Searcharxiv⌕ Search

arXiv subjects

Azam A. Imomov

Publications and source records attributed to Azam A. Imomov.

6 recordsLinked to original sources
arXiv2023

On asymptotic normality of the total progeny in the positive recurrent Q-processes

We examine the population growth system called Q-processes. This is defined by the Galton-Watson Branching system conditioned on non-extinction of its trajectory in the remote future. In this paper we observe the total progeny up to time $n$ in the Q-process. By analogy with branching systems, this variable is of great interest in studying the deep properties of the Q-process. We find that the sum total progeny as a random variable approximates the standard normal distribution function under a second moment assumption for the initial Galton-Watson system offspring law. We estimate the speed rate of this approximation.

math.PR↗
arXiv2023

Further remarks on the explicit generating function expression of the invariant measure of critical Galton-Watson Branching Systems

Consider the critical Galton-Watson branching system with infinite variance of the offspring law. We provide an alternative arguments against what Slack~{\cite{Slack68}} did when it seeked for a local expression in the neighborhood of point $1$ of the generating function for invariant measures of the branching system. So, we obtain the global expression for all $s\in[0,1)$ of this generating function. A fundamentally improved version of the differential analogue of the Basic Lemma of the theory of critical branching systems is established. This assertion plays a key role in the formulation of the local limit theorem with explicit terms in the asymptotic expansion of local probabilities. We also determine the decay rate of the remainder term in this expansion.

math.PR↗
arXiv2020

Renewed Limit Theorems for the discrete-time Branching Process and its Conditioned Limiting Law interpretation

Our principal aim is to observe the Markov discrete-time process of population growth with long-living trajectory. First we study asymptotical decay of generating function of Galton-Watson process for all cases as the Basic Lemma. Afterwards we get a Differential analogue of the Basic Lemma. This Lemma plays main role in our discussions throughout the paper. Hereupon we improve and supplement classical results concerning Galton-Watson process. Further we investigate properties of the population process so called Q-process. In particular we obtain a joint limit law of Q-process and its total state. And also we prove the analogue of Law of large numbers and the Central limit theorem for total state of Q-process. Keywords: Branching process; transition function; Q-process; invariant measures; ergodic chain; total states; joint distribution; limit theorem.

math.PR↗