Andrey Andreyevich Markov (14 June 1856–20 July 1922) was a Russian mathematician best known for his contributions to probability theory, particularly the study of dependent random variables. His investigations established the mathematical framework now called a Markov chain and helped develop the theory of stochastic processes. He also contributed to number theory and mathematical analysis, continuing research traditions associated with his teacher, Pafnuty Chebyshev. (mathshistory.st-andrews.ac.uk)
Education and academic career
Markov was born in Ryazan, Russia, and attended secondary school in Saint Petersburg. He entered Saint Petersburg University in 1874 and graduated in 1878, receiving a gold medal for an essay on integrating differential equations using continued fractions. His teachers included Chebyshev, Aleksandr Korkin, and Yegor Zolotarev. (mathshistory.st-andrews.ac.uk)
He received his master’s degree in 1880 for a dissertation on binary quadratic forms with positive determinant and his doctorate in 1884 for work on applications of continued fractions. These degrees belonged to the contemporary Russian academic system and should not be equated mechanically with modern degrees bearing the same names. He became extraordinary professor in 1886 and ordinary professor in 1893, and a full member of the Russian Academy of Sciences in 1896. Although formally retired in 1905, he continued teaching. His son, also named Andrey Andreyevich Markov, became a mathematician. (mathshistory.st-andrews.ac.uk)
Dependence and probability limit laws
A central problem in Markov’s probability research concerned the assumptions needed for statistical regularities to emerge. The law of large numbers describes conditions under which averages approach their expected values as the number of observations increases. Independence supplies an important sufficient condition, but Markov investigated how comparable results could hold when observations influence one another. (cs.cornell.edu)
His work beginning in 1906 introduced systematic treatment of chains of dependent trials. The significance was not that every dependent sequence obeys a law of large numbers, but that independence is not indispensable: appropriately restricted dependence can also permit limiting regularity. He additionally worked on the central limit theorem, which concerns the limiting distribution of suitably normalized sums rather than simply the convergence of averages. These investigations extended the Chebyshev school’s program of rigorous probability theory. (ebsco.com)
Markov chains and the Markov property
In modern notation, a discrete-time process (X_0,X_1,\ldots) has the Markov property when, for histories having positive probability,
[ \Pr(X_{n+1}=j\mid X_n=i,X_{n-1},\ldots,X_0)
\Pr(X_{n+1}=j\mid X_n=i). ]
Thus, once the present state is known, earlier states provide no additional information about the next state’s probability distribution. This is a statement about conditional probabilities, not a claim that successive observations are independent or that the next state is predetermined. (math.dartmouth.edu)
For a time-homogeneous chain with finitely many states, transition probabilities are represented by a transition matrix (P), with entries (p_{ij}\geq0) and each row summing to one. The entry (p_{ij}) gives the probability of moving from state (i) to state (j). Powers (P^n), calculated through matrix multiplication, describe transitions over (n) steps. (math.dartmouth.edu)
A stationary distribution is a probability row vector (\pi) satisfying (\pi P=\pi). Starting the chain with this distribution leaves its state probabilities unchanged over time. Stationarity does not itself guarantee that every initial distribution converges to it; convergence requires additional conditions. These modern formulations make precise the distinction between a chain’s local transition rules and its long-term behavior. (mpaldridge.github.io)
Statistical analysis of literary text
In 1913 Markov published a statistical investigation of Alexander Pushkin’s Eugene Onegin. He examined a sequence of 20,000 letters, excluding the Russian hard and soft signs, and classified the remaining letters as vowels or consonants. The sample comprised the first chapter and sixteen stanzas of the second. He counted individual categories and adjacent combinations to examine dependence between successive letters. (math.purdue.edu)
The investigation connected abstract probability with literary text through a deliberately simplified representation. Instead of modeling meanings, words, or grammatical structure, Markov reduced the sequence to two categories. His analysis illustrated how observed frequencies could reveal dependence and how a chain could approximate that dependence. It did not establish that the full structure of written language is exactly a first-order Markov process. The distinction between an empirical approximation and an exact stochastic model is essential to interpreting the study. (math.purdue.edu)
Inequalities and other mathematical work
The probabilistic Markov inequality states that, for a nonnegative random variable (X) with finite expected value and any (a>0),
[ \Pr(X\geq a)\leq \frac{\mathbb E[X]}{a}. ]
It bounds a tail probability without requiring a particular probability distribution. Applied to the squared deviation from the mean, it yields Chebyshev’s inequality, whose bound depends on the variance. Neither inequality requires independent observations. (ocw.mit.edu)
A different result, belonging to approximation theory, concerns derivatives of a polynomial. In 1889 Markov proved that a polynomial (p) of degree at most (n) satisfies
[ \max_{-1\leq x\leq1}|p'(x)| \leq n^2\max_{-1\leq x\leq1}|p(x)|. ]
His brother Vladimir extended the result to higher derivatives in 1892. These results are known as the Markov brothers’ inequality and are distinct from the probabilistic inequality sharing the family name. (cambridge.org)
Publications and later development
Markov’s Calculus of Probabilities first appeared in 1900. Its successive editions presented probability through rigorous arguments and incorporated developments in his research; its contents included expectation, limit theorems, and least-squares methods. His teaching and publications helped transmit Chebyshev’s approach to probability to later mathematicians. (old.maa.org)
The theory originating in Markov’s dependent sequences subsequently expanded beyond his original discrete setting. Andrey Kolmogorov developed foundations for a general theory of Markov processes during the 1930s, placing these investigations within a broader mathematical treatment of random evolution. (mathshistory.st-andrews.ac.uk)