scholarly journals Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yanhua Zhang ◽  
Zhenlong Gao
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Branching Process ◽  
Time Process ◽  
Rate Of Growth ◽  
Continuous Time Process ◽  
Watson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.

scholarly journals Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5803-5808 ◽  
Author(s):  
Zhenlong Gao ◽  
Lina Qiu
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Continuous Time ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Time Process ◽  
Continuous Time Process ◽  
Watson Process

Consider a continuous time process {Yt=ZNt, t ? 0}, where {Zn} is a supercritical Galton-Watson process and {Nt} is a renewal process which is independent of {Zn}. Firstly, we study the asymptotic properties of the harmonic moments E(Y-rt) of order r > 0 as t ? ?. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.

Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean

1979 ◽  
Vol 11 (1) ◽  
pp. 31-62 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Transition Probabilities ◽  
Time Process ◽  
Slowly Varying Function ◽  
Continuous Time Process ◽  
Population Sizes ◽  
Rate Of Decay ◽  
Watson Process ◽  
Reversed Time

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean

1979 ◽  
Vol 11 (01) ◽  
pp. 31-62 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Transition Probabilities ◽  
Time Process ◽  
Slowly Varying Function ◽  
Continuous Time Process ◽  
Population Sizes ◽  
Rate Of Decay ◽  
Watson Process ◽  
Reversed Time

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process. The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

Stochastic ordering of random processes with an imbedded point process

1991 ◽  
Vol 28 (3) ◽  
pp. 553-567 ◽  
Author(s):  
François Baccelli
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Random Sequence ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Time Process ◽  
Continuous Time Process ◽  
Partial Orderings ◽  
Stationary Probabilities

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

1976 ◽  
Vol 8 (2) ◽  
pp. 278-295 ◽  
Author(s):  
Michael Sze
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Continuous Process ◽  
Regularly Varying Functions ◽  
Original Process ◽  
Additional Information ◽  
Embedding Technique ◽  
Extinction Probabilities ◽  
Watson Process ◽  
The Given

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

scholarly journals Mixed Spectra for Stable Signals from Discrete Observations

2021 ◽  
Vol 12 (05) ◽  
pp. 21-44
Author(s):  
Rachid Sabre
Keyword(s):  
Spectral Density ◽  
Continuous Time ◽  
Continuous Measurement ◽  
Spectral Measurement ◽  
Polynomial Kernel ◽  
Time Process ◽  
Discrete Observations ◽  
Continuous Part ◽  
Continuous Time Process ◽  
Discrete Measurement

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

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