## On Associated Polynomials and Decay Rates for Birth-Death Processes (2001)

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@MISC{Doorn01onassociated,

author = {Erik A. Van Doorn},

title = { On Associated Polynomials and Decay Rates for Birth-Death Processes },

year = {2001}

}

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### Citations

590 | An Introduction to Orthogonal Polynomials - Chihara - 1978 |

298 | The Laplace Transform - Widder - 1941 |

176 | The Problem of Moments - Shohat, Tamarkin - 1943 |

62 |
An Introduction to Orthogonal Polynomials (Gordon and
- Chihara
- 1978
(Show Context)
Citation Context ...s satisfy the recurrence relation ˜Pn(x) =(x − cn+1) ˜ Pn−1(x) − dn+1 ˜ Pn−2(x), n > 1, ˜P0(x) =1, ˜ P1(x) =x − c2. (1.2) Associated polynomials are sometimes called numerator polynomials (in Chihara =-=[5]-=- for example) because they are the numerators of the convergents of certain continued fractions (the denominators of which are the polynomials Pn(x)). Clearly, also the associated polynomials { ˜ Pn(x... |

62 |
Recherches sur les fractions continues
- Stieltjes
- 1918
(Show Context)
Citation Context ...ique, since Shohat and Sherman [20] have shown that the Hmp’s corresponding to {Pn(x)} and { ˜ Pn(x)} are determined simultaneously. 1Associated polynomials appear already in Stieltjes’ seminal work =-=[22]-=- and have been studied by many authors since then (see, for example, Sherman [19], Chihara [5], Belmehdi [1], Van Assche [23], Peherstorfer [16], Ronveaux and Van Assche [17], and the references cited... |

47 |
The differential equations of birth-anddeath processes, and the Stieltjes moment problem
- Karlin, McGregor
- 1957
(Show Context)
Citation Context ...n, n ∈ S} and death rates {µn, n ∈ S}, all strictly positive except µ0 ≥ 0. When µ0 > 0 the process may evanesce by escaping from S, via state 0, to an ignored absorbing state −1. Karlin and McGregor =-=[11]-=- have shown that the transition probabilities pij(t) ≡ Pr{X(t) =j | X(0) = i}, t ≥ 0, i,j ∈ S, can be represented as pij(t) =πj ∫ ∞ Here {πn} are constants given by 0 e −xt Qi(x)Qj(x)ψ(dx), t ≥ 0, i,j... |

41 |
The classification of birth and death processes
- KARLIN, MCGREGOR
- 1957
(Show Context)
Citation Context ...omials {Pn(x)} of the previous sections. For by letting cn+1 = λn + µn and dn+2 = λnµn+1, n ≥ 0, (4.3) we readily see that Pn(x) =(−1) n λ0λ1 ...λn−1Qn(x), n > 0. It is known (see Karlin and McGregor =-=[12]-=- and Chihara [6]) that the Hmp associated with the polynomials {Qn(x)} is determined if and only if ∞∑ ( n∑ (λkπk) −1 ) 2 = ∞. (4.4) πn+1 n=0 k=0 So, assuming (4.4) to prevail, the spectral measure ψ ... |

14 |
Weak convergence of conditioned processes on a countable state space
- Jacka, Roberts
- 1995
(Show Context)
Citation Context ...(0) = i}, i ∈ S. (4.9) Using [14, Equation (3.7)] it is not difficult to show that this limit exists and αi,−1 = α = ξ1, i ∈ S. (4.10) This result follows also from Theorem 3.2.2 of Jacka and Roberts =-=[10]-=-, which gives, in a more general setting, sufficient conditions for equality of the decay rates αi,−1 and α. Next assuming µ0 = 0, the question arises what the relation between α and the right-hand si... |

13 |
Assche, Orthogonal polynomials, associated polynomials and functions of the second kind
- Van
- 1991
(Show Context)
Citation Context ...eously. 1Associated polynomials appear already in Stieltjes’ seminal work [22] and have been studied by many authors since then (see, for example, Sherman [19], Chihara [5], Belmehdi [1], Van Assche =-=[23]-=-, Peherstorfer [16], Ronveaux and Van Assche [17], and the references cited there). The theme of this paper is related to that of [17], namely, we shall be interested in the problem of obtaining infor... |

12 |
On the associated orthogonal polynomials
- Belmehdi
- 1990
(Show Context)
Citation Context ...ermined simultaneously. 1Associated polynomials appear already in Stieltjes’ seminal work [22] and have been studied by many authors since then (see, for example, Sherman [19], Chihara [5], Belmehdi =-=[1]-=-, Van Assche [23], Peherstorfer [16], Ronveaux and Van Assche [17], and the references cited there). The theme of this paper is related to that of [17], namely, we shall be interested in the problem o... |

12 |
Indeterminate symmetric moment problems
- Chihara
- 1982
(Show Context)
Citation Context ... the previous sections. For by letting cn+1 = λn + µn and dn+2 = λnµn+1, n ≥ 0, (4.3) we readily see that Pn(x) =(−1) n λ0λ1 ...λn−1Qn(x), n > 0. It is known (see Karlin and McGregor [12] and Chihara =-=[6]-=-) that the Hmp associated with the polynomials {Qn(x)} is determined if and only if ∞∑ ( n∑ (λkπk) −1 ) 2 = ∞. (4.4) πn+1 n=0 k=0 So, assuming (4.4) to prevail, the spectral measure ψ of the birth-dea... |

