Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

The Apparitions of the Virgin Mary in Medjugorje: The convergence of Croation nationalism on her apparitions

This article concerns the alleged apparitions of the Virgin Mary in one of the most popular, 'active' apparitional sites in the world: Medjugorje in Bosnia and Herzegovina. The connection between nationalist discourse and apparitions has often been observed and noted in the literature on nationalism; however, the examples of this connection are scattered in the literature and the question why the apparitional phenomenon so easily lends itself to co-option into nationalist discourse has never been addressed. This article explores this question by showing that what binds the two phenomena together is the idea of 'chosenness' and 'specialness', which in turn can be theoretically linked to discussions about national election in the literature on nationalism. This article illustrates the convergence of nationalist and apparitional discourses by drawing on a selected number of examples of how the apparitions in Medjugorje have been appropriated by Croatian nationalist discourse.

Income Ranking and convergence with physical and human capital and income equality

This paper presents an overlapping generations model with physical and human capital and income inequality. It shows that inequality impedes output growth by directly harming capital accumulation and indirectly raising the ratio of physical to human capital. The convergence speed of output growth equals the lower of the convergence speeds of the relative capital ratio and inequality, and varies with initial states. Among economies with the same balanced growth rate but different initial income levels, the ranking of income can switch in favor of those starting from low inequality and a low ratio of physical to human capital, particularly if the growth rate converges slowly. (C) 2004 Elsevier B.V. All rights reserved.

Convergence analysis of a discrete-time single-unit gradient ICA algorithm

We revisit the one-unit gradient ICA algorithm derived from the kurtosis function. By carefully studying properties of the stationary points of the discrete-time one-unit gradient ICA algorithm, with suitable condition on the learning rate, convergence can be proved. The condition on the learning rate helps alleviate the guesswork that accompanies the problem of choosing suitable learning rate in practical computation. These results may be useful to extract independent source signals on-line.

Searching for convergence in phylogenetic Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) is a methodology that is gaining widespread use in the phylogenetics community and is central to phylogenetic software packages such as MrBayes. An important issue for users of MCMC methods is how to select appropriate values for adjustable parameters such as the length of the Markov chain or chains, the sampling density, the proposal mechanism, and, if Metropolis-coupled MCMC is being used, the number of heated chains and their temperatures. Although some parameter settings have been examined in detail in the literature, others are frequently chosen with more regard to computational time or personal experience with other data sets. Such choices may lead to inadequate sampling of tree space or an inefficient use of computational resources. We performed a detailed study of convergence and mixing for 70 randomly selected, putatively orthologous protein sets with different sizes and taxonomic compositions. Replicated runs from multiple random starting points permit a more rigorous assessment of convergence, and we developed two novel statistics, delta and epsilon, for this purpose. Although likelihood values invariably stabilized quickly, adequate sampling of the posterior distribution of tree topologies took considerably longer. Our results suggest that multimodality is common for data sets with 30 or more taxa and that this results in slow convergence and mixing. However, we also found that the pragmatic approach of combining data from several short, replicated runs into a metachain to estimate bipartition posterior probabilities provided good approximations, and that such estimates were no worse in approximating a reference posterior distribution than those obtained using a single long run of the same length as the metachain. Precision appears to be best when heated Markov chains have low temperatures, whereas chains with high temperatures appear to sample trees with high posterior probabilities only rarely. [Bayesian phylogenetic inference; heating parameter; Markov chain Monte Carlo; replicated chains.]