What is the stationary distribution in a Markov chain and why is it important for MCMC?

A single probability distribution, such as a bell curve, describes uncertainty at one instant but cannot capture temporal dynamics. Markets evolve continuously: returns arrive sequentially, volatility changes, and extreme events cluster.

Conditions: The system being modeled changes over time.; Returns, volatility, or event clustering exhibit temporal structure.

Markov chains are used not because markets are memoryless, but because they capture persistence, regime behaviour, and long-run risk with minimal assumptions. They sit between pure randomness and full determinism, offering a balance of being simple enough to analyse yet rich enough to matter.

Conditions: Used as a modelling framework rather than as an exact description of markets.; The goal is parsimonious representation of persistence and regime changes.

An absorbing state is a state that, once entered, cannot be left. It is characterized by a transition probability of 1 to itself.

Conditions: Transition probability from the state to itself must be exactly 1.; The system eventually enters an absorbing state if reachable.

The burn-in period consists of the initial samples generated by an MCMC algorithm before it has converged to its stationary distribution. These samples must be discarded because they are influenced by the arbitrary starting point and do not accurately represent the target distribution.

Conditions: Applies to MCMC sampling.; Requires discarding initial samples before using the rest for inference.; The chain starts from an arbitrary point.

Volatility clustering arises from simple state dependence in a Markov chain. It does not require heavy-tailed return distributions or complex non-linear dynamics.

Conditions: The underlying process follows a Markov chain with distinct states.; Returns within each state follow the same distribution.