## “Mathematics Days in Sofia”

Section “Numerical Analysis, Operations Research, Probability and Statistics”

### Participants

#### Invited Speakers

**Dieter Mitsche**, University Lyon 1, France**Geno Nikolov**, Sofia University “St. Kliment Ohridski”, Bulgaria**George Yanev**, University of Texas Rio Grande Valley, USA

#### Contributors

**Alessandro Barbiero**, Università degli Studi di Milano, Italy**Alexander Klump**, Institute of Mathematics and Informatics, Bulgaria**Detelina Kamburova**, Institute of Mathematics and Informatics, Bulgaria**Hristo Sariev**, Institute of Mathematics and Informatics, Bulgaria**Lyuben Lichev**, University Jean Monnet, France**Martin Minchev**, Sofia University “St. Kliment Ohridski”, Bulgaria**Mohammed Beggas**, University of El-Oued, Algeria**Pando Georgiev**, Institute of Mathematics and Informatics and Sofia University “St. Kliment Ohridski”, Bulgaria

### Program and Abstracts

Motivated by Krioukov et al.’s model of random hyperbolic graphs for real-world networks, and inspired by the analysis of a dynamic model of graphs in Euclidean space by Peres et al., we introduce a dynamic model of hyperbolic graphs in which vertices are allowed to move according to a Brownian motion maintaining the distribution of vertices in hyperbolic space invariant. For different parameters of the speed of angular and radial motion, we analyze tail bounds for detection times of a fixed target and obtain a complete picture, for very different regimes, of how and when the target is detected: as a function of the time passed, we characterize the subset of the hyperbolic space where particles typically detecting the target are initially located.

We overcome several substantial technical difficulties not present in Euclidean space, and provide a complete picture on tail bounds. On the way, we obtain also new results for the time more general continuous processes with drift and reflecting barrier spent in certain regions, and we also obtain improved bounds for independent sums of Pareto random variables.

Joint work with Marcos Kiwi and Amitai Linker.

The problem of characterization of probability distributions can be described as follows. It is known that a family of distributions F possesses certain property P. Is it true, conversely, that a distribution has the property P only if it is a member of F? If so, then P characterizes the family F.

In this talk, we will discuss characterizations of the exponential distribution. More specifically, we will present recent results regarding characterization properties of the exponential distribution involving order statistics, record values, and sums of random variables. Let us point out that the contributions of Bulgarian mathematicians in the area of probability characterizations can be traced back to a 1961 paper by Apostol (a.k.a. Shefa) Obretenov.

A count distribution obtained as a discrete version of the continuous half-logistic distribution is introduced. It is derived by assigning to each non-negative integer value a probability proportional to the corresponding value of the density function of the parent model. Statistical properties of this new distribution, in particular related to its shape, moments, and reliability concepts, are described. Parameter estimation, which can be carried out resorting to different methods including maximum likelihood, is discussed and a numerical comparison between the methods, based on Monte Carlo simulations, is presented. The applicability of the proposed distribution is proved on a real dataset, which has been already fitted by other well-established count distributions. In order to increase the flexibility of this counting model, a generalization is finally suggested, which is obtained by adding a shape parameter to the continuous one-parameter half-logistic and then applying the same discretization technique, based on the mimicking of the density function.

The inverse first-passage time problem for a stochastic process X(t), t≥0, consists of the following question. Given a distribution on the positive real numbers, does there exist a function b such that the first-passage time τ = inf{t > 0 : X_t ≥ b(t)} has this given distribution? In this talk we will give conditions on the process X(t), t ≥ 0, under which the answer is affirmative. For a Markov process we present further additional conditions under which the solutions of the

inverse first-passage time problem are unique. These conditions include Lévy processes with infinite activity or unbounded variation and diffusions on an interval with appropriate behavior at the boundaries. This extends the results in the inverse first-passage time problem for Brownian motion to a class of processes with discontinuous paths and, for example, allows a new range of applications.

Joint work with Mladen Savov.

This is a joint work with Rumen Marinov and Nadia Zlateva.