In this paper, we study properties of random walks on finite groups and later use them to obtain the limiting braid length expectation and component number of braid closure in a model of random braids, which is constructed by lifting elements of random walk on a Coxeter group to a braid group.

This paper explores the asymptotic behavior of solutions to multi-dimensional stochastic differential equations with interactions (SDEWI). By integrating interaction terms reflecting the distribution of all participating particles, SDEWI provides a complex model that captures dynamic changes across a system of particles. Theoretical insights are substantiated by the derivation of conditions under which the solutions exhibit specific asymptotic properties. This work extends previous research by confirming the shift-compactness criterion and establishing conditions for the existence of asymptotic limits, thereby offering a deeper understanding of the interaction dynamics within stochastic systems.

In the paper we suggest a new construction of stochastic flows of kernels in a locally compact separable metric space M. Starting from a consistent sequence of Feller transtition function (P (n) : n ≥ 1) on M we prove existence of a stochastic flow of kernels K=(K s,t : -∞; < s ≤ t < ∞) in M, such that distributions of n-point motions of K are determined by P (n) . Presented construction allows to find a single idempotent measurable presentation p of distributions of all kernels K s,t from a flow, and to construct a flow that is invariant under p and is jointly measurable in all arguments.

In this article, we will construct an approximation of Gaussian white noise based on the sequence of Bernoulli random variables and define Wick products and the stochastic exponent for the Bernoulli case. Here we will propose a method to calculate the expectations of Wick products for Bernoulli noise using diagrams, that converge to Feynman diagrams in the Gaussian case. We will prove that orthogonal polynomials for Bernoulli noise converge to Hermite polynomials, which form an orthogonal system in the Gaussian case.

In this research paper, we define a kernel estimator of the mode based on the recursive kernel density estimator developed by [23]. In addition, we establish its almost sure convergence under strong mixing hypothesis. Finally, we corroborate these theoretical results through numerical simulations.

In recent years, in view of theory of empirical processes, authors have become more interested in the uniform analogue of the three fundamental theorems: the uniform law of large numbers of Glivenko-Cantelli type, the uniform central limit theorem for Donsker type and the functional law of the iterated logarithm (LIL). In this paper, under the bracketing entropy conditions, the uniform law of large numbers, uniform central limit theorem and the uniform LIL of Strassen type have been investigated in the case of length-biased and type I censoring.

In this paper, we extend the dual divergence approach to general semiparametric models and study dual divergence estimators for semiparametric models. Asymptotic properties such as consistency, asymptotic normality of the proposed estimators are deeply investigated by mean the sophisticated modern empirical theory. We investigate the exchangeably weighted estimators in this setting and establish the consistency. We finally consider the functional M-estimator and obtain its weak convergence result.