The proof of the central limit theorem and Fourier
analysis I.
Back to the series
of problems in Probability Theory
This series of problems contains a detailed discussion of the
central limit theorem which is one of the most important results of
probability theory. The discussion is based on the Fourier analysis,
and I gave a much more detailed explanation about the relation of
Fourier analysis and limit theorem problems in probability theory
than it is usually done in the probability literature. I tried to
show that the application of the Fourier analysis in the study of
limit theorems is natural, and the basic methods and ideas of
Fourier analysis yield a considerable help in the proofs.
This series of problem also contains the result which explains how
the convegence of the Fourier transforms (characteristic functions)
of probability measures imply the convergence of these measures.
A (considerably shorter) continuation will contain results about the
speed of convergence in limit theorems and certain expansions for the
distribution of normalized partial sums of indepentent random variables.
These results are based on the fact that a good asymptotic formula for
the Fourier transform of probability distributions also yield a good
asymptotic formula for the original distributions.
This series of problem contains 49 problems together with their
solutions. Beside this, to make it self-contained, it has an Appendix
where the formula about the inverse Fourier transform and Weierstrass
second approximation theorem is proved together with some results
closely related to them. This work also contains several problems
which do not belong directly to the theory of the central limit
theorem, but I inserted them into this text, because either I hope
that they can better explain some ideas or methods of the theory or
they appear as natural questions when we want to understand better
the content of certain results.
This series of problems starts with the proof of the local limit
theorems by means of a standard formula expressing the coefficients
of a Fourier series with the help of this Fourier series. I wanted
to show that this method is more powerful than one may think, and it
helps to investigate the general problem about limit theorems. Then
the notion of characteristic functions and convolution operators are
introduced and their most important properties are proved. The
continuation contains the definition of convergence, tightness and
relative compactness of sequences of probabiltity measures together
with the most important results related to these notions. After this
a fairly detailed discussion about the relation between the properties
of a measure and its Fourier transform is given. These results enabled
us to prove the most general form of the central limit theorem
together with its converse. This series of problems also contains the
multi-dimensional version of the central limit theorem. At the end
some problems and ideas are discussed which appear naturally in the
investigation of the central limit theorem. But their detailed
discussion will be made in subsequent works.
Finally, I would mention some parts of this text which may be
interesting for the reader and which cannot be found in usual text
books on probability. I gave a proof of the Stirling formula which
is a natural adaptation of the method applied in the proof of local
central limit theorems (problem 2), formulated a necessary and
sufficient condition for the tightness (or relative compactness) of
probability distributions on the real line with the help of their
characteristic functions (problem 22) and presented a more detailed
discussion about the relation between the properties of a measure and
its Fourier transform than it is done in usual works on probability
theory (problems 27--34).
83 pages
Content:
Introduction: pages 1--2
Formulation of the problems and the explanatation of the ideas behind
them: pages 3--35
Further problems and remarks: pages 36--39
Solution of the problems: pages 40--77
Appendix: pages 78--83
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