Tail Estimates Research Articles

Regularity of random attractors for non-autonomous stochastic discrete complex Ginzburg-Landau equations

In this paper, we consider the asymptotic behaviour of non-autonomous stochastic discrete complex Ginzburg-Landau equations with additive noise in weighted space . The existences of tempered random attractors for this equation in spaces and are proved respectively by tail estimates, which implies that the obtained -random attractor is compact and attracting in the topology of space. The main difficulty here is the lack of compactness on infinite lattices. To deal with this, we introduce a common embedding space of and and derive some tail-estimates of solutions.

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

Global attractor of the fractional damping wave equation on ℝ3

ABSTRACT The paper studies the well-posedness and the longtime behaviors of the fractional damping wave equation on with critical nonlinearity, we firstly prove the well-posedness of weak solutions via an established Strichartz estimate; Secondly, by the uniformly tail estimate and the energy identity, we prove that the corresponding semigroup possesses a global attractor in the energy space .

LIL for the Length of the Longest Increasing Subsequences

Let X1, X2, ⋯, Xn, ⋯ be a sequence of i.i.d. random variables uniformly distributed on [0; 1], and denote by Ln the length of the longest increasing subsequences of X1, X2, ⋯, Xn. Consider the poissonized version Hn based on Hammersley’s representation in the 2-dimensional space. A law of the iterated logarithm for Hn is established using the well-known subsequence method and Borel-Cantelli lemma. The key technical ingredients in the argument include superadditivity, increment independence and precise tail estimates for the Hn’s. The work was motivated by recent works due to Ledoux (J. Theoret. Probab.31, (2018)). It remains open to establish an analog for the Ln itself.

Probability inequalities and tail estimates for metric semigroups

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathscr{G}$ a Banach space.) Using our prior work [Ann. Prob. 2017] that extends the Hoffmann-Jorgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the "correct" universal constants in several cases, extending results in the Banach space literature by Johnson-Schechtman-Zinn [Ann. Prob. 1985], Hitczenko [Ann. Prob. 1994], and Hitczenko and Montgomery-Smith [Ann. Prob. 2001]. Our results also hold more generally, in a very primitive mathematical framework required to state them: metric semigroups $\mathscr{G}$. This includes all compact, discrete, or (connected) abelian Lie groups.

Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

We investigate the asymptotic behavior of the (relative) extrapolation error associated with some estimators of extreme quantiles based on extreme-value theory. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In constrast, the extrapolation error associated with Weissman estimator has a common behavior over the whole Frechet maximum domain of attraction. First order equivalents of the extrapolation error are then derived and their accuracy is illustrated numerically.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on $ (n+1) $-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the $ (n+1) $-dimensional unbounded thin domains collapse onto the $ n $-dimensional space $ \mathbb{R}^n $. Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.

Quenched tail estimate for the random walk in random scenery and in random layered conductance II

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.

Combined tail estimation using censored data and expert information

We study tail estimation in Pareto-like settings for datasets with a high percentage of randomly right-censored data, and where some expert information on the tail index is available for the censored observations. This setting arises for instance naturally for liability insurance claims, where actuarial experts build reserves based on the specificity of each open claim, which can be used to improve the estimation based on the already available data points from closed claims. Through an entropy-perturbed likelihood, we derive an explicit estimator and establish a close analogy with Bayesian methods. Embedded in an extreme value approach, asymptotic normality of the estimator is shown, and when the expert is clair-voyant, a simple combination formula can be deduced, bridging the classical statistical approach with the expert information. Following the aforementioned combination formula, a combination of quantile estimators can be naturally defined. In a simulation study, the estimator is shown to often outperform the Hill estimator for censored observations and recent Bayesian solutions, some of which require more information than usually available. Finally we perform a case study on a motor third-party liability insurance claim dataset, where Hill-type and quantile plots incorporate ultimate values into the estimation procedure in an intuitive manner.

