We consider a semi-Markov process with values in Rd and create a scheme of series with the help of time and space scaling. We prove the weak convergence in a Skorokhod space of such scaled processes to a limiting diffusion process under conditions on the distribution of jumps and holding times of the original semi-Markov process.

The article explores a generalization of the Goldstein-Kac model, specifically a model of random evolution on a complex plane, with the velocity that decreases over time. This process simulates the motion of a particle in a force field, among other phenomena. Limit theorems describing the distribution of the absorbing point for this process have been derived. Additionally, nonlinear integral equations for functionals of the process have been obtained, and the existence and uniqueness of their solutions have been proven.

We construct and study an information warfare model, with an internal conflict integrated into it (interregional migration) based on the example of the Lotka-Voltaire model with cyclical migration, as well as its research using a software tool. For this purpose, the behavior of the spread of one and several information threats within the same community is described, the principle of conflict is described, a model is built and its behavior is studied on various examples, and conclusions are drawn regarding the importance of the influence of internal conflict on the model and regarding methods of predicting the results.

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

We discuss a generalization of Goldstein-Kac model on a complex plane and apply probabilistic approach to construct solutions of the corresponding Cauchy problem for complex-analytic initial conditions. The method is based on reconstruction of complex-analytic functions by combination of power functions, for which corresponding solutions are the moments of evolution process.As soon as in the hydrodynamic limit the equation for our model approximates a Schrödinger-type equation, the solutions constructed for pre-limit Cauchy problem may approximate solutions for corresponding Cauchy problem for a Schrödinger-type equation.

Abstract An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

This study investigates a complex system that describes a non-trivial epidemiological model with integrated internal conflict (interregional migration) on the example of cyclic migration using the software. JetBrains PyCharm Community Edition 2020.3.3, a free and open-source integrated development environment (IDE) in the Python programming language, was chosen as the software development tool. The Matplotlib 3.5 library was used to display the modelling results graphically. The integration of internal conflict into the model revealed significant and notable changes in its behavior. This study’s results prove that not only the characteristics of the interaction factors but also the size of the values determine the direction of migration concerning relation to competitors.

A nested occupancy scheme in a random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in a random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper, we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that n balls have been thrown, denote by K-n(j) the number of occupied boxes in the jth level and call the level j intermediate if j = j(n) -> infinity and j(n) = o(log n) as n -> infinity. We prove a multidimensional central limit theorem for the vector (K-n(vertical bar j(n)u(1)vertical bar),..., K-n(vertical bar j(n)u(l)vertical bar)(,) properly normalized and centred, as n -> (,)infinity where j(n) -> infinity and j(n) = o((log n)(1/2)). The present paper continues the line of investigation initiated in the article [D. Buraczewski, B. Dovgay, and A. Iksanov, On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I, Electron. J. Probab. 25(123) (2020), pp. 1- 24] in which the occupancy of intermediate levels j(n) -> infinity, j(n) = o((log n)(1/3)) was analysed.

This study is focused on the construction and analysis of a complex epidemiological practical model built on the basis of the Susceptible-Infected-Removed (SIR) model. The examples illustrate the behavior of the practical model in various scenarios and also compare this model and a similar model, taking into account migration. The nature of the behavior of the model is determined by parameters such as the rate of spread of infection, the coefficients of recovery, mortality, the intergroup transition and others with different values of influence.

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for convergent perpetuities $\sum_{k\geq 0}b^{\xi_1+\ldots+\xi_k}\eta_{k+1}$ as $b\to 1-$. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.

We construct and analyze a continuous evolutionary model that describes the conflict interaction of two complex systems with nontrivial internal structures. External conflict interactions are modeled on the additional influence of random factors. The dynamics of the external conflict is similar to the Lotka–Volterra model, namely, the information warfare model. We interpret the new model of information warfare as the impact of rare events that quickly change certain perceptions of a large number of people. As a result, the number of proponents of different ideas makes stochastic leaps, which we can see using the Levy approximation scheme. We assume that the new model is more natural, because important news now has a rapid impulse impact on the audience through information channels and social networks.

Conditions for asymptotic dissipativity are established for the merged system of stochastic differential equations with Markov switchings and impulse perturbations under conditions of Levy approximation. In particular, it is analyzed how the behavior of the boundary process depends on the pre-limiting normalization of a stochastic evolution system in the ergodic Markovian environment under the conditions of Lévy approximation.

Double merging of phase space for the stochastic evolutionary system is performed. The case is considered where system’s perturbations are determined by the impulse process at the Poisson approximation scheme. The limiting process under such conditions has two components: deterministic shift and Poisson jump addition.

The regular as well as singular component of the asymptotic expansion of a functional constructed from a semi-Markov random evolution is found, and regularity of the initial data is shown in Theory Probab. Math. Statist.

We construct and study a discrete time model describing the conflict interaction between two complex systems with non-trivial internal structures. The external conflict interaction is based on the model of alternative interaction between a pair of non-annihilating opponents. The internal conflict dynamics is similar to the one of Lotka-Volterra model, namely information warfare model. We show that the typical trajectory of the complex system converges to an asymptotic attractive cycle. We propose an interpretation of our model in terms of migration processes.

The authors analyze asymptotic dissipation of pre-limit normalized stochastic evolutionary system in ergodic Markov environment, which significantly influences the behavior of the limiting process.

We propose new methods for the investigation of a model of stochastic evolution with Markov switchings capable of separation of the diffusion component and big jumps of the perturbing process in the limiting equation. Big jumps of this type may describe seldom catastrophic events in various applied problems. We consider the case where the system is perturbed by an impulsive process in the nonclassical approximation scheme. Special attention is given to the asymptotic behavior of the generator of the analyzed evolutionary system.

This is a short survey of the joint author results concerning the large deviations problem for some stochastic processes of random evolution published in the papers

Let (ξ 1, η 1), (ξ 2, η 2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J 1-topology on the Skorokhod space of \(n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})\) was proved under the assumption that contributions of \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) to the limit are comparable and that n −1/2(ξ 1+… + ξ [n⋅]) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ 1+… + ξ [n⋅], properly normalized without centering, is attracted to a centered stable Lévy process, a process with jumps. As a consequence, weak convergence normally holds in the M 1-topology. We also provide sufficient conditions for the J 1-convergence. For completeness, less interesting situations are discussed when one of the sequences \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) dominates the other. An application of our main results to divergent perpetuities with positive entries is given.

The methods proposed in the paper allow us to investigate the model of stochastic evolution, which includes Markov switchings, and to identify big jumps of disturbing process in the limiting equation. Big jumps of this type may describe rare catastrophic events in different applied problems. We consider the case where system disturbance is defined by impulse process in nonclassical approximation scheme. Particular attention is paid to the asymptotic behavior of the generator of the evolutionary system under examination.