I. Rahimov

Random sums and branching stochastic processes

I. Sums of a Random Number of Random Variables.- x1.1. Sampling sums of dependent variables and mixtures of infinitely divisible distributions.- x1a. Sums of a random number of random variables.- x1b. Multiple sums of dependent random variables.- x1c. Sampling sums from a finite population.- x1.2. Limit theorems for a sum of randomly indexed sequences.- x2a. Sufficient conditions.- x2b. Necessary and sufficient conditions.- x2c. An application.- x1.3. Necessary and sufficient conditions and limit theorems for sampling sums.- x3a. Convergence theorems.- x3b. The rate of convergence.- II. Branching Processes with Generalized Immigration.- x2.1.Classical models of branching processes.- x1a. Bellman-Harris processes.- x1b. Moments and extinction probabilities.- x1c. Asymptotics of non-extinct Ion probability and exponential unit distribution.- x1d. Branching processes with stationary immigration.- x1e. Continuous tine branching processes with immigration.- x2.2 General branching processes with reproduction dependent immigration.- x2a. The model.- x2b. The main theorem.- x2c. The proof of the twin theorem.- x2d. Applications of the main theorem.- x2.3.Discrete time processes.- x3a. The model.- x3b. Limit theorems for discrete time processes.- x3c. Some examples.- x3d.Randomly stopped immigration.- x2.4.Convergence to Jirina processes and transfer theorems for branching processes.- x4a. The model.- x4b. The main theorem and corollaries.- x4c. The proof of the main theorem.- III. Branching Processes with Time-Dependent Immigration.- x3. 1.Decreasing immigration.- x1a. The main theorem.- x1b. The proof of the main theorem.- x1c. State-dependent immigration.- x3.2.Increasing immigration.- x2a. The process with Infinite variance.- x2b. The process with finite variance.- x3.3.Local limit theorems.- x3a. Occupation of an increasing state.- x3b. Occupation of a fixed state.- IV. The Asymptotic Behavior of Families of Particles in Branching Processes.- x4.1. Sums of dependent indicators.- x1a. Sums of functions of independent random variables.- x1b. Sampling sums of dependent indicators.- x4.2.Family of particles in critical processes.- x2a. The model.- x2b. Limit theorems.- x4.3.Families of particles in supercritical and subcritical processes.- x3a. Supercritical processes.- x3b. Subcritical processes.- References.

Functional limit theorems for critical processes with immigration

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.

Deterministic Approximation of a Sequence of Nearly Critical Branching Processes

Abstract We consider a sequence of discrete time nearly critical branching processes with time-dependent immigration. Using a martingale approach, we prove that when the immigration mean tends to infinity depending on the time of immigration, the suitable normalized sequence can be approximated in Skorokhod metric by a deterministic process. Consequences related to the maxima and the total progeny of the process will be discussed.

Validity of the bootstrap in the critical process with a non-stationary immigration

In the critical branching process with a stationary immigration, the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n≥1 and are regularly varying sequences with non-negative exponents α and β, respectively. We prove that if α(n)→∞ and β (n)=o(nα2(n)) as n→∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.

Estimation of the population mean using random selection in ranked set samples

The use of ranked set sample to estimate the population mean is well known for its advantages over usual methods using simple random sample. In this paper we generalize the random selection in ranked set sampling proposed by Li et al. (J. Statist. Plann. Inf. 76 (1999) 185) to come up with estimator of the population mean. It will be shown that this estimator is more practical and more efficient in some cases.

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 α and β, respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if β <2α, 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.

Asymptotically Normal Estimators for the Offspring Mean in the Branching Process with Immigration

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a branching process with stationary immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with norming factor n 3/2. We study the process with a non degenerate offspring distribution and time-dependent immigration, whose mean and variance vary regularly with non negative exponents, α and β, respectively. We propose new weighted CLSE using more flexible weights and prove that if β < 1 + 2α, it is asymptotically normal with two different norming factors and if β > 1 + 2α, its limiting distribution is not normal but can be expressed in terms of certain functionals of the time-changed Wiener process. When β = 1 + 2α, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.

Random Sums of Independent Indicators and Generalized Reduced Processes

We consider a random sum of independent and identically distributed Bernoulli random variables. We prove several limit theorems for this sum under some natural assumptions. Using these limit theorems a generalized version of the reduced critical Galton-Watson process will be studied. In particular we find limit distributions for the number of individuals in a given generation the number of whose descendants after some generations exceeds a fixed or increasing level. An application to study of the number of “big” trees in a forest containing a random number of trees will also be discussed.

APPROXIMATION OF EXCEEDANCE PROCESSES IN LARGE POPULATIONS

We consider a population of n individuals. Each of these individuals generates a discrete time branching stochastic process. We study the number of ancestors S(n,t) whose offspring at time t exceeds level θ(t), where θ(t) is some positive valued function. It is proved that S(n,t) may be approximated as t → ∞ and n → ∞ by some stochastic processes with independent increments. AMS Subject Classification: Primary 60J80; Secondary 60G70, 60F05

Maximal number of direct offspring in simple branching processes

The number Yn of offspring of the most productive particle in the nth generation of a Bienayme-Galton-Watson process is considered. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems when the offspring mean is finite are proved using some convergence results for branching processes as well as transfer theorems for maxima.