Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Abstract

A novel approach is employed and developed to derive transition probabilities for a simple time-inhomogeneous birth-death process. Algebraic probability theory and Lie algebraic treatments make it easy to treat the timeinhomogeneous cases. As a result, an expression based on the Charlier polynomials is obtained, which can be considered as an extension of a famous Karlin-KcGregor representation for a time-homogeneous birth-death process. I. INTRODUCTIONBirth-death processes have been widely used in various contexts including physics, biology, and social sciences 1-3 ; it is a continuous-time Markov chain with discrete states on non-negative integers. Not only for their applicability to modeling various phenomena, but also for their rich mathematical structures, the birth-death processes have been studied well. For example, the birth-death processes have discrete states, so that a treatment based on generating functions are useful 1 ; using the generating function approach, various quantities, including transition probabilities, can be derived. However, it has been shown that orthogonal polynomials give beautiful representations for the transition probabilities. [4][5][6][7][8][9] The expression for the transition probabilities, so-called Karlin-McGregor spectral representation, is based on a sequence of orthogonal polynomials and a spectral measure. Because the orthogonal polynomials have deep relationship with continued fractions, it would be natural to consider that the birth-death process could be dealt with by using the continued fractions. Actually, numerical algorithms based on the continued fractions have been proposed (for this topic, for example, see the recent paper by Crawford and Suchard. 10 ) However, most of the above discussions are basically for time-homogeneous cases, in which rate constants for the birth-death processes are time-independent. In contrast, studies for time-inhomogeneous cases are not enough. While the generating function approach have been applied to time-inhomogeneous birth-death processes, 3,11 it has not been known even whether transition probabilities for the time-inhomogeneous birth-death processes can be described in terms of the orthogonal polynomials or not. The time-inhomogeneous cases are sometimes important in mathematical modeling of external influences. In addition, in a practical sense, concise expressions for the transition probabilities are demanded; for example, in time-series data analysis for bioinformatics, rapid evaluation of the transition probabilities is needed. While we can use various Monte Carlo simulations in order to deal with the time-inhomogeneous cases, it is important to try to find concise expressions and easy calculations for the transition probabilities.In the present paper, we show that it is possible to describe transition probabilities in terms of orthogonal polynomials at least in a simple time-inhomogeneous birth-death process. The birth-death process has only a state-independent birth rate and linearly-state-dependent death rate. For a time...

Did the ever dead outnumber the living and when? A birth-and-death approach

“…A particular solvable case. It turns out that in several case there is an explicit solution to (12), illustrating the general conclusions just discussed. Let us give an example.…”

“…Remark: We note from the above differential equation (12) and the expression of (α, β, γ) in the transition matrix that, as functions of ζ…”

“…for some α i (t) obeying a Wei-Norman type non-linear differential equation [8] and L i Pauli matrices. For recent further developments, see [12].…”

“…Let ρ > 0 be a fixed number. When λ d (t) = ρλ b (t) , (12) can be solved explicitly while observing…”

Computational methods for birth‐death processes

“…Beyond these simple results related to the interpretation of (11), the formalism developed by Karlin and McGregor (1957b) makes possible deep analytic insight into the behavior of general BDPs, including recurrence times and first passage times. Notably, a similar spectral representation for the transition probabilities of time-inhomogeneous linear BDPs has been derived recently (Ohkubo, 2014).…”