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Page 1

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First I describe the problem of this talk.

We define a set of random sums $S_n(f)$ with the help of a sequence 
of i.i.d. random variables and a class of functions with some 
good properties in the way as it is written down on this page.
These random sums are indexed by the elements of our class of 
functions. 

For all positive numbers $v$ we want to give a good bound
on the probability that the supremum of our random 
sums $S_n(f)$ is larger than this number. We take the 
supremum for all functions $f$ in our class of functions.

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Page 2

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By some classical results there is a concentration point $v_0$
with the following property. The above probability 
is almost one if $v$ is smaller than $v_0$, while it begins to 
decrease fast for $v$ greater than $v_0$. We want to find a 
good estimate on this concentration point. 

Our problem has a natural Gaussian version about the estimation 
of the supremum of Gaussian random variables. This Gaussian
problem is solved. Under some restrictions a similar estimate 
holds in our problem too. Our goal is to find a general 
solution which also holds without these restrictions. 

To formulate these results I recall the notion of classes of 
functions with polynomially increasing covering numbers. 
%I give its definition  in two steps on the next page.

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Page 3

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Here is this definition. I do not read it out. I only remark
that this notion is a natural version  of  
Vapnik--\v{C}ervonenkis classes if we are working with a
class of functions and not with a class of sets.

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Page 4

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First I show a simple example that may help to understand 
the behaviour of the concentration point $v_0$.

In this example we take i.i.d. random variables with uniform 
distribution on the unit interval $[0,1]$. To define our class
of functions we fix a number $\sigma$, and first we consider the 
indicator functions of disjoint subintervals of the interval 
$[0,1]$ with length $\sigma$. We take as many of them as it is 
possible. Then we define our class of functions by taking the 
normalized versions of these  indicator functions. In such a way 
we get a model that satisfies  our conditions. We want to give 
a good lower bound on the concentration point $v_0$ in this model.
%It depends on the parameter $\sigma$.

%\vfill\eject

Page 5

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The result on this page gives a good estimate on $v_0$. The 
number hat $u(\sigma)$ is a good lower bound for it. 
I explain the content of this result on the next page.

\vfill\eject

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Page 6

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In this result we considered three different cases. In case three
we have a large parameter $\sigma$, and we get such an estimate for
hat $u(\sigma)$ as in the Gaussian case. In case two $\sigma$ is 
smaller. In this case a Poissonian and not a Gaussian approximation 
gives the right choice for hat $u(\sigma)$. Finally, in case one 
$\sigma$ is very small, and a trivial consideration gives the 
right bound on hat $u(\sigma)$.

Next I formulate the main result of this paper in a Theorem and in
its Extension. Their right formulation could be found by means of
the above example. 

%They give an estimate suggested by the above 
%Example and some classical results in probability theory.

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Page 7

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The theorem is formulated here. Three different cases appear in it,
similarly to the result about our Example. The result in cases one
and two is satisfactory, but to get a complete picture in case three we
need an additional result. This is formulated in the Extension of the
Theorem on the next page.

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Page 8

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I discuss the content of these results. In cases one and two 
the theorem itself gives a good estimate. It holds for numbers 
$v$ suggested by our example. It is a non-Gaussian bound 
suggested by Bennett's inequality. In case three the theorem 
and its extension together give a good estimate in the right domain. 
%where we can expect to have it. 
In the extension we get a good 
Gaussian estimate if the number $v$ is not too large, while in 
the remaining cases the original theorem gives the right bound.

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Page 9

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Here I gave a short overview of this paper. A more detailed version
can be found on my homepage at the given address. It also explains
the main ideas of the proof.



This longer version explains the ideas behind the proof. The 
most important of them is the application of a new 
Vapnik--\v{C}ervonenkis type argument to control the irregular 
contributions to the supremum we investigate. Actually, the
main subject of this reseach was to find this new method.

Thank you for your attention.


%Finally, I would say some words on the proof. It is based on my
%paper `Sharp estimate on the supremum of a class of sums of small 
%i.i.d. random variables.' With its help I could reduce the problem 
%to the estimation of the supremum of finitely many partial 
%sums $S_n(f)$. This reduced problem can be solved by  
%classical methods.

%The main point of the proof was this reduction. We have 
%separated in it the regular and irregular contributions to the 
%supremum  we wanted to bound. 
%%There are good methods to control the regular contributions. 
%The hard part of the problem was to give a good bound on
%the irregular contributions. This demanded new methods, and 
%the main subject of this research was to find them.

%\medskip
%I wrote a more detailed version of this talk, and its address you
%can find on this page. 


\bye
