On some inequalities connected to the strong law of large numbers

The note that is published here was written by A. N. Kolmogorov more than 40 years ago (the author dates it April 1962). At that time I was a graduate student of Andrei Nikolaevich [Kolmogorov] and was investigating the possibility of generalizing and amplifying the well-known Chebyshev inequality. Some of my results I presented at Kolmogorov's seminar at the Moscow State University. In 1962, on Bernstein's request, I wrote a commentary on his work “On some modifications of the Chebyshev inequality” (this paper can be found in the fourth volume of Bernstein's collected works). Andrei Nikolaevich approached my work (which was published by the MphTI press the same year) with interest. During my next visit to Komarovka, when I was reporting the work in progress on my doctoral thesis, he gave me a short manuscript and asked me to read it. The idea of that note was close to the one contained in my published work and in my commentary on Bernstein's work. After some time I asked Andrei Nickolaevich whether he planed prepare that note for publication. He said that he did not plan to do so in the near future. The manuscript remained in my archive.
This note does not contain a fundamental result, as was usually the case with most of Kolmogorov's other works. However, it presents an opportunity of learning what this great scientist thought and worked on during a fruitful period of his career. In this respect, this short work is certainly valuable to both experts and new practitioners in the field of probability theory.
The manuscript contains formula (9), where $\varepsilon>0$ and $p\in (0,1)$ and $\mu_n$ denotes the number of successes in $n$ Bernoulli trials with the probability of success $p$. For $p=\frac12$ the manuscript contains the more precise inequality (8).
It should be mentioned that similar inequalities appear in several textbooks published at later dates: A. A. Borovkov, Probability Theory, Gordon and Breach, United Kingdom, 1998; A. N. Shiryaev, Probability, Springer-Verlag, Berlin, New York, 1984.
In particular, the first textbook gives the following inequalities:
$$ P(\mu_n-np\ge \varepsilon)\le e^{-nH(p+\varepsilon/n)},\qquad P(\mu_n-np\le -\varepsilon)\le e^{-nH(p-\varepsilon/n)}, $$
where $H$ is some function that satisfies ${H(x)\ge 2x^2}$. The second textbook gives the inequality $P(|\mu_n/n-p|\ge\varepsilon)$ $\le 2e^{-2n\varepsilon^2}$. A more careful analysis of Kolmogorov's technique may lead to the inequality $P(\sup_{k\ge n}|\mu_k/k-p|\ge \varepsilon)\le 2e^{-2n\varepsilon^2}$ for all $p\in (0,1)$.
Finally, I express gratitude to V. Yu. Korolev and V. M. Kruglov for their help in preparing this manuscript for publication.
V. M. Zolotarev