Exact Asymptotics Research Articles

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

For {BH(t)=(BH,1(t),…,BH,d(t))⊤,t≥0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\varvec{B}_{H}(t)= (B_{H,1}(t) ,\\ldots ,B_{H,d}(t))^{{\ op }},t\\ge 0\\}$$\\end{document}, where {BH,i(t),t≥0},1≤i≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{B_{H,i}(t),t\\ge 0\\}, 1\\le i\\le d$$\\end{document} are mutually independent fractional Brownian motions, we obtain the exact asymptotics of P(∃t≥0:ABH(t)-μt>νu),u→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb P (\\exists t\\ge 0: A \\varvec{B}_{H}(t) - \\varvec{\\mu }t >\\varvec{\ u }u), \\ \\ \\ \\ u\\rightarrow \\infty ,$$\\end{document}where A is a non-singular d×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\ imes d$$\\end{document} matrix and μ=(μ1,…,μd)⊤∈Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mu }=(\\mu _1,\\ldots , \\mu _d)^{{\ op }}\\in \\mathbb {R}^d$$\\end{document}, ν=(ν1,…,νd)⊤∈Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ u }=(\ u _1, \\ldots , \ u _d)^{{\ op }} \\in \\mathbb {R}^d$$\\end{document} are such that there exists some 1≤i≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le i\\le d$$\\end{document} such that μi>0,νi>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _i>0, \ u _i>0.$$\\end{document}

Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of △nv=0 for a discretization △ of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.

Residual finiteness growth in virtually abelian groups

A group G is called residually finite if for every non-trivial element g∈G, there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g, for example by using the word norm ‖g‖G if the group G is assumed to be finitely generated. The residual finiteness growth RFG:N→N is then defined as the smallest function such that if ‖g‖G≤r, there exists a morphism φ:G→Q to a finite group Q with |Q|≤RFG(r) and φ(g)≠eQ.Although upper bounds have been established for several classes of groups, exact asymptotics for the function RFG are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals logk for some k∈N, where the value k is given by an explicit expression. As an application, we show that for every m≥1 and every 1≤k≤m, there exists a group G containing a normal abelian subgroup of rank m and with RFG≈logk.

The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on Rd. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation

On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation

Factors associated with the quality of life of women undergoing radiotherapy.

To evaluate the skin characteristics and quality of life of patients with breast cancer undergoing radiotherapy. Cross-sectional study conducted with 60 women. The classification scales of skin changes resulting from exposure to ionizing radiation (RTOG) and the validated versions in Portuguese of those that classified skin types (Fitzpatrick), symptoms (RISRAS) and quality of life (DLQI) were applied. in the period between December 2021 and October 2022. For data analysis, Fisher's Exact Test, Chi-Square and Asymptotic General Independence Test were used. 100% of patients had skin irritation. As the treatment progressed and the radiodermatitis appeared or worsened, there was a tendency for the intensity of signs and symptoms to increase, such as: sensitivity, discomfort or pain, itching, burning and heat, dry and wet desquamation, which may have impacted the quality of life and reflected in other aspects, such as: shopping activities or outings (p=0.0020), social activities or leisure activities (p=0.0420). Radiodermatitis is a common condition that affects women with breast cancer undergoing radiotherapy, skin characteristics and quality of life of patients affected during this treatment.

Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

Asymptotic Behavior of Adversarial Training in Binary Linear Classification.

Adversarial training using empirical risk minimization (ERM) is the state-of-the-art method for defense against adversarial attacks, that is, against small additive adversarial perturbations applied to test data leading to misclassification. Despite being successful in practice, understanding the generalization properties of adversarial training in classification remains widely open. In this article, we take the first step in this direction by precisely characterizing the robustness of adversarial training in binary linear classification. Specifically, we consider the high-dimensional regime where the model dimension grows with the size of the training set at a constant ratio. Our results provide exact asymptotics for both standard and adversarial test errors under general lq -norm bounded perturbations ( q ≥ 1 ) in both discriminative binary models and generative Gaussian-mixture models with correlated features. We use our sharp error formulae to explain how the adversarial and standard errors depend upon the over-parameterization ratio, the data model, and the attack budget. Finally, by comparing with the robust Bayes estimator, our sharp asymptotics allow us to study the fundamental limits of adversarial training.

