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\noindent
{\bf General limit theorems for sums of independent random variables
and infinitely divisible distributions.}

\medskip\noindent
Here we discuss the question when the appropriately normalized partial sums 
of independent random variables or more generally the sums of the 
random variables in a triangular array have a limit distribution, 
and also describe the limit. This work consists of three parts that can be
found by clicking to the appropriate part below.

\medskip

\href{http://www.renyi.hu/~major/probability/divisib1.pdf}{\Rd The first 
part of the work.}

\href{http://www.renyi.hu/~major/probability/divisib2.pdf}{\Rd The second 
part of the work.}

\href{http://www.renyi.hu/~major/probability/divisib3.pdf}{\Rd The third 
part of the work.}

\medskip
For the sake of security I also send the internet address of these files.
You can use them if you cannot find these files by clicking to the 
appropriate place. They are:

The first part: 
http://www.renyi.hu/$\,\tilde{}\,$major/probability/divisib1.pdf 

The second part: 
http://www.renyi.hu/$\,\tilde{}\,$major/probability/divisib2.pdf 

The third part: 
http://www.renyi.hu/$\,\tilde{}\,$major/probability/divisib3.pdf 


\medskip
\href{http://www.renyi.hu/~major/probability/divisib1.pdf}{\Rd The first part}
is of introductory character. Here we introduce the most important notions, 
formulate the questions we are interested in, and recall some classical 
results. We show how the so-called infinitely distributed random variables, 
the random variables whose distributions are the natural (and right) 
candidates for the limit distribution in these limit theorems, can be 
constructed as the (regularized) sums of the elements of a Poisson process. 
We also show that infinitely divisible distributions can be characterized 
by means of the L\'evy--Hinchin formula. At the end  we also construct 
infinitely divisible processes with nice trajectories. 
\href{http://www.renyi.hu/~major/probability/divisib1.pdf}{\Rd The first part}
also has an Appendix which contains some useful results like a simple 
construction of a Poisson process and limit theorems with Poissonian limit.
 
\medskip 
\href{http://www.renyi.hu/~major/probability/divisib2.pdf}{\Rd In the second 
part}
we deal with the question when the normalized partial sums of independent 
random variables, or more generally the sums of the random variables in
the same row of a triangular array converge in distribution.
We present a result which gives a necessary and sufficient condition
for the existence of a limit distribution if the sequence of random
variables or the triangular array satisfies the so-called uniform smallness
condition, and also describe the limit distribution in this case. It turns 
out that the limit is always an infinitely divisible distribution. The hard 
part of the problem is to show that the sufficient condition given for the
existence of a limit distribution is at the same time a necessary
condition. We discuss the content of this condition in more detail and
also show how the most important classical limit theorems can be
obtained as special cases of the result discussed in this work.
 
We also try to explain the main ideas of the proof. An important step
in it is the introduction of the so-called associated distributions
of the distribution functions of the summands and the proof of a 
statement that says that the convergence of independent random 
variables with these associated distributions is closely related to 
the original limit problem. The associate distributions are infinitely 
divisible. So to understand limit theorems for sums of independent 
random variables it is useful to study the special problem when the 
sums of independent random variables with infinitely divisible 
distributions have a limit.

\medskip
\href{http://www.renyi.hu/~major/probability/divisib3.pdf}{\Rd In the third 
part}
first we prove the sufficiency part of the main result in the second part 
with a different method. We make a good coupling of the independent random
variables for whose sums we want to prove a limit theorem with
independent random variables which have the so-called associated distributions 
to the distributions of the original random variables. This coupling shows
that the sums of the original random variables and the sums of the
random variables with these associated distributions have the same
limit behaviour. This proof helps us to understand better the picture
behind the limit theorem we discuss. Its method also enables us
to prove a functional limit theorem version of this result. 

There is a functional central limit theorem in probability theory 
which says that the conditions of the central limit theorem also imply 
a stronger result called the functional central limit theorem. Our 
functional limit theorem states that in the case of general (not 
necessarily central) limit theorems for sums of independent random 
variables or triangular arrays a similar result holds. If the random 
sums converge to an infinitely divisible distribution, then under 
some not too restrictive and natural additional conditions such a 
sequence of random broken lines can be constructed from our random 
variables in a natural way which converges weakly to the infinitely 
divisible process corresponding to the infinitely divisible random 
variable appearing in the limit theorem. This is a result that is 
generally not discussed in the literature.
 

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