A unified stability theory for classical and monotone Markov chains

In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

LetPbe the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vectorπT, and let(n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n+ 1) x (n+ 1) northwest corner truncation ofParbitrarily, with invariant probability vector(n)πT. We derive computableV-norm bounds on the error betweenπTand(n)πTin terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 asntends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

A functional form for a particular coefficient of ergodicity

Seneta, in a recent paper, presented a general treatment of the concept of ‘coefficient of ergodicity', τ (P), for a finite stochastic matrix P. In this paper, a functional form for τ ∞(P) in terms of the attributes of P is determined. It is shown that, by increasing the dimension of P, τ ∞(Ρ) can assume any large value. In view of this, τ ∞ will be practically useful only in the case τ ∞(P) ≦ τ 1(P), where τ 1(P) is the well-known Dobrushin or delta coefficient.

A functional form for a particular coefficient of ergodicity

Seneta, in a recent paper, presented a general treatment of the concept of ‘coefficient of ergodicity', τ (P), for a finite stochastic matrix P. In this paper, a functional form for τ∞(P) in terms of the attributes of P is determined. It is shown that, by increasing the dimension of P, τ∞(Ρ) can assume any large value. In view of this, τ∞ will be practically useful only in the case τ∞(P) ≦ τ1(P), where τ1(P) is the well-known Dobrushin or delta coefficient.

Stability and exponential convergence of continuous-time Markov chains

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

Stability and approximation of invariant measures of Markov chains in random environments

We consider finite-state Markov chains driven by stationary ergodic invertible processes representing random environments. Our main result is that the invariant measures of Markov chains in random environments (MCREs) are stable under a wide variety of perturbations. We prove stability in the sense of convergence in probability of the invariant measure of the perturbed MCRE to the original invariant measure. We also develop a new numerical scheme to construct rigorous approximations of the invariant measures, which converge in probability as the resolution of the scheme increases. This numerical approach is illustrated with an example of a random walk in a random environment.

Static and dynamic expression of life history traits in the Northern Fulmar (Fulmarus glacialis)

Understanding the static and dynamic expression of life history traits is a prerequisite for the development of a causal theory of the evolution of aging and of life histories. We analyzed the statics and dynamics of reproduction and survival in a wild population of the Northern Fulmar, Fulmarus glacialis (Procellaridae). Survival rate is most influenced by year as compared to age and cohort. When temporal variation is ignored, survival rate increases slowly with age and then declines more rapidly at late ages. Survival rate contingent upon reproductive "stratum" (producing an egg, hatching an egg, fledging a hatchling) also exhibits this pattern. Survival and reproduction have a positive static association in that survival rate increases as the apparent energy allocated to reproduction increases (as indexed by stratum). There is a broad distribution of realized lifetime reproductive success, which could be due to "fixed" heterogeneity, with some individuals always having low survival and reproduction and others always having high survival and reproduction, or be due to "dynamic" heterogeneity, with all individuals having the same expected reproductive and survival rates. Analysis of stochastic stratum dynamics indicates that individuals do not remain long in any given stratum and suggest that the variation among individuals with respect to lifetime reproductive success is due to dynamic heterogeneity. The probability of producing an egg increases with age for both sexes, whereas the probability of producing a fledgling initially declines with age and then increases. These results underscore the necessity of understanding the static and dynamic expression of demographic traits when making a causal claim about their evolution.

Coefficients of ergodicity with respect to vector norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l 1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.