Interior of the Integral of a Set-Valued Mapping and Problems with a Linear Control System

The dependence of the radius of a ball centered at zero inscribed in the values of the integral of a set-valued mapping on the upper integration limit is studied. For some types of integrals, exact asymptotics of the radius with respect to the upper limit are found when the upper limit tends to zero. Examples of finding this radius are considered. The results obtained are used to derive new sufficient conditions for the uniformly continuous dependence of the minimum time and solution-point in the linear minimum time control problem on the initial data. We also consider applications in some algorithms with a reachability set of a linear control system

Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game

Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game

Extremes of reflecting Gaussian processes on discrete grid

For {X(t),t∈Gδ} a centered Gaussian process with variance σ2 and stationary increments on a discrete grid Gδ={0,δ,2δ,...}, where δ>0, we investigate the stationary reflected processQδ,X(t)=sups∈[t,∞)∩Gδ⁡(X(s)−X(t)−c(s−t)),t∈Gδ with c>0. We derive the exact asymptotics of P(supt∈[0,T]∩Gδ⁡Qδ,X(t)>u) and P(inft∈[0,T]∩Gδ⁡Qδ,X(t)>u), as u→∞, with T>0. It appears that φ=limu→∞⁡σ2(u)u determines the asymptotics, leading to three qualitatively different scenarios: φ=0, φ∈(0,∞) and φ=∞.

Cramér–Lundberg asymptotics for spectrally positive Markov additive processes

This paper studies the Cramér–Lundberg asymptotics of the ruin probability for a model in which the reserve level process is described by a spectrally-positive light-tailed Markov additive process. By applying a change-of-measure technique in combination with elements from Wiener-Hopf theory, the exact asymptotics of the ruin probability are expressed in terms of the model primitives. In addition a simulation algorithm of generalized Siegmund type is presented, under which the returned estimate of the ruin probability has bounded relative error. Numerical experiments show that, compared to direct estimation, this algorithm greatly reduces the number of runs required to achieve an estimate with a given accuracy. The experiments also reveal that our asymptotic results provide a good approximation of the ruin probability even for relatively small initial surplus levels.

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

. This article investigates the Parisian ruin probability for a class of Gaussian processes with power-asymmetric behavior of the variance near the unique optimal point. We derive the exact asymptotics as the initial capital tends to infinity and extend the previous result[8] to the case when the length of the Parisian interval is on the Pickands scale. As a primary application, we extend the recent result[13] on the many inputs proportional reinsurance fractional Brownian motion risk model to the Parisian ruin.

Fluctuations, bias, variance and ensemble of learners: exact asymptotics for convex losses in high-dimension *

From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the ‘double-descent’ phenomenon.

Noisy linear inverse problems under convex constraints: Exact risk asymptotics in high dimensions

Noisy linear inverse problems under convex constraints: Exact risk asymptotics in high dimensions

On the Convergence Rate in “The Exact Asymptotics” for Random Variables with a Stable Distribution

On the Convergence Rate in “The Exact Asymptotics” for Random Variables with a Stable Distribution

Leftward, rightward, and complete exit-time distributions of jump processes.

First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.

Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.

On the Convergence Rate in Precise Asymptotics

We study some aspects of estimation of the convergence rate in the so-called “exact asymptotics.” In particular, we obtain asymptotic expansions in powers of $\varepsilon$ of sums of the form $\sum_{n\ge 1} n^s\,\mathbf P(\xi_{\alpha}> \varepsilon n^{\delta})$, where a random variable $\xi_{\alpha}$ has a stable distribution with an exponent $\alpha\in (0, 2]$, $\delta>0$, $s\in \mathbf R$.

Oscillations of BV measures on unbounded nested fractals

Motivated by recent developments in the theory of bounded variation functions on unbounded nested fractals, this paper studies the exact asymptotics of functionals related to the total variation measure associated with unions of n -cells. The oscillatory behavior observed implies the non-uniqueness of BV measures in this setting.