Maximum entropy distribution with fractional moments for reliability analysis

An important task in reliability engineering is the approximation of the probability distribution function from limited sample data, both in terms of modeling of input random variables and response quantities from numerical simulations. Good estimation of the distribution tails is crucial, but challenging. In principle, the maximum entropy density with prescribed moments produces the least biased distribution, but it suffers from several drawbacks. Recently, maximum entropy with fractional moments has become popular as the fractions can be optimized, allowing a reduced number of moments to accurately characterize the distribution. However, the optimization problem is difficult to solve efficiently because the objective function is non-convex and non-continuous. This paper outlines a new method based on maximum entropy and fractional moments. Genetic algorithm is found to be a robust method for finding the global optimal solution for the fractional orders. The number of moments, which controls the bias-variance trade-off, is also optimized using the Akaike information criterion to avoid an overcomplex model. A new location variable is introduced, thus allowing negative data to be modeled. The proposed method is tested on several examples, including theoretical distributions, a truss system, and wind turbine loads. To investigate the bias and variance errors of the exceedance probabilities predicted by the proposed method, for each application, many datasets collected from the same population are analyzed. The speedup over Monte Carlo simulation is calculated from the variance error. The proposed method is shown to have very low bias errors and high speedups for the examples examined.

Weighted bounded mean oscillation applied to backward stochastic differential equations

We deduce conditional Lp-estimates for the variation of a solution of a BSDE. Both quadratic and sub-quadratic types of BSDEs are considered, and using the theory of weighted bounded mean oscillation we deduce new tail estimates for the solution (Y,Z) on subintervals of [0,T]. Some new results for the decoupling technique introduced in Geiss and Ylinen (2019) are obtained as well and some applications of the tail estimates are given.

Extreme significant wave height of tropical cyclone waves in the South China Sea

Abstract. Extreme significant wave heights are assessed in the South China Sea (SCS), as assessments of wave heights are crucial for coastal and offshore engineering. Two significant factors include the initial database and assessment method. The initial database is a basis for assessment, and the assessment method is used to extrapolate appropriate return-significant wave heights during a given period. In this study, a 40-year (1975–2014) hindcast of tropical cyclone waves is used to analyse the extreme significant wave height, employing the peak over threshold (POT) method with the generalized Pareto distribution (GPD) model. The peak exceedances over a sufficiently large value (i.e. threshold) are fitted; thus, the return-significant wave heights are highly dependent on the threshold. To determine a suitable threshold, the sensitivity of return-significant wave heights and the characteristics of tropical cyclone waves are studied. The sample distribution presents a separation that distinguishes the high sample from the low sample, and this separation is within the stable threshold range. Because the variation in return-significant wave heights in this range is generally small and the separation is objectively determined by the track and intensity of the tropical cyclone, the separation is selected as a suitable threshold for extracting the extreme sample in the tropical cyclone wave. The asymptotic tail approximation and estimation uncertainty show that the selection is reasonable.

Coalescence of geodesics in exactly solvable models of last passage percolation

Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R−c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (−k2/3, k2/3) and v4 = (k2/3, −k2/3), we establish the optimal tail estimate P(|v|1>Rk)≍R−2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187–3206 (2016)] who proved the corresponding lower bound.

Tail Index Estimation: Quantile Driven Threshold Selection

The selection of upper order statistics in tail estimation is notoriously difficult. Most methods are based on asymptotic arguments, like minimizing the asymptotic mse, that do not perform well in finite samples. Here we advance a data driven method that minimizes the maximum distance between the fitted Pareto type tail and the observed quantile. To analyse the finite sample properties of the metric we organize a horse race between the other methods. In most cases the finite sample based methods perform best. To demonstrate the economic relevance of choosing the proper methodology we use daily equity return data from the CRSP database and find economic relevant variation between the tail index estimates.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on (n + 1)-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the (n + 1)-dimensional unbounded thin domains collapse onto the n-dimensional space Rn. Here, the tail estimates are utilized to deal with the noncompactness of Sobolev embeddings on unbounded domains.