12 | Many server queueing processes with Poisson input and exponential service times - Karlin, McGregor - 1958 |

11 | Limiting conditional distributions for birth-death processes - Kijima, Nair, et al. - 1995 |

9 | Assche, Upward extension of the Jacobi matrix for orthogonal polynomials
- Ronveaux, Van
- 1996
(Show Context)
Citation Context ...n Stieltjes’ seminal work [22] and have been studied by many authors since then (see, for example, Sherman [19], Chihara [5], Belmehdi [1], Van Assche [23], Peherstorfer [16], Ronveaux and Van Assche =-=[17]-=-, and the references cited there). The theme of this paper is related to that of [17], namely, we shall be interested in the problem of obtaining information about the measure ψ from (partial) knowled... |

9 |
On the numerators of the convergents of the Stieltjes continued fractions
- Sherman
- 1933
(Show Context)
Citation Context ...n(x)} and { ˜ Pn(x)} are determined simultaneously. 1Associated polynomials appear already in Stieltjes’ seminal work [22] and have been studied by many authors since then (see, for example, Sherman =-=[19]-=-, Chihara [5], Belmehdi [1], Van Assche [23], Peherstorfer [16], Ronveaux and Van Assche [17], and the references cited there). The theme of this paper is related to that of [17], namely, we shall be ... |

9 | Some estimates of the rate of convergence for birth and death processes - Zeifman |

8 | Markov’s theorem revisited
- Berg
- 1994
(Show Context)
Citation Context ... ∈ C\supp( ˜ ψ), respectively, which are analytic in their domains of definition. Indeed, a classical result in the theory of continued fractions (Shohat and Sherman [20], Sherman [19], see also Berg =-=[2]-=-) tells us that the two transforms are related as F (z) = 1 z − c1 − d2 ˜ . (2.7) F (z) 3So if ˜ ψ (and hence ˜ F (z)) is completely known, we can use (2.7) to find F (z), and then apply the Stieltje... |

8 | Birth-death processes and associated polynomials
- Doorn
(Show Context)
Citation Context ... mn ≡ ∫ ∞ 0 x n ψ(dx) and ˜mn ≡ ∫ ∞ 0 x n ˜ ψ(dx), n ≥ 0, and their relations. As an aside we note that moments of negative orders (and their relevance for birth-death processes) have been studied in =-=[9]-=-). We first observe that the system of equations ∫ ∞ −∞ P0(x)ψ(dx) =1, ∫ ∞ −∞ Pn(x)ψ(dx) =0, n ≥ 0, can be solved recursively for the moments mn, n =0, 1,.... In this way we find, for example, m0 =1 m... |

6 |
On the rate of convergence in birth-and-death processes
- Callaert
- 1974
(Show Context)
Citation Context ...d ergodic. We are interested in the exponential rate of convergence (or decay rate) ofpij(t) to its limit pj, thatis,inthe quantities 1 αij ≡−lim t→∞ t log |pij(t) − pj|, i,j ∈ S. (4.6) From Callaert =-=[3]-=- (see also [7]) we know that these limits exist, and that α ≡ α00 ≤ αij, i,j ∈ S, (4.7) with equality whenever µ0 > 0, and inequality subsisting for at most one value of i or j when µ0 =0. Thequantity... |

6 | Representations for the rate of convergence of birth-death processes - Doorn |

5 |
Finite perturbations of orthogonal polynomials
- Peherstorfer
- 1992
(Show Context)
Citation Context ...d polynomials appear already in Stieltjes’ seminal work [22] and have been studied by many authors since then (see, for example, Sherman [19], Chihara [5], Belmehdi [1], Van Assche [23], Peherstorfer =-=[16]-=-, Ronveaux and Van Assche [17], and the references cited there). The theme of this paper is related to that of [17], namely, we shall be interested in the problem of obtaining information about the me... |

4 |
Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space
- Martínez, Ycart
- 2001
(Show Context)
Citation Context ...nd lim y→σ− ∫ ∞ 0 dψ(x) y − x ≥ 0. (ii) If X is not ergodic, then γ ≥ α with equality subsisting if and only if ξ1 = σ. It is interesting to relate this corollary to recent work of Martínez and Ycart =-=[15]-=-. Rather than α – the decay rate of p00(t) – these authors study (in a more general setting) αtv ≡ sup{x ≥ 0: ∑ |p0j(t) − pj| = O(e −xt )ast→∞}, j∈S that is, the decay rate as t goes to infinity of th... |

4 |
On the numerators of the continued fraction λ1 x
- Shohat, Sherman
- 1932
(Show Context)
Citation Context ...the real axis. We will denote this measure by ˜ ψ and refer to it as the associated measure. Our assumption that the Hmp for {Pn(x)} is determined implies that ˜ ψ is unique, since Shohat and Sherman =-=[20]-=- have shown that the Hmp’s corresponding to {Pn(x)} and { ˜ Pn(x)} are determined simultaneously. 1Associated polynomials appear already in Stieltjes’ seminal work [22] and have been studied by many ... |