Anatoliy Volodymyrovych Skorokhod (Ukrainian: Анатолій Володимирович Скороход; September 10, 1930 – January 3, 2011) was a Soviet and Ukrainian mathematician known for his fundamental contributions to probability theory, the theory of stochastic processes, and mathematical statistics.
Biography
Skorokhod was born in Nikopol, Dnipropetrovsk region, Ukraine, to parents who were both school teachers. He graduated from secondary school with a gold medal in 1948 and entered the Mechanics and Mathematics Faculty of Kyiv University. He graduated in 1953 and undertook postgraduate research at Moscow State University under the supervision of Eugene Dynkin, submitting his Candidate dissertation (equivalent to a Ph.D.), Limit Theorems for Random Processes, in 1956.
He returned to Kyiv University as a lecturer from 1956 to 1964. In 1962, he was awarded his Doctorate for the thesis Stochastic Differential Equations and Limit Theorems for Random Processes. In 1964, he became Head of the Department of the Theory of Random Processes at the Institute of Mathematics of the Ukrainian Academy of Sciences. From 1993 until his death, he was a professor at Michigan State University in the United States.
He was elected a Corresponding Member of the Ukrainian Academy of Sciences in 1967, an Academician in 1985, and a Member of the American Academy of Arts and Sciences in 2000. He received the Ukrainian State Prize in Science and Technology in 1982 and again in 2003.
Mathematical Contributions
Skorokhod authored over 450 scientific works, including more than 40 monographs and books. Many fundamental concepts in probability theory bear his name:
- Skorokhod space – The space of càdlàg functions (right-continuous with left limits) equipped with the Skorokhod topology, which makes it a separable metric space (Polish space). This space is essential for the study of stochastic processes with discontinuities.
- Skorokhod topology – A topology (specifically the J₁ topology) on the space of functions without discontinuities of the second kind, introduced to prove limit theorems for stochastic processes.
- Skorokhod's representation theorem – A result stating that if a sequence of probability measures converges weakly, then there exists a single probability space on which random variables with those distributions can be constructed to converge almost surely.
- Skorokhod's embedding theorem – A method for representing sums of independent random variables as values of a Wiener process at random stopping times.
- Skorokhod integral – An extension of the Itô stochastic integral to non-adapted (anticipating) integrands, also known as the extended stochastic integral.
- Skorokhod problem – The problem of solving a stochastic differential equation with a reflecting boundary condition, leading to the concept of reflected Brownian motion.
- Skorokhod's method of a single probability space – A direct probabilistic method for proving limit theorems.
His research spanned limit theorems for random processes, stochastic differential equations, probability distributions in infinite-dimensional spaces, Markov processes, and the asymptotic behavior of dynamical systems under random perturbations.
Selected Works
Skorokhod co-authored a comprehensive three-volume treatise The Theory of Stochastic Processes with Iosif Gikhman, as well as Introduction to the Theory of Random Processes and Stochastic Differential Equations. His other notable monographs include Studies in the Theory of Random Processes (1961), Integration in Hilbert Space (1974), Random Processes with Independent Increments (1991), and Random Perturbation Methods with Applications in Science and Engineering (2002, with F. C. Hoppensteadt and H. Salehi).
Legacy
Skorokhod supervised more than 50 graduate students, 17 of whom became Doctors of Sciences. He played a decisive role in forming the Kyiv school of probability theory. During the Soviet era, he faced political sanctions for his participation in a 1968 letter defending constitutional rights, which resulted in a fifteen-year ban on attending international conferences. This led to the belief among some foreign scientists that "Skorokhod" was a collective pseudonym for a group of mathematicians, analogous to the French group "Bourbaki."