The tail that wags: differences in effective right tail coverage and estimates of wealth inequality

This paper focuses on the sensitivity of survey-based estimates of wealth inequality to the quality of the measurement of the upper tail of the distribution. Using data from the 2013 Survey of Consumer Finances (SCF), it develops a series of illustrative examples to highlight some of the problems in making comparisons of wealth inequality measures when there are specific defects in the measurement of the upper tail of the distribution. The results presented strongly indicate that in the absence of effective controls on the measurement of the upper tail of the wealth distribution, great caution should be the rule in the interpretation of most commonly used measures of wealth inequality from a given survey, comparison of such measures across the waves of the survey, and perhaps even more strongly, comparison across independently designed and managed surveys. A graphical decomposition of the 2013 SCF wealth distribution provides additional insight into the underlying cause of the sensitivity of the inequality measures. The paper concludes with a brief outline of a research program for improving the ability of surveys to provide more meaningful estimates of inequality measures.

Tailfindr: alignment-free poly(A) length measurement for Oxford Nanopore RNA and DNA sequencing.

Polyadenylation at the 3′-end is a major regulator of messenger RNA and its length is known to affect nuclear export, stability, and translation, among others. Only recently have strategies emerged that allow for genome-wide poly(A) length assessment. These methods identify genes connected to poly(A) tail measurements indirectly by short-read alignment to genetic 3′-ends. Concurrently, Oxford Nanopore Technologies (ONT) established full-length isoform-specific RNA sequencing containing the entire poly(A) tail. However, assessing poly(A) length through base-calling has so far not been possible due to the inability to resolve long homopolymeric stretches in ONT sequencing. Here we present tailfindr, an R package to estimate poly(A) tail length on ONT long-read sequencing data. tailfindr operates on unaligned, base-called data. It measures poly(A) tail length from both native RNA and DNA sequencing, which makes poly(A) tail studies by full-length cDNA approaches possible for the first time. We assess tailfindr’s performance across different poly(A) lengths, demonstrating that tailfindr is a versatile tool providing poly(A) tail estimates across a wide range of sequencing conditions.

Upper Semicontinuity of Global Attractors for Singularly Perturbed Plate Equations

In the paper, upper semicontinuity of global attractors of singularly perturbed plate equations on an unbounded domain with small positive parameter, is considered. Under suitable assumptions, the equations possess a family of global attractors in natural energy space, and the corresponding singular limit equation, i.e., the parabolic equation, possesses a global attractor, which can be naturally embedded into a compact set of the natural energy space. Using the idea of tail estimates, the author established the upper semicontinuity of the family of global attractors to the compact set in the natural energy space (even more regular space) with respect to the Hausdorff semidistance, as the perturbation parameter tends to zero. The results obtained are new and are not common in the existing literature.

The continuous Anderson hamiltonian in d ≤ 3

We construct the continuous Anderson hamiltonian on (−L,L)d driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension d≤3 and relies on the theory of regularity structures [13]: it yields a self-adjoint operator in L2((−L,L)d) with pure point spectrum. In d≥2, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for d=3 these estimates show that the eigenvalues do not have exponential moments.

On the missing log in upper tail estimates

In the late 1990s, Kim and Vu pioneered an inductive method for showing concentration of certain random variables X. Shortly afterwards, Janson and Ruciński developed an alternative inductive approach, which often gives comparable results for the upper tail P(X≥(1+ε)EX). In some cases, both methods yield upper tail estimates which are best possible up to a logarithmic factor in the exponent, but closing this narrow gap has remained a technical challenge. In this paper we present a BK-inequality based combinatorial sparsification idea that can recover this missing logarithmic term in the upper tail.As an illustration, we consider random subsets of the integers {1,…,n}, and prove sharp upper tail estimates for various objects of interest in additive combinatorics. Examples include the number of arithmetic progressions, Schur triples, additive quadruples, and (r,s)-sums.