Аспірантський воркшоп

Abstract: I will discuss a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by t and monotone in t. If the expectation b and the variance a of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of a. If the expectation grows faster than the variance, while the ratio log b/log a remains bounded, then the normalization in the LIL includes the single logarithm of a (so that the LIL becomes a law of the single logarithm). Finally, I will present an application of the LIL to the number of occupied boxes and related quantities in Karlin’s occupancy scheme.
The talk is based on a joint paper with Dariusz Buraczewski (Wroclaw) and Oleksandr Iksanov (Kyiv).

Abstract: This work continues the study of regression models of mixtures with variable concentrations (MVC) previously described by R.E. Maiboroda. In his works, Maiboroda describes models with fixed mixing probabilities or their parametric models. In these models, mixing probabilities are either known or can be estimated from auxiliary data, such as those that are not used to fit the regressor parameters. The purpose of this work is to describe such a method for which the presence of auxiliary data is not mandatory. Such a model is built using a combination of the mixture model of experts (MoE) and MVC. The disadvantage of MoE is the dependence of parameter fitting on initial approximations and imbalance in decision-making by the router. In this model, the task of the router is to estimate the mixing probabilities for the MVC model. Direct application of loaded functionals with minimax weights from MVC does not give obvious advantages, since weights loss functions are difficult to minimize due to the need to find derivatives over the elements of the inverse matrices. That is why in this work, differential constraints of such functionals are constructed, and the problem of differentiating inverse matrices is replaced by a simpler problem of differentiating the eigenvalues of positive definite symmetric matrices. In this case, the derived loss function additionally allows balancing between the uniform distribution in the router’s decisions and the quality of the experts’ predictions.

Abstract: This presentation discusses ongoing work on limit theorems for generalized convex hulls, specifically K-hulls. Let K be a compact convex body in the d-dimensional Euclidean space, and let X_1, X_2, …, X_n be points sampled uniformly at random from K. Marynych and Molchanov (2022) showed that the set of all translations of K that keep the entire sample inside the translated body converges in distribution to the zero cell Z of a certain Poisson hyperplane tessellation. Equivalently, the polar set to the set of allowed translations converges to the polar set Z°, which is also the convex hull of a certain Poisson point process.

We investigate how this convergence behaves when translations are required to contain only “most” of the sample points. Specifically, if we allow translations that contain all but at most m points, we prove convergence to the so-called m-point peeling of Z°. We also study the effect of removing “extreme points” from the sample. To that end, we introduce the notion of a contributing set to a family of sets, which generalizes the classical notion of an extreme point of a convex set. We show that removing sets that contribute to the family of admissible translations leads to convergence to the convex-hull peeling of Z°. This operation has been studied extensively by Calka et al.

The talk is based on joint work with Alexander Marynych.

Анотація: Розглядається оцінювання параметра зсуву в процесах Орнштейна-Уленбека, керованих темперованим дробовий броунівським рухом (ТДБР) або темперованим дробовий броунівським рухом другого роду (ТДБРII). У вже відомих результатах в основному розглядають оцінювання для неперервних спостережень. Дане дослідження присв’ячене більш реалістичній ситуації – дискретним даним. Встановлено строгу конзистентність дискретизованої оцінки найменших квадратів для схеми спостережень, у якій інтервал між спостереженнями прямує до нуля, у той час коли часовий горизонт зростає. Водночас, було отримано обмеження майже напевно для приростів обох процесів (ТДБР та ТДБРII).

Abstract: This talk focuses on the distributional properties of generalized counting processes models with time change. In particular, for time change we use the following subordinators: compound Poisson-Gamma process; inverse compound Poisson-Exponential process; Bessel subordinator; Gamma process subordinated to a Poisson process with a drift; Bessel transform of a Gamma process; Inverse Gaussian process.

For each model with time change, we derive explicit expressions for their probability mass functions, the governing systems of difference-differential equations, and their first two moments. The resulting distributions are expressed in closed form in terms of special functions, such as the generalized Wright function, the Mittag-Leffler function, and modified Bessel functions.

Анотація: Розглядаються стохастичні диференціальні рівняння, для яких випадковий вплив задано інтегралом за загальною стохастичною мірою. Особливість таких мір полягає в тому, що для них вимагається лише σ-адитивність за ймовірністю. В доповіді розглянуто декілька класів рівнянь, для яких доведено існування та єдиність розв’язків, їх неперервність за Гельдером по часовій та просторовій змінній. Доведення проводиться з використанням допоміжних тверджень про властивості інтегралів за стохастичними мірами, які також наводяться в доповіді.