MARKOV, ANDREY ANDREYEVICH

born June 14, 1856, Ryazan, Russia died July 20, 1922, Petrograd [now St. Petersburg] Russian mathematician who helped to develop the theory of stochastic processes, especially those called Markov chains. Based on the study of the probability of mutually dependent events, his work has been developed and widely applied in the biological and social sciences. As a child Markov had health problems and used crutches until he was 10 years old. In 1874 he enrolled at the University of St. Petersburg (now St. Petersburg State University), where he earned a bachelor's degree (1878), a master's degree (1880), and a doctorate (1884). In 1883, as his station in life improved, he married his childhood sweetheart, the daughter of the owner of the estate that his father managed. Markov became a professor at St. Petersburg in 1886 and a member of the Russian Academy of Sciences in 1896. Although he officially retired in 1905, he continued to teach probability courses at the university almost to his deathbed. While his early work was devoted to number theory and analysis, after 1900 he was chiefly occupied with probability theory. As early as 1812 the French mathematician Pierre-Simon Laplace had formulated the first central limit theorem, which states, roughly speaking, that probabilities for almost all independent and identically distributed random variables converge rapidly (with sample size) to the area under an exponential function. (See also normal distribution.) In 1887 Markov's teacher Pafnuty Chebyshev outlined a proof of a generalized central limit theorem. Using a different approach, Chebyshev's student Aleksandr Lyapunov proved the theorem under weakened hypotheses in 1901. Eight years later Markov succeeded in proving the general result rigorously using Chebyshev's method. While working on this problem, he extended both the law of large numbers (which states that the observed distribution approaches the expected distribution with increasing sample size) and the central limit theorem to certain sequences of dependent random variables forming special classes of what are now known as Markov chains. These chains of random variables have found numerous applications in modern physics. One of the earliest applications was to describe Brownian motion, the small, random fluctuations or jiggling of small particles in suspension. Another frequent application is to the study of fluctuations in stock prices, generally referred to as random walks.