##  [Markov Chain](/markov-chain-0) 

 Definition

A stochastic process that progresses through a discrete set of states in discrete time (or in a discrete-indexed sequence) with the memoryless property that the probability distribution of the next state depends only on the current state, not on the earlier history.

 

 

 

 

 

 





## Principle

Principle

The law of evolution at one step is conditional only on the present state; transitions are described by a matrix or kernel of conditional probabilities.

 

 

 

 

 





## Demonstration

Demonstration

A simple random walk on the integers where at each step the process moves +1 or −1 with fixed probabilities; its one-step transition probabilities fully determine multi-step distributions by repeated application of the transition matrix.

 

 

 

 

## Misapplication

Misapplication

Treating data with long-range dependence or explicit history dependence as a Markov chain and using single-step transition probabilities to predict multi-step behavior.

 

 

 

 

 





## Consequence

Consequence

Enables analysis using matrix powers, stationary (steady-state) distributions, mixing times and first-step analysis without tracking full histories.

 

 

 

 

## Reversal

Reversal

A history-dependent stochastic process in which future probabilities require one or more past states (e.g., higher-order or non-Markovian processes).

 

 

 

 

 





## Boundary

Boundary

Applies to processes with a well-defined state space and one-step transition probabilities; excludes processes where transitions depend on unobserved history or where state space and transition law are not defined.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Distinguished from continuous-time Markov processes: chains are discrete-indexed and use transition matrices, whereas continuous models use generators and continuous-time transition kernels.

 

 

 

 

 





## Synthesis

Synthesis

A Markov chain is a discrete-time stochastic system on a defined state set whose single-step transition law, represented by a matrix or kernel, suffices to determine all future distributions because of the memoryless property.