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

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

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

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

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

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

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

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

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.