scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (2) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n3/2, in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (02) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (02) ◽  
pp. 526-542
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (3) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

Functional limit theorems for critical processes with immigration

2007 ◽  
Vol 39 (4) ◽  
pp. 1054-1069 ◽  
Author(s):  
I. Rahimov
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Asymptotically Normal ◽  
Covariance Functions ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator ◽  
Mean And Variance ◽  
The Mean

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (03) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n 3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

scholarly journals Functional limit theorems for critical processes with immigration

2007 ◽  
Vol 39 (04) ◽  
pp. 1054-1069 ◽  
Author(s):  
I. Rahimov
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Asymptotically Normal ◽  
Covariance Functions ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator ◽  
Mean And Variance ◽  
The Mean

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

ON ASYMPTOTIC RELATIONS FOR THE CRITICAL GALTON – WATSON PROCESS

2021 ◽  
pp. 5-16
Author(s):  
K.E. Kudratov ◽  
◽  
Y.M. Khusanbaev ◽  
Keyword(s):  
Single Particle ◽  
Branching Process ◽  
Critical Case ◽  
Random Number ◽  
Branching Processes ◽  
Factorial Moment ◽  
Subcritical Case ◽  
Watson Process ◽  
Third Moment ◽  
Number Of Particles

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

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