.) Now the same argument as above yields rigorously
Combining the two previous expressions with the inequality for and
Since the inequality holds for every , it follows that
The deﬁnition of the Fourier transform and the inversion Theorem are adapted from William Feller, Introduction to Probability Theory and It Applications, Volume II, Second Edition, by W. Feller, J. Wiley and Sons, pages 507-511. The main results of the section are adapted from Statistical Independence in Probability, Analysis, and Number Theory by Mark Kac, Carus Mathematical Monographs, Number 12, Chapter 3, Sections 1 and 2, pages 36-41. 
is not absolutely integrable on .
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