**Boris Vladimirovich Gnedenko**was born in Simbirsk but his family, who were of Ukrainian origin, moved to Kazan when he was only three years old and it was in Kazan that he received his school education. He was a brilliant school pupil. In 1927, when he was only fifteen years old, Gnedenko tried to enter the University of Saratov. However, he was too young to meet the official entry requirements and only after he kept arguing his case did Lunacharskii, the Soviet Minister of Education, personally recommend that he be allowed to enter university.

Gnedenko himself wrote of his reasons for studying mathematics at university. Most mathematicians study the subject because they develop such a deep love of the topic. Gnedenko, however, studied mathematics because [11]:-

It is worth noting at this stage that Gnedenko's choice of mathematics did not save him from having the most severe problems with the Soviet authorities later in his life, as we describe below. It is also worth commenting that, even if he did not choose mathematics through a love of the subject when he was young, he certainly showed throughout his career that he deeply loved the topic. He graduated in 1930 after only three years of university study. This was because the normal five year course was shortened to three years on account of a decision by the government. They also introduced a strange method of examining students, namely to divide them into groups and have only the group leader sit the examinations for the whole group. Gnedenko, as might be expected, was one of the group leaders who took the examinations and his whole group were awarded the high grades which he scored. He did not feel satisfaction, however, rather a feeling of humiliation at the whole procedure.... as an abstract science[it]was beyond the understanding of the Communist Party functionaries, thus remaining out of their control. He therefore selected mathematics as his life study.

From 1930 Gnedenko taught at the Textile Institute in Ivanovo. This town, east of Moscow, was a centre for the textile industry and it had figures highly in the history of mathematics with people such at Luzin and Khinchin teaching at the polytechnic there. It was during this period that Gnedenko published his first papers on probability and statistics. These resulted from problems he studied concerning the reliability of the machines used in textile manufacture and his first papers were published in 1933. He became deeply interested in probability theory after attending seminars by Kolmogorov and Khinchin.

In 1934 Gnedenko decided to resume his university studies at postgraduate level. He was awarded a scholarship which allowed him to undertake research at the Institute of Mathematics at Moscow State University. There he studied under Kolmogorov and Khinchin. Kalashnikov writes in [11] that Gnedenko :-

Gnedenko made other friendships at this time, becoming a close friend of Slutsky and other mathematicians, and his enthusiasm for his studies was not spoilt by the extremely difficult conditions [19]:-... cared for them dearly and held them in the highest affection all his life. ... Kolmogorov was a connoisseur of art, and[Gnedenko and Kolmogorov]talked at length about ancient Russian icons and architecture, poetry and history.

Although Khinchin supervised Gnedenko's studies initially, he left in 1935 to spend two years at Saratov University, and Kolmogorov then took over as Gnedenko's supervisor. In June 1937 Gnedenko was examined on his doctoral dissertation on the theory of infinitely divisible distributions. After the award of his doctorate he was appointed as an assistant researcher in the Mathematics Institute of Moscow State University.... during his first year he and eleven friends shared a room that was so bitterly cold that water left in a glass overnight froze solid.

During the summer of 1937 Gnedenko went on a hiking expedition to the Caucasus along with some fellow researchers. Although Kolmogorov did not spend as long as the others on the trip, he did take part for a while. It was a period when the researchers were able to talk about their mathematical ideas and profit greatly. However, they talked about many subjects and in the discussions on politics Gnedenko showed that he had little liking for the Soviet policies. This was to have the most severe consequences for him.

After taking up his post as a research assistant in November 1937, Gnedenko was conscripted into the Red Army on 1 December. He was sent to Bryansk but on 5 December he was arrested. He had been denounced by one of the members of the Mathematics Department who had been with him on the summer trip. Gnedenko was imprisoned with 120 other prisoners in a cell built for six people and was constantly interrogated about statements he had made on the summer trip. His interrogators demanded that he [24]:-

Without warning he was released after six months and managed to return to his parent's home. After a great deal of difficulty, with strong support from Kolmogorov and Khinchin (which of course was a brave move by these two who could have paid dearly for giving such support), he was reinstated to his post of Research Assistant in 1938. He retained a "black mark" on his record indicating that he was not to be trusted and, because of this, he was not allowed to join the Soviet army in 1941 when the German forces attacked. In other respects, however, he was able to live a normal life despite this terrifying start to his career.... confirm that Kolmogorov was the ringleader of a group of "enemies of the people" centred in the mathematics department. Though interrogated daily over a six-month period, held in grim condition, and promised his release if he cooperated, he refused to admit even the possibility of such an interpretation, knowing that there could be no hard evidence, and that the fate of all, himself included, depended on his resolution.

From 1938 Gnedenko lectured at Moscow State University. He married Natalia Konstantinova in 1939 and they had two sons. He submitted his doctoral dissertation in 1941 and was awarded the doctorate in 1942. Of course this was during World War II and for most of the war years he undertook military related research. There was a period when he was evacuated from Moscow and worked further east in the Soviet Union.

