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

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