arXiv subjects

Svyatoslav Novikov

Publications and source records attributed to Svyatoslav Novikov.

8 recordsLinked to original sources

arXiv2026

Branch-stationary max-stable fields on rooted trees

In this contribution we study max-stable random fields on the rooted tree under shifts to descendant subtrees. Branch-Brown--Resnick stationarity is characterised through homogeneous spectral classes, punctured tail measures, and local spectral tail fields. For lognormal representers, it is equivalent to invariance of the variogram under addition of a common prefix. We give Gaussian, max-autoregressive, regenerative cascade, and free-group cluster constructions, and show that summability on countably branching trees need not satisfy a zero--one law. We also derive the associated branch-invariant extreme-value and Archimax copulas.

math.PR↗

arXiv2026

High Minima of Gaussian Processes: Overshoots and Minimizer Locations

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $σ_*^2$ denote the minimum covariance energy associated with $X$, and assume that $σ_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $σ_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

math.PR↗

arXiv2026

Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels

We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form $$ \mathbb{P}\{Γ_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $Γ$ belongs to a broad class satisfying natural monotonicity, scaling, no-atom, and continuity assumptions. The class covers classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals, and, through a local-footprint extension, shrinking-window Parisian persistence functionals. We obtain exact asymptotics in the stationary case and in the non-stationary case when the inverse generalized variance has a unique minimizer at the boundary point. The non-stationary theorem covers the regimes $α<β$, $α=β$, and $α>β$, which lead respectively to Pickands-type, Piterbarg-type, and deterministic limiting constants. We also derive conditional limit laws for the first exceedance time. The main claims are stated in the body of the paper, while the proofs and auxiliary estimates are collected in the appendices.

math.PR↗

arXiv2026

Extremes of Gaussian fields with a product term in the variance

We study the high excursion probability of a centered Gaussian field on a square. Writing $σ$ and $r$ for its standard deviation and correlation function, we assume that $σ$ has a unique maximum at the corner $\boldsymbol{0}=(0,0)$ and \[ 1-σ(\boldsymbol{t}) \sim t_1^β+t_2^β+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0} \] in $\mathbb R_+^2$. The local correlation is assumed to satisfy \[ 1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^α+|t_2-s_2|^α, \qquad 0<α<β. \] This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range $a<β/2$, the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case.

math.PR↗

arXiv2024

Random eigenvalues of nanotubes

The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in \textit{graphene} and \textit{carbon $(p,q)$--nanotubes}. The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths $k\in\mathbb{N}_0$ on their respective graph representations. Considering that a carbon $(p,q)$--nanotube can be thought of as a graphene sheet rolled up in a matter determined by the \textit{chiral vector} $(p,q)$, our findings are based on the study of \textit{random eigenvalues} of both the hexagonal and triangular lattices presented in \cite{bille2023random}. This study reveals that for any given \textit{chiral vector} $(p,q)$, the sequence of counts of closed paths forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including probability density function (PDF) and moment generating function (MGF), are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the \textit{circumference} of a $(p,q)$--nanotube approaches infinity, i.e., $p+q\rightarrow \infty$, the $(p,q)$--nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length $k$, indicating weak convergence of the underlying distributions.

math.SP↗

arXiv2024

Probability of entering an orthant by correlated fractional Brownian motion with drift: Exact asymptotics

For $\{B_H(t)= (B_{H,1}(t), \ldots, B_{H,d}(t))^\top,t\ge0\}$, where $\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $$ \mathbb P (\exists t\ge 0: A B_{H}(t) - μt >νu), \ \ \ \ u\to\infty, $$ where $A$ is a non-singular $d\times d$ matrix and $μ=(μ_1,\ldots, μ_d)^\top\in R^d$, $ν=(ν_1, \ldots, ν_d)^\top \in R^d$ are such that there exists some $1\le i\le d$ such that $μ_i>0, ν_i>0.$

math.PR↗

arXiv2021

Polynomial method for perfect 2-colourings of circulant graphs

In this paper we prove that if an infinite circulant graph with $k$ distances has a perfect $2$-colouring with parameters $(b, c)$, then $b + c \leq 2k + \frac{b+c}{q^t}$ for all positive integers $t$ and primes $q$ satisfying $\frac{b+c}{gcd(b,c)}\vdots q^t$. In addition, we show that if $b + c = q^t$, then this necessary condition becomes sufficient for the existence of perfect $2$-colourings in circulant graphs.

math.CO↗

arXiv2020

Remarks on asymptotic independence

In this paper we introduce several natural definitions of asymptotic independence of two sequences of random elements. We discuss their basic properties, some simple connections between them and connections with properties of weak dependence. In particular, the case of tight sequences is considered in detail. Finally, in order to clarify the relationships between different definitions, we provide some counterexamples.

math.PR↗