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First I formulate the problem I am speaking about.

We have a sequence of i.i.d. random variables $\xi_1,\xi_2,\dots,\xi_n$
on a space $X$ and a class of functions with some good properties
defined on the same space $X$. We define with their help the 
random variables $f(\xi_i)$ and their normalized sums $S_n(f)$
in the form as it is written down on this page.

We want to give a good estimate on the tail distribution of the supremum
of the normalized random sums $S_n(f)$ if we take the supremum for 
all functions $f$ in our class of functions.

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Page 2

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The normalized random sums $S_n(f)$ we are considering are 
asymptotically normal. Hence it is natural to consider a 
Gaussian version of our problem, where we estimate the 
supremum of a class of Gaussian random variables. This
Gaussian problem can be solved. 
%The estimate we get 
%depends on a metric denoted by $d_2$ on this page.

The result and method of proof applied in the solution of 
this Gaussian problem are also valid for non-Gaussian random 
variables if they satisfy formula~(1) on this page.

The question arises whether this observation helps us in 
solving our original problem.

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Page 3

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In our problem formula~(1) holds only under some slight 
restrictions. Our main problem is to overcome the difficulties 
caused by these restrictions. To settle this problem we 
have to impose new conditions and to find new methods.

These new conditions are formulated with the help of a notion 
called the class of functions with polynomially increasing covering 
numbers. I give its definition  in two steps on the next page.

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Page 4

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Here is this definition. I do not read it out, I only explain 
its main content. We demand in it that the class of functions 
we are working with has an arbitrarily dense subset with 
relatively few elements in the $L_1(\nu)$ norm for all 
probability measures $\nu$. The main point in this definition 
is that we demand this property for all probability measures.

\vfill\eject

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Page 5

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In former works our problem was investigated by means of a method 
called the symmetrization argument. In its application we introduce 
the randomized version of our random sums. Its definition is 
explained on this page. Some results enable us to replace the 
investigation of the supremum of our random sums by the 
investigation  of their randomized versions. But in some cases 
this method does not work well. It yields a weak estimate if 
our random variables have a very small second moment. On the 
other hand, we want to give a good estimate in the general case, 
which covers such situations, too.

The next theorem, the main result of this paper, provides a 
useful estimate to achieve this goal.

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Page 6

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Here is this result. Let me remark that in this theorem we work 
with random sums whose terms may have non-zero expectation. 
On the other hand, their absolute value is very small.

There are simple examples which show that this result is sharp. 
In particular, it gives a good lower bound on the levels~$u$ 
for which estimate~(2) holds.

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Page 7

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The above `Main theorem' is interesting for us not as 
a new inequality, but as a new tool to study our problem. Observe 
that formula~(2) holds in the Main Theorem for relatively large 
parameters~$\rho$, for parameters that are polynomially 
decreasing functions of the sample size~$n$. This enables us
to work out a new Vapnik--\v{C}ervonenkis type argument, which
yields a better estimate than earlier results. 
%proved by means of the symmetrization argument.

This better estimate is proved in my paper ``Sharp tail distribution
estimates for the supremum of sums of i.i.d. random variables''.

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Page 8

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Here I could give only a very short explanation of this paper. 
I wrote a more detailed version which can be found on my 
homepage at the given address.

I also gave the address of another talk about the solution 
of the general problem with the help of the Main theorem.

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Here I spoke about the problem discussed in this paper and also
about earlier approaches to solve it. I tried to explain the 
novelty value of my result, and the reason why it may give 
a better estimate than previous methods.

Thank you for your attention.

\bye
