2,095 research outputs found
Quasi-stationary distributions for birth-death processes with killing
The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state (killing) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains largely intact as long as killing is possible from only finitely many states. In particular, the existence of a quasi-stationary distribution is ensured in this case if absorption is certain and the state probabilities tend to zero exponentially fast
Birth-death processes with killing: orthogonal polynomials and quasi-stationary distributions
The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state ({\em killing}) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains intact as long as killing is possible from only finitely many states, but breaks down otherwise
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Spectral properties of unbounded Jacobi matrices with almost monotonic weights
We present an unified framework to identify spectra of Jacobi matrices. We
give applications to long-standing conjecture of Chihara concerning one-quarter
class of orthogonal polynomials, to the conjecture posed by Roehner and Valent
concerning continuous spectra of generators of birth and death processes and to
spectral properties of operators studied by Janas, Moszy\'nski and Pedersen.Comment: 15 page
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
On the local time of random walks associated with Gegenbauer polynomials
The local time of random walks associated with Gegenbauer polynomials
is studied in the recurrent case: $\alpha\in\
[-\frac{1}{2},0]\alpha\bbN$.Comment: 12 page
Extinction probability in a birth-death process with killing
We study birth-death processes on the non-negative integers where is an irreducible class and an absorbing state, with the additional feature that a transition to state may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence as of the probability of absorption at time , and relate it to the common rate of convergence of the transition probabilities which do not involve state . Finally, we derive upper and lower bounds for the probability of absorption at time by applying a technique which involves the logarithmic norm of an appropriately defined operator. \u
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
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