In 1945, on the recommendation of Kolmogorov, Gnedenko was elected to the Ukrainian Academy of Sciences. He left Moscow and, after a brief spell in Kiev, he became professor at Lvov University. There he met Banach and [11]:-

In 1949 Gnedenko was appointed as Head of the Physics, Mathematics and Chemistry Section of the Ukrainian Academy of Sciences in Kiev and he became Director of the Kiev Institute of Mathematics. He held these posts until 1960 when he returned to Moscow University, becoming Head of the Department of Probability Theory in 1966. He held this post for thirty years until his death.... he retained strong impressions of this meeting all his life.

Gnedenko produced a remarkable number of papers and books during his lifetime. Certainly there are over 200 items in his list of publications, but this number increases substantially if further editions of books, translations into different languages etc. are taken into account. In fact he wrote several important books which we shall say a little about.

In 1949 he published a work, jointly with Kolmogorov, *Limit Distributions for Sums of Independent Random Variables* which contains a description of much of his early research. The book is based on courses given by Gnedenko and Kolmogorov at Moscow and Lvov universities. It has three parts, the first part consisting of three introductory chapters while the second part is on general limit theorems. Included in this second part are sections on: general limit theorems for sums with independent summands; the concept of infinitely small summands; conditions necessary and sufficient that their sums have a given limiting distribution; convergence to the normal, Poisson, and unit distributions; and limit theorems for cumulative sums. The third part of the work is on summands with a common distribution function and includes discussion of principal limit theorems and convergence to the normal law.

One of Gnedenko's most famous books is *Course in the Theory of Probability* which first appeared in 1950. Written in a clear and concise manner, the book was very successful in providing a first introduction to probability and statistics. It has gone through six Russian editions and has been translated into English, German, Polish and Arabic. In 1966, along with I V Kovalenko, he published *Introduction to queuing theory*. A reviewer described the work as follows:-

In his early work Gnedenko had been interested in probability as an abstract topic. However, in his later work his interests turned more towards applications to areas such as reliability and quality control. He wrote an important textThis is an attractively written systematic exposition of the basic probabilistic methods of congestion theory. At the same time the book has the unusual merit of going far enough into explicit computations to be of practical interest to the applied operations analyst. The book is written on a mature level, but little probability theory beyond very elementary concepts or very intuitive ones is presupposed.

*Mathematical methods of reliability*in 1965 with Belyaev and Solov'yev.

Gnedenko was not only interested in research into mathematical topics, but he was also interested in the teaching of these topics. For example, with Solov'yev, he wrote the elementary work *Mathematics and reliability theory* in 1982 which aimed to give a popular account of the mathematical theory of reliability. In an elementary way the book describes the basic notions of reliability, lifetime distributions, redundant systems, renewal and maintenance theory, inclusion-exclusion principle, and the estimation of reliability.

Another work, this time aimed at secondary school pupils, was *An introduction to the speciality of mathematics* published in 1991. The book was written for pupils who love mathematics and want to become mathematicians. The topics covered in this book are interesting in themselves and also, in the present context, because they tell us a good deal about Gnedenko's approach to mathematics. The topics include: the role of mathematics in science, technology, and life; a definition of "mathematics"; the abstract nature of the subject; mathematics as the language of science; interesting problems from various mathematical sciences; mathematical models; mathematical education; the history of mathematics including an appendix on Moscow University.

Gnedenko's interest in the history of mathematics extended well beyond his text aimed at secondary school pupils. He published much on this topic (we list at least twelve articles in the References sections of the archive authored by Gnedenko) including the important *Outline of the History of Mathematics in Russia* which was not published until 1946 although he wrote it before the start of World War II. It is a fascinating book which looks at the history of mathematics in Russia in its cultural background. For example in discussing the pre-eighteenth century, Gnedenko considers the cultural influence of the medieval church and the role of the calendar. He discusses the founding of the St Petersburg Academy of Sciences, paying particular attention to the life and work of Euler. The work of many famous mathematicians is discussed in detail such as that of Lobachevsky, Bunyakovsky, Ostrogradski, Chebyshev, Markov, Lyapunov, and Kovalevskaya. Perhaps given Gnedenko's treatment in prison just before he wrote the book, it is not surprising that he makes every effort to be "politically correct" and plays down contributions by western mathematicians. For example he even manages to discuss non-euclidean geometry and Lobachevsky's contributions without even mentioning Bolyai.

To complete our account of Gnedenko we should give a feeling for his personality. For this we quote from [11] (which is itself quoted in [24]). Gnedenko's favourite topics of conversation were:-

... the inter-relationship of mathematics and its fields of application, problems of education, history, books, poetry, art, and many others. Speaking on any topic, Gnedenko did not insist on his insights although he did not hide them. He knew many interesting stories and told them with a deep sense of humour. Gnedenko enjoyed classical music, and had a large collection of records. Sometimes, when he was especially proud of some new purchase, he would propose that you listen to it. His taste in music was traditional, and he was not afraid to confess that he could not understand some modern composers. On other occasions, he would ask you to look through a new album of painting of old masters. This was, in fact, the method of communication typical of the older generation of Russian intellectuals.

**Article by:** *J J O'Connor* and *E F Robertson*

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