VI.88 Andrei Nikolaevich Kolmogorov

b. Tambov, Russia, 1903; d. Moscow, 1987

Analysis; probability; statistics; algorithms; turbulence

Kolmogorov was one of the greatest mathematicians of the twentieth century. His work was distinguished both by its great depth and power, and by its breadth: he made important contributions to several different areas. He is most famous for his work on probability theory, and is widely regarded as having been the greatest probabilist ever.

Kolmogorov’s mother, Mariya Yakovlena Kolmogorova, died in childbirth; his father, Nikolai Matveevich Kataev, an agronomist, worked for the Ministry of Agriculture after the Revolution and died in the Denikin offensive in the Civil War in 1919. Kolmogorov was brought up by his mother’s sister Vera, whom he regarded as his mother and who lived to see her adopted son’s success.

After his childhood in Tunoshna, near Yaroslavl on the Volga, Kolmogorov became a student of mathematics at Moscow University in 1920. His teachers included Aleksandrov, Lusin, Urysohn, and Stepanov. Kolmogorov’s first work, published in 1923 when he was still nineteen, gave an example of a (Lebesgue integrable) function whose FOURIER SERIES [III.27] diverges almost everywhere. (This is in contrast to the classical theorems giving regularity conditions on a function that are sufficient for its Fourier series to converge to it.) This famous and unexpected result made him a celebrity, all the more so when in 1925 he sharpened “almost everywhere” to “everywhere.”

Kolmogorov became a postgraduate student in 1925, studying under Lusin. Also in 1925, he published his first work on probability theory, in collaboration with Alexander Yakovlevich Khinchin (Khintchine, Hincin), on the “three series theorem.” This classical result gives a necessary and sufficient condition for the convergence of a random series with independent terms, namely the convergence of three nonrandom series. The paper also contains the Kolmogorov inequality on maxima of independent sums. By the time of his doctorate in 1929, Kolmogorov had written eighteen mathematical papers: on analysis, on probability, and on intuitionist logic, an indication of his lifelong interest in the foundations of mathematics. He became a professor at Moscow University in 1931.

Also in 1931, Kolmogorov published his famous paper on analytic methods in probability theory. This deals with Markov processes in continuous time, with the state space continuous or discrete (in which case one speaks of a Markov chain). The Chapman–Kolmogorov equations, and the Kolmogorov forward and backward differential equations, date from this paper. Diffusions are also treated, developing earlier work by Bachelier.

The whole subject of modern probability theory was given a firm foundation by Kolmogorov’s epoch-making monograph Grundbegriffe der Wahrscheinlichkeitsrechnung of 1933 (later translated as Foundations of Probability Theory). Before this time, probability had lacked a rigorous mathematical foundation, and indeed some authors had believed that it was impossible to provide one. However, the relevant mathematical theory, MEASURE THEORY [III.55], had been introduced by LEBESGUE [VI.72] in 1902, in connection with his theory of the integral. Measure theory also provides a firm foundation for the mathematics of length, area, and volume. By the 1930s, the subject had been freed from its origins in Euclidean space. Kolmogorov treated probability simply as a measure of total mass 1, events as measurable sets, RANDOM VARIABLES [III.71 §4] as measurable functions, etc. The decisive technical innovation was his treatment of conditioning, which used the then-recent Radon–Nikodým theorem (whereby conditional expectations became Radon-Nikodým derivatives). The Grundbegriffe also contains two further key results. The first is the Daniell–Kolmogorov theorem, basic to the definition of a STOCHASTIC PROCESS [IV.24]. The second is Kolmogorov’s STRONG LAW OF LARGE NUMBERS [III.71 §4]. When we repeatedly toss a fair coin, we expect the observed frequency of heads to tend to the expected frequency, a half. Some restriction is needed to make precise mathematical sense out of this intuition. It was known before Kolmogorov that the qualification needed here is that convergence takes place with probability 1 (“almost surely,” or “a.s.”). Kolmogorov generalized this result from coin tossing to repeated replication of any random experiment. One needs the expected value (often called the mean) to exist, in the technical sense of measure theory. Then the average value in a sample, the sample mean, converges to the expectation, the population mean, with probability 1.

Further work by Kolmogorov on probability theory followed in the 1930s and 1940s. He worked on limit theorems, on infinite divisibility, on the Kolmogorov–Petrovskii-Piscunov equation governing the wave of advance of an advantageous gene, and on linear prediction of stationary stochastic processes. This application, which led to the “Kolmogorov-Wiener filter,” was motivated by wartime applications to fire control problems.

This last work led Kolmogorov naturally to path- breaking work on turbulence in 1941, including the Kolmogorov “two-thirds power” law. This work has been profoundly important subsequently, as the problem of understanding turbulence is a central one in fluid dynamics.

Motivated by questions of the stability of the solar system, and related DYNAMICAL SYSTEMS [IV.14], Kolmogorov published in 1954 his work on mechanics and invariant tori, work that developed into the subject of “KAM theory” (for Kolmogorov, Arnold, and Moser).

Kolmogorov’s axiomatization of probability theory can be regarded as a solution of (part of) Hilbert’s sixth problem, to put probability and mechanics onto a rigorous footing. In 1956 and 1957, Kolmogorov solved another of Hilbert’s problems, the thirteenth. His solution gave a surprising structure theorem, by which a function of many variables can be built up from functions of few variables by means of basic operations. He showed that a continuous function of any number of real variables may be built up by combining (using the operations of addition and of taking a function of a function) a finite number of functions of only three real variables. He regarded this work as his most technically difficult accomplishment.

In the 1960s, Kolmogorov turned his attention to foundational questions: in mathematics, in probability theory, and in INFORMATION THEORY [VII.06] and the theory of algorithms. He introduced the concept now called “Kolmogorov complexity.” He gave a new approach to randomness, quite different from that in his earlier work on probability theory. Here, random sequences are identified as sequences of maximal complexity. His later work was dominated by his lifelong interest in teaching, and in particular his involvement in special schools for particularly gifted pupils.

Kolmogorov’s Selected Works comprise three volumes: Mathematics and Mechanics, Probability and Statistics, and Information Theory and Algorithms.

He was widely honored, both within the Soviet Union and outside. He was married, with no children.

Further Reading

Kendall, D. G. 1990. Obituary, Andrei Nikolaevich Kolmogorov (1903-1987). Bulletin of the London Mathematical Society 22(1):31–100.

Shiryayev, A. N., ed. 2006. Selected Works of A. N. Kolmogorov. New York: Springer.

Shiryayev, A. N., and others. 2000. Kolmogorov in Perspective. History of Mathematics, volume 20. London: London Mathematical Society.

Nicholas Bingham