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We add the equation $\pi(a)+\pi(b)=1$ because $\pi$ is a valid probability distribution, meaning its elements must sum to 1. This provides the second independent equation needed to solve the underdetermined system created by the redundant balance equations.

Conditions: $\pi$ is a probability distribution over the states.; The state space has exactly two states, a and b.

We add the equation $\pi(a)+\pi(b)=1$ because $\pi$ is a valid probability distribution, meaning its elements must sum to 1. This provides the second independent equation needed to solve the underdetermined system created by the redundant balance equations.

Conditions: $\pi$ is a probability distribution over the states.; The state space has exactly two states, a and b.

The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.

The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.

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.

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.

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.

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.

Pure random sampling is inefficient in high dimensions because the volume of the space grows exponentially, making it nearly impossible to randomly hit regions of high probability. MCMC solves this by using dependent sampling via a Markov chain, which guides exploration and naturally concentrates samples in regions of high probability.

Conditions: The target distribution is defined over a high-dimensional space.; As dimensionality increases, random sampling becomes inefficient.; MCMC uses a Markov chain designed so that its stationary distribution is the target distribution.

Pure random sampling is inefficient in high dimensions because the volume of the space grows exponentially, making it nearly impossible to randomly hit regions of high probability. MCMC solves this by using dependent sampling via a Markov chain, which guides exploration and naturally concentrates samples in regions of high probability.

Conditions: The target distribution is defined over a high-dimensional space.; As dimensionality increases, random sampling becomes inefficient.; MCMC uses a Markov chain designed so that its stationary distribution is the target distribution.