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Alexei Stepanov

Publications and source records attributed to Alexei Stepanov.

At least 19 recordsLinked to original sources
arXiv2026

Kendall Correlation Coefficient for non-Identically Distributed Variables

In the present paper, we discuss for the first time the theoretical Kendall correlation coefficient for non-identical bivariate data. In the non-identical case, we first introduce a theoretical Kendall correlation coefficient $\tau_n$ and show that the expected value of the rank Kendall correlation coefficient $\tilde{\tau}_n$ is equal to $\tau_n$. We then prove that $\tilde{\tau}_n$ converges in probability to $\tau=\lim_{n\rightarrow\infty} \tau_n$. These facts enable us to state that $\tau_n$ is a correctly defined theoretical Kendall correlation coefficient for the non-identical case. We also support our theoretical results by simulation experiments.

math.ST↗
arXiv2025

On Rank Correlation Coefficients

In the present paper, we propose a new rank correlation coefficient $r_n$, which is a sample analogue of the theoretical correlation coefficient $r$, which, in turn, was proposed in the recent work of Stepanov (2025b). We discuss the properties of $r_n$ and compare $r_n$ with known rank Spearman $\rho_{S,n}$, Kendall $\tau_n$ and sample Pearson $\rho_n$ correlation coefficients. Simulation experiments show that when the relationship between $X$ and $Y$ is not close to linear, $r_n$ performs better than other correlation coefficients. We also find analytically the values of $Var(\tau_n)$ and $Var(r_n)$. This allows to estimate theoretically the asymptotic performance of $\tau_n$ and $r_n$.

math.ST↗
arXiv2024

On Correlation Coefficients

In the present paper, we discuss the Pearson, Spearman, Kendall correlation coefficients and their statistical analogues. We propose a new correlation coefficient r and its statistical analogue. The coefficient r is based on Kendal's and Spearman's correlation coefficients. A new extension of the Pearson correlation coefficient is also discussed. We conduct simulation experiments and study the behavior of the above correlation coefficients. We observe that the behavior of Pearson's sample correlation coefficient can be very different from the behavior of the rank correlation coefficients, which, in turn, behave in a similar way. The question arises: which correlation coefficient better measures the dependence rate? We try to answer this question in the final conclusion.

math.ST↗
arXiv2021

A Note on the Borel-Cantelli Lemma

In this short note, we discuss the Barndorff-Nielsen lemma, which is a generalization of well-known Borel-Cantelli lemma. Although the result stated in the Barndorff-Nielsen lemma is correct, it does not follow from the argument proposed in the corresponding proof. In this note, we show this and offer an alternative proof of this lemma. We also propose a new generalization of Borel-Cantelli lemma.

math.PR↗
arXiv2018

Subgroups of Chevalley groups of types $B_l$ and $C_l$ containing the group over a subring and corresponding carpets

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A. Stepanov considered the symplectic case (i. e. the case of simply connected group of type $\Phi=C_l$) in characteristic 2. In this article we extend their result for groups with arbitrary weight lattice of types $B_l$ and $C_l$. Similarly to the work of Nuzhin that handles the case of an algebraic extension $R$ of a nonperfect field $K$ of bad characteristic, we use in the description a special kind of carpet subgroups. In the second half of the article we study Bruhat and Gauss decompositions for these carpet subgroups.

math.GR↗
arXiv2018

Normal structure of isotropic reductive groups over rings

The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the reductive group scheme G is defined over an arbitrary commutative ring, its isotropic rank is at least 2, and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)<=...<=C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R) to G(R/I). The main ingredients of the proof are the "level computation" by the first author and the universal localization method developed by the second author.

math.GR↗
arXiv2015

A new look at the decomposition of unipotents and the normal structure of Chevalley groups

