Revisiting Offspring Maxima in Branching Processes

We present a progress report for studies on maxima related to offspring in branching processes. We summarize and discuss the findings on the subject that appeared in the last ten years. Some of the results are refined and illustrated with new examples.Comment: To appear in PLISKA Studia Mathematica Bulgaric

Characterizations of exponential distribution via conditional expectations of record values

We prove that the exponential distribution is the only one which satisfies a regression identity. This identity involves conditional expectation of the sample mean of record values given two record values outside of the sample

Limit Theorems for Branching Processes with Random Migration Components

The classical Bienaymé-Galton-Watson (BGW) branching process can be interpreted as mathematical model of population dynamics when the members of an isolated population reproduce themselves independently of each other according to a stochastic law.The work is supported by the Bulgarian National Foundation for Scientific Investigations, grant MM-704/97

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

On characterization of the exponential distribution via hypoexponential distributions

The sum of independent, but not necessary identically distributed, exponential random variables follows hypoexponential distribution. We study a situation when the rate parameters of the exponential variables are not all different from each other. We obtain a representation for the Laplace transform of the hypoexponential distribution in the case of two repeated parameter values. Applying this decomposition, we prove a characterization of the exponential distribution

Exponential and Hypoexponential Distributions: Some Characterizations

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n ≥ 2, X1, X2, . . . , Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj ’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables