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Additional info for Mathematical aspects of mixing times in Markov chains
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N)) denote the diagonal matrix with entries drawn from π(·). The matrix M = √ √ −1 ( π) P( π) is a symmetric matrix because M (x, y) = π(x)P(x, y) π(y)P(y, x) π(x) P(x, y) = = = M (y, x) . π(y) π(x)π(y) π(x)π(y) It follows from the spectral theorem that since P is similar to a symmetric real matrix then it has a real valued eigenbasis. In this eigenbasis, suppose v is a left eigenvector w a right eigenvector, with corresponding eigenvalues λv and λw . Then, λv v w = (vP)w = v(Pw) = λw v w .
N . 4. Let E {x} then ˆ n πS (y) , Pn (x, y) = E n where πS (y) = on set S by π. 1S (y)π(y) π(S) denotes the probability distribution induced Proof. ˆ n )({x}, y) = E ˆ n πS (y) Pn (x, y) = (Pn Λ)({x}, y) = (ΛK n The final equality is because Λ(S, y) = πS (y). 2. 1) that if a distance dist(µ, π) is convex in µ then the worst initial distribution is a point mass. Given the preceding lemma it is easy to show a distance bound for all such convex distances. 5. Consider a finite Markov chain with stationary distribution π.
8. For a non-empty subset S ⊂ Ω the first Dirichlet eigenvalue on S is given by E(f, f ) λ1 (S) = inf Var(f ) f ∈c+ 0 (S) where c+ 0 (S) = {f ≥ 0 : supp(f ) ⊂ S} is the set of non-negative functions supported on S. The spectral profile Λ : [π∗ , ∞) → R is given by Λ(r) = inf π∗ ≤π(S)≤r λ1 (S). The spectral profile is a natural extension of spectral gap λ, and we will now see that it can be used to improve on the basic bound E(f, f ) ≥ λVar(f ) used earlier. 2. 9. For every non-constant function f : Ω → R+ , E(f, f ) ≥ 1 Λ 2 4(Ef )2 Var f Var(f ) .