Variance estimators in critical branching processes with non-homogeneous immigration

The asymptotic normality of conditional least squares estimators for the offspring variance in critical branching processes with non-homogeneous immigration is established, under moment assumptions on both reproduction and immigration. The proofs use martingale techniques and weak convergence results in Skorokhod spaces.Comment: Accepted for publication in Math Population Studie

Statistical inference for partially observed branching processes with immigration

Copyright © 2017 Applied Probability Trust. In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the \u27skipped\u27 version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution\u27s mean is asymptotically normal under additional assumptions

Asymptotic inference for non-supercritical partially observed branching processes

© 2017 Elsevier B.V. To estimate the offspring mean of a branching process one needs observed population sizes up to some generation. However, in applications very often not all individuals existing in the population are observed. Therefore the question about possibility of estimating the population mean based on partial observations is of interest. In existing literature this problem has been studied assuming that the process never becomes extinct, which is possible only in supercritical case. In the paper we consider it in subcritical and critical processes with a large number of initial ancestors. We prove that the Harris type ratio estimator remains consistent, if we have observations of a binomially distributed subsets of the population. To obtain the asymptotic normality of the estimator we modify the estimator using a “skipping” method. The proofs use the law of large numbers and the central limit theorem for random sums in the case when the number of terms and the terms in the sum are not independent

Magnetized Particle Motion Around Black Hole in Braneworld

We investigate the motion of a magnetized particle orbiting around a black hole in braneworld placed in asymptotically uniform magnetic field. The influence of brane parameter on effective potential of the radial motion of magnetized spinning particle around the braneworld black hole using Hamilton-Jacobi formalism is studied. It is found that circular orbits for photons and slowly moving particles may become stable near $r = 3M$. It was argued that the radii of the innermost stable circular orbits are sensitive on the change of brane parameter. Similar discussion without Weil parameter has been considered by de Felice et all in Ref. \refcite{rs99,98}.Comment: 10 pages, 1 figure, accepted for publication in Mod. Phys. Lett.

Conditional least squares estimators for the offspring mean in a subcritical branching process with immigration

Consider a Bienayme-Galton-Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents and , respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if \u3c2, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n 1/2, which is not the case in the process with stationary immigration. © 2012 Copyright Taylor and Francis Group, LLC

Extremes of geometric variables with applications to branching processes

We obtain limit theorems for the row extrema of a triangular array of zero-modified geometric random variables. Some of this is used to obtain limit theorems for the maximum family size within a generation of a simple branching process with varying geometric offspring laws.Comment: 12 pages, some proofs are added to the published versio

Random sums and partially observed branching processes

© Nova Science Publishers, Inc. In the paper we consider a random sum of a double array of independent random variables. We provide limit theorems for the joint distribution of the random sum and the number of summands in various assumptions on the asymptotic behavior of the number of terms. Further, we apply these limit theorems in study of the following modification of a discrete-time branching process. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals change their offspring distributions. Using our limit theorems for the random sum we derive asymptotic distributions for the vector of inspected and partially observed branching processes in cases when the inspected process is subcritical, critical and supercritical

Estimation of the offspring mean in a supercritical branching process with non-stationary immigration

In this paper, we consider the conditional least squares estimator (CLSE) of the offspring mean of a branching process with non-stationary immigration based on the observation of population sizes. In the supercritical case, assuming that the immigration variables follow known distributions, conditions guaranteeing the strong consistency of the proposed estimator will be derived. The asymptotic normality of the estimator will also be proved. The proofs are based on direct probabilistic arguments, unlike the previous papers, where functional limit theorems for the process were used. © 2011 Elsevier B.V