The asymptotic behavior of a divergent linear birth and death process

1976 ◽  
Vol 8 (2) ◽  
pp. 315-338 ◽  
Author(s):  
Martha J. Siegel
Keyword(s):  
Asymptotic Behavior ◽  
High Probability ◽  
Arrival Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Time Zero ◽  
First Arrival ◽  
First Arrival Time

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

The asymptotic behavior of a divergent linear birth and death process

1976 ◽  
Vol 8 (02) ◽  
pp. 315-338 ◽  
Author(s):  
Martha J. Siegel
Keyword(s):  
Asymptotic Behavior ◽  
High Probability ◽  
Arrival Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Time Zero ◽  
First Arrival ◽  
First Arrival Time

We consider a birth and death process with Q-matrix of rates q m,m + 1 = mβ, q m,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 &lt; δ &lt; β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m &lt; n and m is large that the first-arrival time is close to log n/m with high probability.

First arrival time picking for microseismic data based on DWSW algorithm

2018 ◽  
Vol 22 (4) ◽  
pp. 833-840 ◽  
Author(s):  
Yue Li ◽  
Yue Wang ◽  
Hongbo Lin ◽  
Tie Zhong
Keyword(s):  
Arrival Time ◽  
Microseismic Data ◽  
First Arrival ◽  
First Arrival Time ◽  
Arrival Time Picking

scholarly journals ANALYSES OF THE AGE OF GENES AND THE FIRST ARRIVAL TIMES IN A FINITE POPULATION

Genetics ◽  
1983 ◽  
Vol 105 (4) ◽  
pp. 1041-1059
Author(s):  
Takeo Maruyama ◽  
Paul A Fuerst
Keyword(s):  
Arrival Time ◽  
Mutant Gene ◽  
Intermediate Frequency ◽  
Lag Phase ◽  
Reversal Theory ◽  
Infinite Allele Model ◽  
First Arrival ◽  
Mean And Variance ◽  
The Mean ◽  
First Arrival Time

ABSTRACT The age of a mutant gene is studied using the infinite allele model in which every mutant is new and selectively neutral. Based on a time reversal theory of Markov processes, we develop a method of mathematical analysis that is considerably simpler for calculating the various statistics of the age than previous methods. Formulas for the mean and variance and for the distribution of age are presented together with some examples of relevance to cases in natural populations.—Theoretical studies of the first arrival time of an allele to a specified frequency, given an initially monomorphic condition of the locus, are presented. It is shown that, beginning with an allele that has frequency p = 1 or an allele with frequency p = 1/2N, there is an initial lag phase in which there is virtually no chance of an allele with a specified intermediate frequency appearing in the population. The distribution of the first arrival time is also presented. The distribution shows several characteristics that are not immediately obvious from a consideration of only the mean and variance of first arrival time. Especially noteworthy is the existence of a very long tail to the distribution. We have also studied the distribution of the age of an allele in the population. Again, the distribution of this measure is shown to be more informative for several questions than are the mean and variance alone.

Application of Stieltjes theory for S-fractions to birth and death processes

1983 ◽  
Vol 15 (03) ◽  
pp. 507-530 ◽  
Author(s):  
G. Bordes ◽  
B. Roehner
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Special Class ◽  
Jacobi Matrix ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Death Process ◽  
Nearest Neighbour ◽  
Birth And Death Process ◽  
Birth And Death Processes

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix. To this aim we use some results of Stieltjes theory for S-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete. The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.

The asymptotic behaviour of a divergent linear birth and death process

1978 ◽  
Vol 15 (1) ◽  
pp. 187-191 ◽  
Author(s):  
John Haigh
Keyword(s):  
Asymptotic Behaviour ◽  
Death Rate ◽  
High Probability ◽  
Birth Rate ◽  
Mean Value ◽  
Death Process ◽  
Applied Probability ◽  
Birth And Death Process ◽  
Initial Size

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

Sign in / Sign up

Export Citation Format

Share Document