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 LévyHinchin 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 socalled 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 LévyHinchin 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.
