On associated polynomials and decay rates for birth-death processes

We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two measures, and their moments. As an application we analyse the relation between two decay rates connected with a birth-death process. \u

Associated polynomials and birth-death processes

We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials, with a view to applications in the setting of birth-death processes. In particular, we relate the supports of the two measures, and their moments of positive and negative orders. Our results indicate how the prevalence of recurrence or $\alpha$-recurrence in a birth-death process can be recognized from certain properties of an associated measure. \u

Connectivity of circulant digraphs

An explicit expression is derived for the connectivity of circulant digraphs

On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

A sufficient condition is obtained for a discrete-time birth-death process to possess the strong ratio limit property, directly in terms of the one-step transition probabilities of the process. The condition encompasses all previously known sufficient conditions

The indeterminate rate problem for birth-death processes

A birth-death process is completely determined by its set of rates if and only if this set satisfies a certain condition C, say. If for a set of rates R the condition C is not fulfilled, then the problem arises of characterizing all birth-death processes which have rate set R (the indeterminate rate problem associated with R). We show that the characterization may be effected by means of the decay parameter, and we determine the set of possible values for the decay parameter in terms of JR. A fundamental role in our analysis is played by a duality concept for rate sets, which, if the pertinent rate sets satisfy C, obviously leads to a duality concept for birth-death processes. The latter can be stated in a form which suggests the possibility of extension in the context of indeterminate rate problems. This, however, is shown to be only partially true

Asymptotic period of an aperiodic Markov chain

We introduce the concept of asymptotic period for an irreducible and aperiodic, discrete-time Markov chain X on a countable state space, and develop the theory leading to its formal definition. The asymptotic period of X equals one - its period - if X is recurrent, but may be larger than one if X is transient; X is asymptotically aperiodic if its asymptotic period equals one. Some sufficient conditions for asymptotic aperiodicity are presented. The asymptotic period of a birth-death process on the nonnegative integers is studied in detail and shown to be equal to 1, 2 or infinity. Criteria for the occurrence of each value in terms of the 1-step transition probabilities are established.Comment: 19 page

On the α-classification of birth-death and quasi-birth-death processes

In several recent papers criteria for the α-classification of birth-death and quasi-birth-death processes have been proposed. In this paper the relations between the various criteria are brought to light

Weighted sums of orthogonal polynomials related to birth-death processes with killing

We consider sequences of orthogonal polynomials arising in the analysis of birth-death processes with killing. Motivated by problems in this stochastic setting we discuss criteria for convergence of certain weighted sums of the polynomials