The eigentime identity for continuous-time ergodic Markov chains

Abstract

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑n≥1 λn-1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1 λn-1 < ∞ if and only if α > 0.

Eigentime identity for transient Markov chains

“…For an irreducible finite Markov chain, discrete or continuous time, it will always be ergodic. In [7], this identity was extended to continuous-time ergodic Markov chains on infinitely countable state space.…”

Uniform ergodicity of continuous-time controlled Markov chains: A survey and new results

“…6.1). Bounds on R and δ in Definition 2.20 for strong 1-exponential ergodicity are studied in Mao (2004Mao ( , 2006 for the particular case of reversible Markov chains.…”

“…For reversible Markov chains, lower bounds on α, stated in terms of the eigenvalues of a suitably defined operator (which is related to the infinitesimal matrix Q) are proposed in Mao (2004). If x is a birth and death process (that is, with q ij = 0 provided that |i − j | > 1) which is, in addition, strongly 1-exponentially ergodic, then it is necessarily reversible.…”

Eigentime identity for asymmetric finite Markov chains

“…The eigentime identity (1.3) is then generalized to continuous-time reversible Markov chains on countable state spaces, both in the ergodic case ( [10]) and in the transient case ( [11]). In the ergodic case, T is also connected to the strong ergodicity and the spectral property of Q in Hilbert space L 2 (π).…”

“…In the ergodic case, T is also connected to the strong ergodicity and the spectral property of Q in Hilbert space L 2 (π). For reversible Markov chains on a countable state space, it was proved in [10] that T < ∞ implies that the essential spectrum of Q in L 2 (π) is empty. For the birth and death process on Z + , T < ∞ is equivalent to the strong ergodicity.…”