Searcharxiv⌕ Search

arXiv subjects

Vyacheslav M. Abramov

Publications and source records attributed to Vyacheslav M. Abramov.

At least 19 recordsLinked to original sources
arXiv2026

Conditions for recurrence and transience for time-inhomogeneous random walks

The present paper extends the earlier results obtained by Abramov [`Conditions for recurrence and transience for time-inhomogeneous birth-and-death processes' \emph{Bull. Aust. Math. Soc.} \textbf{109} (2024), 393--402] for the case of time-inhomogeneous random walks, the increments of which take values in $\mathbb{R}$. By this, we give a full solution of the open problem formulated by Menshikov and Volkov [`Urn-related random walk with drift $ρx^α/t^β$' \emph{Electron. J. Probab.}, \textbf{13} (2008), paper No. 31, 944--960] that was partially solved in the aforementioned paper by Abramov.

math.PR↗
arXiv2026

The convergence of power matrices

For the class of $d\times d$ matrices $B=[b_{i,j}]$ with complex nonzero entries satisfying $\sum_{i=1}^{d}|b_{i,j}|=1$, we provide the conditions for the convergence of power matrices $B^n$ to a nonzero limit matrix.

math.CA↗
arXiv2026

Some notes on Lamperti's recurrence of stochastic sequences

The present study provides another look on Lamperti's theorem on recurrence or transience of stochastic sequences. We establish connection between Lamperti's theorem and the recent result by the author [V. M. Abramov, Theor. Probab. Math. Stat. 112 (2025), 1--15].

math.PR↗
arXiv2025

Classes of functions in the set of circulant matrices and their characterization

In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend the recently obtained characterization theorems in [Li, B. Q. Two elementary properties of entire functions and their applications. Amer. Math. Monthly 122(2) (2015), 169--172] and [Li, B. Q. A characterization of rational functions. Amer. Math. Monthly. 132(3) (2025), 269--271] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems lead to the generalization of the variant of the fundamental theorem of algebra established recently in [Abramov, V. M. Extension of the fundamental theorem of algebra for polynomial matrix equation with circulant matrices. Amer. Math. Monthly. DOI: 10.1080/00029890.2024.2440298].

math.CA↗
arXiv2025

A solution to a Paul Erdos problem

Paul Erdos posed the following question: Is there a prime number $p>5$ such that the residues of $2!$, $3!$,\ldots, $(p-1)!$ modulo $p$ all are distinct? In this short note, we prove that there are no such prime numbers.

math.NT↗
arXiv2023

Conditions for recurrence and transience for one family of random walks

A parametric family of two-dimensional random walks $\mathbf{S}_t(a)$ $=\big(S_t^{(1)}(a),$ $S_t^{(2)}(a)\big)$ in the main quarter plane is studied. The components $S_t^{(1)}(a)$ and $S_t^{(2)}(a)$ are assumed to be correlated in the way that is defined exactly in the paper. We derive the conditions on $a$, under which a random walk $\mathbf{S}_t(a)$ is recurrent.

math.PR↗
arXiv2022

Fixed point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices

In [V. M. Abramov, \emph{Bull. Aust. Math. Soc.} \textbf{104} (2021), 108--117] the fixed point equation for an infinite nonnegative Toeplitz matrix has been studied. It was found the conditions for existence of a positive solution and bounded positive solution. However, the proof of the existence of a positive solution was entirely straightforward, not admitting extensions for more general types of matrices. In the present note, we provide an alternative proof for the existence of a positive solution in more general case. The presented proof is based on an application of a variant of the fixed point theorem of M. A. Krasnoselskii. The results are then extended for the equations with infinite matrices of a general type.

math.CA↗
arXiv2022

Crossings states and sets of states in Pólya random walks

We consider the Pólya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.

math.PR↗
arXiv2021

Necessary and sufficient conditions for the convergence of positive series

We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran-De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452-457] and [L. Bourchtein et al, Int. J. Math. Anal. 6(2012), 1847-1869]. The obtained result enables us to extend the known conditions for recurrence and transience of birth-and-death processes given in [V. M. Abramov, Amer. Math. Monthly 127(2020) 444-448].

math.CA↗
arXiv2021

A simple proof of Tong's theorem

We provide a new simple and transparent proof of the version of Kummer's test given in [Tong, J. (1994). Amer. Math. Monthly. 101(5): 450--452]. Our proof is based on an application of a Hardy--Littlewood Tauberian theorem.

math.HO↗
arXiv2021

Extension of the Bertrand-De Morgan test and its application

We provide a simple proof for the extended Bertrand-De Morgan test that was earlier studied in [F. Ďuriš, Infinite Series: Convergence tests. Bachelor thesis, 2009] and [A.A. Tabatabai Adnani, A. Reza and M. Morovati, J. Lin. Topol. Algebra, 2 (3), pp. 141--147 (2013)] and demonstrate an application of that test in the theory of birth-and-death processes and random walks.

math.GM↗
arXiv2020

Fixed point theorem for an infinite Toeplitz matrix

For an infinite Toeplitz matrix $T$ with nonnegative real entries we find the conditions, under which the equation $\boldsymbol{x}=T\boldsymbol{x}$, where $\boldsymbol{x}$ is an infinite vector-column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behavior of convolution type recurrence relations, and can be applied to different problems arising in the theory of stochastic processes and applied problems from other areas.

math.PR↗