Cutoff for Markov Chains: Some Examples and Applications

Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

“…After the cut-off instant, they converge exponentially fast (see Definition 2). This phenomenon has been recognized in various families of Markov processes or chains; see [6], [15], and [18], for example. In [17], Ycart showed that a cut-off occurs for samples of Markov chains on a finite set, both in discrete and continuous time.…”

Central limit theorem for hitting times of functionals of Markov jump processes

“…It is easy to prove that the counting process N n is again a continuous time Markov chain (see [21] p. 296 for the Jackson network interpretation of N n ). Since the X i 's are independent, the distribution of the random vector N n (t) is multinomial with parameters n and p(t).…”

“…Markov chains appear in different domains of applications. They can represent locations of a set of customers in a closed Jackson network [21], the current state of a parallel MCMC algorithm [6], or a description of a coherent system in reliability [3]. We shall use here the reliability interpretation as a support for intuition, although the applicability of our results is not restricted to that area.…”

Cut-off and exit from metastability: two sides of the same coin

“…This phenomenon has been studied using Fourier techniques, spectral properties of transition matrices, and functional analysis techniques (see [9,10,15,16] for reviews). In this work we adopt the probabilistic approach advocated in [11] (see also [19] and the relevant chapter in [20]). …”

“…Dans ce travail nous utilisons une approche probabiliste développée dans [11] (voir aussi [19] et le chapitre correspondant dans [20]). La sortie de la métastabilité implique aussi une convergence abrupte, mais celle-ci a lieu à des instants très aléa-toires.…”