## Mini-symposium

Advances in Stochastics

*dedicated to the contributions of Prof. Nikolay Yanev*

#### in the frame of the International Conference “Mathematics Days in Sofia”

**T**

**he mini-symposium** “Advances in Stochastics” is devoted to the advancements in the probability theory, statistical inference, branching processes, stochastic processes, probabilistic and statistical analysis of discrete data, robust and nonparametric statistics, extreme value modeling, time series analysis, asymptotic methods in stochastics and their applications in many areas of mathematics, epidemiology, biology, ecology and others. The works presented honor the substantial contributions of Professor Nikolay Yanev especially in areas of branching processes theory, statistical inferences in branching processes and applications.

#### Organizing Committee

Maroussia Bojkova (Sofia University and Institute of Mathematics and Informatics, Bulgaria)

Eugenia Stoimenova (Institute of Mathematics and Informatics, Bulgaria)

### Participants

**Assen Tchorbadjieff**, Institute of Mathematics and Informatics, Bulgaria**Eugenia Stoimenova**, Institute of Mathematics and Informatics, Bulgaria**George Yanev**, University of Texas Rio Grande Valley, USA**Ines del Puerto**, Universidad de Extremadura, Spain**Kosto Mitov**, Medical university – Pleven, Bulgaria**Maroussia Bojkova**, Sofia University and Institute of Mathematics and Informatics, Bulgaria**Miguel González Velasco**, University of Extremadura, Spain**Nikolay Yanev**, Institute of Mathematics and Informatics, Bulgaria**Ollivier Hyrien**, Fred Hutchinson Cancer Center, USA**Pavlina Jordanova**, Konstantin Preslavsky University of Shumen, Bulgaria**Penka Mayster**, Institute of Mathematics and Informatics, Bulgaria**Vessela Stoimenova**, Sofia University “St. Kliment Ohridski”, Bulgaria

### Program and Abstracts

The estimation of multitype branching processes requires a large amount of data that cannot always be observed. The presence of hidden data often requires an algorithmic estimation, which can be carried out on the basis of approximation methods like the machine learning approach to parameter estimation. This work demonstrates an approach to the Markov chain Monte Carlo estimation related to the application of the Hamiltonian Monte Carlo method for the parametric statistical estimation of two-type branching processes. The Leapfrog and the Langevin diffusion methods used for its iterations are compared. It is illustrated by computer simulations and numerical results showing its applicability and advantage in the field of branching processes.

The supercritical geometric reproduction of particles in the model of a Markov branching process is the topic of this work. The solution to the Kolmogorov equation is expressed by the Wright function. The series expansion of this representation is obtained by the Lagrange inversion method. The asymptotic behaviour is described by using two different equivalent forms for the Laplace transform. They include computation of the limit distribution and its moments. The exact formula for the asymptotic density is written in terms of the reduced Wright function. In particular, when the ultimate extinction probability q = 1/2, the density of the limit random variable is given by the incomplete Gamma function.

In his seminal paper [3], Nikolay Yanev introduced control branching processes with random control functions, known as φ-branching processes. Available results for this general class of processes, were presented in [2]. Recently, φ-branching processes with continuous time were introduced as well as limit theorems obtained in [1].

Using a renewal process as subordinator, we study branching processes with migration in continuous time. For these processes, we derive limit theorems assuming the offspring mean is one (critical case) and the emigration prevails over the immigration on average.

References

1. Gonzalez, M., Molina, M, del Puerto, I. Yanev, N.M., and Yanev, G.P. Controlled Branching Processes with Continuous Time. J. Appl. Prob. 58,

830–848 (2021).

2. Gonzalez, M., del Puerto, I. and Yanev, G.P.: Controlled Branching Processes. ISTE Ltd. and John Wiley & Sons, Inc., London, (2018).

3. Yanev, N.M.: Conditions of extinction of φ-branching processes with random φ. Theory Prob. Appl. 20, 433-440 (1975).

This year we celebrate 50 years of Nikolay Yanev’s valuable contributions in stochastics. Nowadays, he is a distinguished professor Emeritus at the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. His creative achievements in the field of stochastics are highly appreciated by the entire scientific community. His original scientific contributions related to the theory of branching processes have had a direct impact on the development of generations of followers in the field.

A pioneering scholar and professor of probability and statistics at the Institute of Mathematics and Informatics he has made exceptional advancements in different areas of probability, statistics, and biostatistics, especially in the areas of the theory of branching processes and statistical inference for them and renewal theory. As a tribute to the 50-years anniversary of his dedication to stochastics, this talk presents his contribution to the development of the field, as well as to the creation of the Bulgarian school of branching processes.

Acknowledgements

This research was partially supported by the Bulgarian NSF at the Ministry of Science and Education, grant No. KP-06-H22/3 and grant No 80-10-65/ 24.04.2023 by the Scientific Fund of Sofia University