Supplementary Material of A Matrix Bound for Markov Chains and Its Application to Co occurrence Matrices A Convergence Rate of Co occurrence Matrices

If P is a regular Markov chain, then Q satisfies: 1. Q Q may have zero spectral gap. We prove the fours parts of this Claim one by one. Kronecker product has the mixed-product property. Due to symmetry, it suffices to prove one of the statements. The second inequality follows Markov inequality.