# General limit theorems for sums of independent random variables and infinitely divisible distributions.

Back to the series of problems in Probability Theory.

In this text the problem about possible limit distributions of normalized sums of independent random variables or more generally about limit distribution of sums of random variables in a row from a triangular array (whose elements in the same row are independent) is discussed. There is a remarkable result, --- not really well known even among probabilists --- which gives a complete answer to the question what kind of limit theorems may appear. The only restriction we have to impose is a natural condition called the uniform smallness condition. We give a complete proof of this result and try to explain the ideas behind it. Some interesting applications of this result are also presented. This text differs from the other works on this homepage. It discusses the problem under consideration not in the form of a series of problems. It consists of three parts which can be read independently from each other.

The first part is of introductory character. It contains the natural formulation of the problems we discuss and its relation to some standard problems of probability theory. It is shown how infinitely divisible random variables and stochastic processes, --- the natural candidates for the limit in limit theorems for sums of independent random variables, --- can be constructed in a ``visible'' way by means of Poisson fields and what relation this construction has to a classical result of probability theory, to the Lévy--Hinchin formula. Moreover, this construction supplies a version of infinitely divisible processes with nice trajectories. The idea to study infinitely divisible random variables by means of Poisson processes goes back to Lévy and Ito.

The first part also contains in its Appendix a simple construction of Poisson fields and a limit theorem with Poissonian limit distribution.

The second part contains the necessary and sufficient condition for the existence of a limit distribution for sums of random variables in fixed rows of a triangular array of random variables if the elements in a row of this triangular array are independent, and they satisfy the so-called uniform smallness condition. Also the limit distribution in this limit theorem is described. A detailed proof is given together with the explanation of the ideas behind it. We also present examples which show how some classical results of the probability theory like the necessary and sufficient condition of the central limit theorem or the Lévy--Hinchin formula follows from this result. (The study of stable laws and their domain of attraction is postponed to a later work, because a complete proof of this result also applies some result about slowly varying functions, a subject not discussed here.)

The third part contains the functional limit theorem version of the result proven in the second part. The necessary and sufficient condition for the existence of a limit distribution for the sums of random variables in fixed rows of a triangular array can be expressed by means of certain canonical measures on the real line which are simple transforms of the distribution functions of the random variables we consider. The necessary and sufficient condition for the existence of a limit distribution is that these canonical measures converge to a limit canonical measure. The limit canonical measure also determines the limit distribution.

A natural modification of these notions can be introduced, and it helps us to give the condition of the functional limit theorem. In the functional limit theorem we want that not only the sums of all elements of fixed rows in a triangular array converge in distribution, but we also demand that the distributions of certain random broken lines made from the partial sums of the random variables in fixed rows have a limit in an appropriate function space. To formulate the condition of the functional limit theorem it is useful to introduce canonical measures on the strip of the plane consisting of the points with second coordinate in the interval [0,1] and to define the convergence of such canonical measures. If not only the original but also these new canonical measures on the above strip converge then also the functional limit theorem holds, and the limit process can be described explicitly. A detailed formulation of this result is given in the main text.

In the investigation of the second part we apply the method of characteristic functions i.e. of the Fourier analysis which is based on the observation that the convergence of distribution functions can be well characterized by the convergence of their Fourier transform. On the other hand, the investigation of the third part is based on probabilistic arguments. We exploit that if a sequence of random variables converges in distribution to a probability law, then small perturbations of these random variables also converge, and they have the same limit. In such a way we can prove the functional limit theorem by means of a good coupling argument.

