OAI.MarkovSuperreflexivity.hasNontrivialMarkovType_iff_hasEquivalentUCNorm

The theorem states that for a real normed space X that is complete (a Banach space), X has nontrivial Markov type if and only if X admits an equivalent uniformly convex norm. Nontrivial Markov type means there exist p>1 and K>0 such that for every finite reversible Markov chain on m states {0,…,m−1}, with stationary probability vector π, nonnegative row-stochastic transition matrix P, and detailed balance π(s)P(s,t)=π(t)P(t,s), every map f from the states to X, and every n≥1, the expected value of ‖f(Z_n)−f(Z_0)‖^p along n-step paths, started from π, is at most K^p·n times the corresponding one-step expectation of ‖f(Z_1)−f(Z_0)‖^p. Having an equivalent uniformly convex norm means there is a seminorm q on X and constants a,b>0 with a‖x‖≤q(x)≤b‖x‖ for all x, such that for every ε in (0,2] there is δ>0 with q((x+y)/2)≤1−δ whenever q(x)≤1, q(y)≤1 and q(x−y)≥ε.