Dean Isaacson

On strong ergodicity for nonhomogeneous continuous-time Markov chains

Let X(t) be a nonhomogeneous continuous-time Markov chain. Suppose that the intensity matrices of X(t) and some weakly or strongly ergodic Markov chain X(t) are close. Some sufficient conditions for weak and strong ergodicity of X(t) are given and estimates of the rate of convergence are proved. Queue-length for a birth and death process in the case of asymptotically proportional intensities is considered as an example.

The convergence of Cesaro averages for certain nonstationary Markov chains

If P is a stochastic matrix corresponding to a stationary, irreducible, positive persistent Markov chain of period d>1, the powers Pn will not converge as n --> [infinity]. However, the subsequences Pnd+k for k=0,1,...d-1, and hence Cesaro averages [Sigma]nk-1 Pk/n, will converge. In this paper we determine classes of nonstationary Markov chains for which the analogous subsequences and/or Cesaro averages converge and consider the rates of convergence. The results obtained are then applied to the analysis of expected average cost.

Strongly ergodic Markov chains and rates of convergence using spectral conditions

For finite Markov chains the eigenvalues of P can be used to characterize the chain and also determine the geometric rate at which Pn converges to Q in case P is ergodic. For infinite Markov chains the spectrum of P plays the analogous role. It follows from Theorem 3.1 that ||Pn-Q||[less-than-or-equals, slant]C[beta]n if and only if P is strongly ergodic. The best possible rate for [beta] is the spectral radius of P-Q which in this case is the same as sup{[lambda]: [lambda] |-> [sigma] (P), [lambda] [not equal to];1}. The question of when this best rate equals [delta](P) is considered for both discrete and continous time chains. Two characterizations of strong ergodicity are given using spectral properties of P- Q (Theorem 3.5) and spectral properties of a submatrix of P (Theorem 3.16).

A characterization of geometric ergodicity

SummaryA homogeneous Markov chain on a countable state space can be classified as ergodic, geometrically ergodic, or strongly ergodic. Ergodicity and strong ergodicity have been characterized using the δ-coefficient. In this paper the δ-coefficient is used to characterize geometric ergodicity.

On solutions to min \((X,{\text{ Y}})\mathop = \limits^d aX\) and min \((X,{\text{ Y}})\mathop = \limits^d aX\mathop = \limits^d bY\)

Suppose that the minimum of a pair of independent non-negative random variables X and y has the same distribution, up to a scale factor, as the first of the two random variables. The restricted class of possible distributions for X and Y is identified. If in addition it is required that X and Y have distributions only differing by a scale factor, it is shown under mild regularity conditions that X and Y have Weibull distributions.

On normal characterizations by the distribution of linear forms, assuming finite variance

If X1 and X2 are independent and identically distributed (i. i. d.) with finite variance, then (X1+X2)/[radical sign]2 has the same distribution as X1 if and only if X1 is normal with mean zero (Polya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where [sigma][infinity]i=1aiXi has the same distribution as X1. In particular if at least two of the ais are non-zero and X1 has finite variance, then Laha and Lukacs [8] showed that X1 is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Polya in [9]. The same technique is also used to give a characterization of the exponential distribution.

Ergodicity for countable inhomogeneous Markov chains

Abstract A notion of ergodicity is defined by analogy to homogeneous chains, and a necessary and sufficient condition for it to hold for an inhomogeneous Markov chain is given in terms of matrix products. A comparison to the situation for homogeneous chains is made. A final section discusses the better-known notion of strong ergodicity in relation to the geometric convergence rate.

Creating effective biocontainment facilities and maintenance protocols for raising specific pathogen-free, severe combined immunodeficient (SCID) pigs:

Severe combined immunodeficiency (SCID) is defined by the lack of an adaptive immune system. Mutations causing SCID are found naturally in humans, mice, horses, dogs, and recently in pigs, with the serendipitous discovery of the Iowa State University SCID pigs. As research models, SCID animals are naturally tolerant of xenotransplantation and offer valuable insight into research areas such as regenerative medicine, cancer therapy, as well as immune cell signaling mechanisms. Large-animal biomedical models, particularly pigs, are increasingly essential to advance the efficacy and safety of novel regenerative therapies on human disease. Thus, there is a need to create practical approaches to maintain hygienic severe immunocompromised porcine models for exploratory medical research. Such research often requires stable genetic lines for replication and survival of healthy SCID animals for months post-treatment. A further hurdle in the development of the ISU SCID pig as a biomedical model involved the establishment of facilities and protocols necessary to obtain clean SPF piglets from the conventional pig farm on which they were discovered. A colony of homozygous SCID boars and SPF carrier sows has been created and maintained through selective breeding, bone marrow transplants, innovative husbandry techniques, and the development of biocontainment facilities.

L 1 ergodic behavior of non-negative kernels

Some results that have been obtained in the study of strongly and weakly ergodic behavior of non-homogeneous stochastic kernels are generalized to the case of non-negative kernels. The first generalization simply involves extending the definitions of weakly and strongly ergiodic behavior to the case of non-negative kernels and using the ergodic coefficient which was first defined for stochastic kernels by Dobrushin and extended to non-negative kernels by Blum and Reichaw. It happens that this straightforward extension excludes many cases of non-negative kernels which do exhibit a types of ergodic behavior. In order to study these cases a definition ofL1 weakly and strongly ergodic behavior is given in which normalizing by constants is allowed. Sufficient conditions for these types of ergodic behavior are given.