Lecture
This lecture covers the proof of the convergence rate theorem for an ergodic Markov chain with a finite state space, transition matrix, and stationary distribution satisfying the detailed balance equation. The theorem provides insights into the eigenvalues of the transition matrix and their implications. The lecture also revisits the concept of detailed balance, defining the symmetric matrix Q and exploring its spectral properties. The presentation progresses to discuss orthonormal vectors in the context of the symmetric matrix Q, emphasizing their significance in the theorem's proof. The lecture concludes with a detailed proof of the convergence rate theorem, showcasing the mathematical intricacies involved in establishing the convergence rate of the Markov chain.