Tomorrow depends only on today—history drops out
A Markov chain is a random sequence whose next move depends only on the present state, not the full past. That “memoryless” structure powers weather sketches, text generators, and Markov chain Monte Carlo sampling across Bayesian statistics, finance, and physics.
In probability and statistics a Markov chain—or Markov process—describes events whose transition probabilities depend solely on the state just attained. Discrete-time chains (DTMCs) step on a countable clock; continuous-time chains (CTMCs) evolve in continuous time. They are named for the Russian mathematician Andrey Markov. The Markov property says that, conditional on the present, future and past are independent: forecasts that use only today’s state are as good as those that know the entire history.
Usage of the label varies. Many writers reserve “Markov chain” for discrete time, while others allow continuous time with a countable state space. Applications often use finite or countably infinite states because analysis is cleaner. The process is specified by a state space, a matrix of transition probabilities, and an initial distribution. Exact future states cannot be predicted, but statistical properties can. The classic “drunkard’s walk” on the integers moves +1 or −1 with equal chance; from five, the moves to four and six each have probability one half, independent of how five was reached.
Markov’s first paper on the topic appeared in 1906. He sought an extension of independent sequences after disagreeing with Pavel Nekrasov, who claimed independence was needed for the weak law of large numbers; Markov showed averages along a chain can still converge to a fixed vector under suitable conditions. He later analyzed vowel patterns in Pushkin’s Eugene Onegin and proved a central limit theorem for such chains. Continuous-time relatives include the Poisson process, known earlier, and the Wiener process of Brownian motion. Henri Poincaré studied finite-group chains for card shuffling in 1912; Andrey Kolmogorov’s 1931 work developed much early continuous-time theory, linked to Louis Bachelier’s 1900 market model and Norbert Wiener’s Brownian studies. Today Markov chain Monte Carlo methods sample complex distributions in biology, chemistry, economics, finance, information theory, physics, and speech processing.
Source: Markov chain