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call an element $a\in G(\Phi,R)$ "good", if it lies in the unipotent radical of a parabolic subgroup whereas the conjugate to $a$ by $h$ belongs to another proper parabolic subgroup (here we assume that all parabolics contain a given split maximal torus). Decomposition of unipotents is a representation of a root unipotent element as a product of "good" elements. Existence of such a decomposition implies a simple proof of the normality of the elementary subgroup and description of the normal structure of $G(\Phi,R)$. However, the decomposition is available not for all root systems. In the current article we show that for the proof of the standard normal structure it suffices to construct only one "good" element for the generic element of the scheme $G(\Phi,_-)$, and construct such a "good" element. The question whether "good" elements generate the whole elementary group will be addressed in a separate paper (the first version of the article was in Russian).

math.RA↗
arXiv2015

Subring subgroups in symplectic groups in characteristic 2

In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice splits into a disjoint union of "sandwiches", parametrized by intermediate subrings between $K$ and $A$. In the current article a similar result is proved for $\Phi=B_n$ or $C_n$, $n\ge3$, and $2=0$ in $K$. In this settings one has to introduce more sandwiches, namely, the sandwiches are parametrized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. In particular, this result, generalizes a part of Ya.\,N.\,Nuzhin's theorem of 2013 concerning root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained under the condition that $A$ is an algebraic extension of~$K$.

math.RA↗
arXiv2015

On the Kendall Correlation Coefficient

In the present paper, we first discuss the Kendall rank correlation coefficient. In continuous case, we define the Kendall rank correlation coefficient in terms of the concomitants of order statistics, find the expected value of the Kendall rank correlation coefficient and show that the later is free of n. We also prove that in continuous case the Kendall correlation coefficient converges in probability to its expected value. We then propose to consider the expected value of the Kendall rank correlation coefficient as a new theoretical correlation coefficient which can be an alternative to the classical Pearson product-moment correlation coefficient. At the end of this work we analyze illustrative examples.

math.ST↗
arXiv2015

Bijections preserving commutators and automorphisms of unitriangular group

We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $UT(n,F)$ over a field $F$, where $n$ is a natural number or infinity. PC-maps were recently described up to almost identity PC-maps by M.Chen, D.Wang, and H.Zhai (2011) for finite $n$ and by R.Slowik (2013) for $n=\infty$. An almost identity map is a map, preserving elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if $n=\infty$, then an almost identity map is identity. Together with the result of R.Slowik this shows that any PC-map of $UT(\infty, F)$ is an automorphism.

math.GR↗
arXiv2013

Structure of Chevalley groups over rings via universal localization

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.

math.RA↗
arXiv2012

Local-global principle for congruence subgroups of Chevalley groups

We prove Suslin's local-global principle for principal congruence subgroups of Chevalley groups. Let $G$ be a Chevalley--Demazure group scheme with a root system $\Phi\ne A_1$ and $E$ its elementary subgroup. Let $R$ be a ring and $I$ an ideal of $R$. Assume additionally that $R$ has no residue fields of 2 elements if $\Phi=C_2$ or $G_2$. Theorem. Let $g\in G(R[X],XR[X])$. Suppose that for every maximal ideal $\m$ of $R$ the image of $g$ under the localization homomorphism at $\m$ belongs to $E(R_\m[X],IR_\m[X])$. Then, $g\in E(R[X],IR[X])$. The theorem is a common generalization of the result of E.Abe for the absolute case ($I=R$) and H.Apte--P.Chattopadhyay--R.Rao for classical groups. It is worth mentioning that for the absolute case the local-global principle was obtained by V.Petrov and A.Stavrova in more general settings of isotropic reductive groups.

math.RA↗
arXiv2012

On Strong Convergence for Maxima

In the present paper, a generalization of the first part of the Borel-Cantelli lemma is obtained by the recent work of Balakrishnan and Stepanov (2010). This generalization is further applied to derive strong limit results for the sequence of maxima.

math.PR↗
arXiv2012

Commutator width in Chevalley groups

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

math.RA↗
arXiv2011

On the Borel-Cantelli Lemma

In the present note, we generalize the first part of the Borel-Cantelli lemma. By this generalization, we obtain some strong limit results.

math.PR↗