Markovketju

Markovketju is the Finnish term for a Markov chain, a mathematical model describing a sequence of random states in which the probability of each next state depends only on the current state (the Markov property). Markov chains can be defined in discrete time with a countable or finite state space and a transition matrix, or in continuous time by a generator matrix. They are a special case of stochastic processes.

Formally, a discrete-time Markov chain is a sequence {Xn} of random variables on a state space S such that P(Xn+1 = j | Xn = i, Xn−1, …) = P(Xn+1 = j | Xn = i) for all states i, j in S. Transition probabilities are often arranged in a matrix P where Pij denotes the probability of moving from state i to state j in one step. Long-run behavior is characterized by concepts like communicating classes, irreducibility, periodicity, recurrence and transience.

A central concept is the stationary (or invariant) distribution π satisfying πP = π, which, when unique and reached from many starting states, describes the long-term proportions of time spent in each state. Ergodicity ensures convergence to this distribution regardless of initial state.

Markovketjut are widely used in fields such as physics, biology, economics, queueing theory, and computer science. They underpin algorithms like Markov Chain Monte Carlo for sampling complex distributions. The class of models is named after Andrey Markov, who introduced the chains in the early 20th century.