Checking Irreducibility and Aperiodicity in Markov Chains
Consider the following Markov chain with states \(S = \{1, 2, 3\}\) and transition matrix: \(P = \begin{pmatrix}0.3 & 0.7 & 0 \\ 0.4 & 0.2 & 0.4 \\ 0 & 0.5 & 0.5 \end{pmatrix}\). Determine whether the Markov chain is irreducible and aperiodic. Justify your answer.
Empirical Analysis of Stationary Distributions in Markov Chains
Consider an empirical transition matrix obtained from a financial market study, representing the probabilities of transitioning between different market states over time. The matrix is given as: \(P = \begin{pmatrix} 0.7 & 0.2 & 0.1 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.5 \end{pmatrix}\) where the states are defined as: – […]
Part b
Discuss the assumptions required for a unique stationary distribution to exist in a Markov chain. Explain whether the empirical data provided supports these assumptions.
Unique Stationary Distribution in Regular Markov Chains
Consider a regular Markov chain with a transition matrix \(T\). Explain why there exists a unique stationary distribution vector \(p\) satisfying \(pT = p\). In your explanation, detail how the properties of irreducibility and aperiodicity (which together define regularity) guarantee this uniqueness.
Uniqueness of the Steady-State Eigenvector
Consider a stochastic matrix representing a Markov chain. Explain the conditions under which the steady-state eigenvector corresponding to \(\lambda=1\) is unique (up to scaling). In your answer, discuss the roles of irreducibility and aperiodicity in ensuring this uniqueness.