arXiv subjects

Pavel Ievlev

Publications and source records attributed to Pavel Ievlev.

5 recordsLinked to original sources

arXiv2026

Laplace Asymptotics near Stratified Minimum Sets

We develop a real Laplace method for integrals $$ I(z)=\int_N a(x)e^{-z f(x)}\operatorname{dvol}_g(x),\qquad z\to\infty, $$ near a compact, possibly stratified, minimum set of the phase. The basic input is an adapted normal-cell disintegration, defined up to null sets, together with piecewise anisotropic dilations and homogeneous limiting models. The framework simultaneously permits finite $C^2$ stratifications; Borel base refinements and $C^1$ adapted charts; measurable base-dependent cells, relative domains, and incident-stratum assignments; different fiber dimensions and weights on different pieces; a continuous phase and a merely measurable amplitude; and limiting models controlled either by ambient coercivity or directly by cellwise exponential tightness. The cells may converge only in weighted measure, or pointwise under an integrable base majorant. To the best of our knowledge, no existing theorem for real Laplace integrals accommodates this full combination of geometric and analytic assumptions. The tradeoff is that the adapted disintegration and limiting models are supplied hypotheses to be verified in each application.

math.CA↗

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 $$ \mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $\Gamma$ 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 $\alpha<\beta$, $\alpha=\beta$, and $\alpha>\beta$, 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↗

arXiv2025

Extremes of Brownian Decision Trees

We consider a Brownian motion with linear drift that splits at fixed time points into a fixed number of branches, which may depend on the branching point. For this process, which we shall refer to as the Brownian decision tree, we investigate the exact asymptotics of high exceedance probabilities in finite time horizon, including: the probability that at least one branch exceeds some high threshold, the probability that the largest distance between branches gets large and the probability that all branches simultaneously exceed some high barrier. Additionally, we find the asymptotics for the probability that all branches of at least one of $M$ independent Brownian decision trees exceed a high threshold.

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↗

arXiv2022

Parisian ruin with power-asymmetric variance near the optimal point with application to many-inputs proportional reinsurance

This paper investigates the Parisian ruin probability for processes with power-asymmetric behavior of the variance near the unique optimal point. We derive the exact asymptotics as the ruin boundary tends to infinity and extend the previous result arXiv:1504.07061 to the case when the length of Parisian interval is of Pickands scale. As a primary application, we extend the recent result arXiv:2010.00222 on the many inputs proportional reinsurance fractional Brownian motion risk model to the Parisian ruin.

math.PR↗