Spectral gap and convex concentration inequalities for birth–death processes

Spectral gap, isoperimetry and concentration on trees

“…This becomes an equality if T = N: That is Chen's variational formula of the spectral gap, re-proved by Liu and Ma [13]. The lower bound of the spectral gap in (3.3) holds even for general graphs: It is due to Diaconis and Stroock [7] for w(e) = 1/Q(e), and to Chen [2] for general length function w.…”

“…However for the example in Remark 2.5, since T = {0, 1} a particular case of birth death processes on a half line, it follows from the result in [13] that…”

“…Instead of using the coupling method and motivated by the work of Djellout and Wu [8] on 1-dimensional diffusions, Liu and Ma [13] re-obtained Chen's variational formula of λ 1 for birth-death processes by calculating explicitly the Lipschitzian norm (−L) −1 Lip of the Poisson operator (−L) −1 . Estimate of λ 1 on trees turns out to be much more difficult.…”

“…The identification of the Lipschitzian norm above turns out to be a powerful tool for concentration, see Guillin et al [11,12], Djellout and Wu [8], Liu and Ma [13], Wu [19], etc. In particular for T = N, Liu and Ma [13] identified (−L) −1…”

Bernstein type’s concentration inequalities for symmetric Markov processes

“…Birth-death processes continued. The following two lemmas are taken from Liu and Ma [35]. For any k ≥ 0, the solution of the above equation (5.3) satisfies the following relation :…”