Markov, Andrei Andreevich (1856–1922)

Much of what is known regarding Andrei Andreevich Markov comes from two sources. The first is a short biography provided by Ahiezer and Volkobyskii, which was written in Russian 25 years after Markov's death. The second resource is an obituary written by Vladimir A. Stekloff (or Steklov, 1864–1927) and Ya. V. Uspenskii.

Markov was born in Ryazan, Russia. He had two sons from two different marriages, both of whom were mathematicians, but especially noted was the first son, Andrei, Jr. (1903–1979). Markov completed his undergraduate work in 1878 and his master's degree in 1880 as a student of A. N. Korkin (d. 1908) and Egor Ivanovich Zolotarev (1847–1878). That same year, Markov accepted a lecturer's position in mathematics at St. Petersburg University. Four years later, he [Page 568]defended his doctoral dissertation, “On Certain Applications of Algebraic Continuous Functions,” under the tutelage of Pafnuty Lvovich Chebyshev (or Tchebichef, 1821–1894).

Markov rose through the professorial ranks from Adjunct (1886), to Extraordinary (1890), and to Ordinary Academician (1896) of the Imperial Academy of Science of St. Petersburg, which was later renamed the Russian Academy of Sciences after the revolution of 1917. Two of his most well-known students were Georgy F. Voronoy (or Voronoi, 1868–1908) and Stanislaw Zaremba (1863–1942). In 1905, Markov retired from the university with the title of Distinguished Professor, but continued lecturing until his death.

Markov worked on analysis, approximation theory, converging series, continuous fractions, integrals, interpolation, number theory, limits, and probability theory. Building on the work of Chebyshev, Markov made major advances in the methods of moments in probability theory. Two of his primary pedagogical works were “Calculus of Finite Differences” and “Calculus of Probabilities.”

Markov is best known for his work on extensions from the law of large numbers that led to the development of Markov chains. It is a set of finite or discrete states (e.g., on vs. off) and an associated matrix that determines the probability of moving from one state to another. The primary feature, called the Markov property, is that the future state is determined by a random process based on the present state, but independent from all previous states, meaning that it has no memory.

A Markov process may be discrete or continuous, such as displacement over continuous time. A Markov process of the nth order means that both the memory and future probability state of the process are fully articulated by n elements. A Markov field pertains to multidimensional space.

A simple example of a Markov chain is a random walk, such as Brownian motion. The Markov chain is the progenitor and special case of stochastic processes, and led to the application of Liouville's theorem (Joseph Liouville, 1809–1882) to the ergodic hypothesis.