It is shown that the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if
Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry–Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.
Let us consider the class
The first upper bounds for the constant
In 1956 C.-G. Esseen [
Consequently,
Since then, a number of upper bounds for
The present paper is devoted to estimation of
In 2007 C. Hipp and L. Mattner published an analytical proof of the inequality
In 2009 the second and third authors of the present paper have suggested the compound method in which a refinement of C.L.T. for i.i.d. Bernoulli random variables was used along with direct calculations [
In 2015 we obtained the bound
Meanwhile, in 2016 J. Schulz [
Let
Note that for fixed
Note also that we can vary the parameter
Lemma By the same method that is used to prove inequality ( An alternative approach, using Poisson approximation, is proposed in the preprint [ An alternative bound is found in the domain Next, the distance
Define
It was proved in [
Some words about bound (
It should be noted here that, according to the method, the quantity
Denote by symbol
The counting algorithm is a triple loop: a loop with respect to the parameter
With the growth of
It follows from [
Dependence of computer time on
computer time: | 3 min | 2 hrs + 5 min | 4 hrs + 50 min | 7 hrs |
Calculations were carried out on the supercomputer Blue Gene/P of the Computational Mathematics and Cybernetics Faculty of Lomonosov Moscow State University. After some changes in the algorithm, the calculations for
The program is written in C+MPI and registered [
Let
Denote
The next statement follows from Theorem It follows from Theorem
It follows from (
The structure of the paper is as follows. The proof of Theorem
In Section
In the third subsection, we give, in particular, the proof of Lemma
We need the following statement, which we give without proof.
Define
Taking into account the difference in the notations, we obtain the statement of Lemma
Further, we will use the following notations:
Using the equalities
Since for
Applying Lemma
Denote
It was proved in [
Let us find another bound for
By the inversion formula for integer random variables,
Since
We have
It is shown in [
Similarly to the proof of Lemma
The lemma follows from the equalities:
Denote
Since
Taking into account the equality
Let
Let
Now let for a point
First, consider the case
Now let
Denote
Note that It is obvious that Taking into account that Inequality ( It follows from the definition of
First we formulate Theorem 1.1 from [
Denote
Graphs of the functions (from top to down):
Recall that
Since
In order to verify the plausibility of the previous numerical result, we estimate the function
Separate the proof of (
Step 1. Note that for
Step 2. We have
Step 3. Let us write up
Thus, for
Step 4. Now consider the function
Let us introduce the following notations:
Some values of
Since
Proceed to the derivation of the values of
Similarly, with more efforts only, we get
Consequently,
1. One can observe from the previous proof that
2. With increasing
The following bound for
Corollary C allows to obtain the same estimate for
First we recall Uspensky’s estimate, published by him in 1937 in [
Uspensky’s result can be formulated in the following form.
A lot of works are devoted to generalizations and refinements of (
In 2005 K. Neammanee [
It follows from (
We may consider
Denote In 2014 V. Senatov obtained non-uniform estimates of the approximation accuracy in the central limit theorem, and, in particular, generalized Uspensky’s result (
Before proving Theorem
By [
If instead of [
Let us indicate such
It follows from Corollary
We thank the following colleagues from Lomonosov Moscow State University for providing the opportunity to use supercomputer Blue Gene/P: V. Yu. Korolev, Head of the Department of Mathematical Statistics of the Faculty of Computational Mathematics and Cybernetics, Professor, I. G. Shevtsova, Assistant Professor of the same Department, A. V. Gulyaev, Deputy Dean of the same Faculty, and S. V. Korobkov, the Data Center administrator.
We also thank our colleagues from Computing Center FEB RAS for the opportunity to use the Center for the Collective Use “Data Center FEB RAS”.
We also would like to thank reviewers for useful comments.