A Markov chain is a collection of random variables conditionally independent of the past.
In other words,
If a Markov sequence of random variates
and the sequence
A simple random walk is an example of a Markov chain.
The Season 1 episode "Man Hunt " (2005) of the television crime drama NUMB3RS
See also Conditional Probability ,
Independent Events ,
Markov
Bridge ,
Markov Order ,
Markov
Sequence ,
Monte Carlo Method ,
Random
Walk
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References Gamerman, D. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Gilks, W. R.; Richardson, S.; and Spiegelhalter,
D. J. (Eds.). Markov
Chain Monte Carlo in Practice. Grimmett,
G. and Stirzaker, D. Probability
and Random Processes, 2nd ed. Harary,
F. Graph
Theory. Kallenberg,
O. Foundations
of Modern Probability. Kemeny,
J. G. and Snell, J. L. Finite
Markov Chains. Papoulis, A.
"Brownian Movement and Markoff Processes." Ch. 15 in Probability,
Random Variables, and Stochastic Processes, 2nd ed. Stewart, W. J. Introduction
to the Numerical Solution of Markov Chains. Veritasium. "The Strange Math That Predicts (Almost)
Anything." Jul. 25, 2025. https://www.youtube.com/watch?v=KZeIEiBrT_w . Referenced
on Wolfram|Alpha Markov Chain
Cite this as:
Weisstein, Eric W. "Markov Chain." From
MathWorld https://mathworld.wolfram.com/MarkovChain.html
Subject classifications
(where the index 
runs through 0, 1, ...) having the property that, given the
present, the future is conditionally independent of the past.
takes the discrete values 
,
..., 
, then
is called a Markov chain (Papoulis 1984, p. 532).
