Nonnormal Limit Research Articles

Distributional hyperspace-convergence of Argmin-sets in convex 𝑀-estimation

In M M -estimation we consider the sets of all minimizing points of convex empirical criterion functions. These sets are random closed sets. We derive distributional convergence in the hyperspace of all closed subsets of the real line endowed with the Fell-topology. As a special case single minimizing points converge in distribution in the classical sense. In contrast to the literature so far, unusual rates of convergence and non-normal limits emerge, which go far beyond the square-root asymptotic normality. Moreover, our theory can be applied to the sets of zero-estimators.

Asymptotic normality for -dependent and constrained -statistics, with applications to pattern matching in random strings and permutations

AbstractWe study (asymmetric) $U$ -statistics based on a stationary sequence of $m$ -dependent variables; moreover, we consider constrained $U$ -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version.Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization.The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.

The metric space of limit laws for $q$-hook formulas

Billey-Konvalinka-Swanson studied the asymptotic distribution of the coefficients of Stanley's \(q\)-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's \(q\)-hook-content formula related to semistandard tableaux and \(q\)-hook length formulas of Björner-Wachs related to linear extensions of labeled forests. We show that, while their coefficients are "generically" asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded \(2\)-norm. The closure of these distributions in the Lévy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.Mathematics Subject Classifications: 05A16 (Primary), 60C05, 60F05 (Secondary)Keywords: Hook length, \(q\)-analogues, major index, semistandard tableaux, plane partitions, forests, asymptotic normality, limit laws, Irwin-Hall distribution

Haezendonck–Goovaerts risk measure with a heavy tailed loss

Recently Haezendonck–Goovaerts (H–G) risk measure has received much attention in (re)insurance and portfolio management. Some nonparametric inferences have been proposed in the literature. When the loss variable does not have enough moments, which depends on the involved Young function, the nonparametric estimator in Ahn and Shyamalkumar (2014) has a nonnormal limit, which challenges interval estimation. Motivated by the fact that many loss variables in insurance and finance could have a heavier tail such as an infinite variance, this paper proposes a new estimator which estimates the tail by extreme value theory and the middle part nonparametrically. It turns out that the proposed new estimator always has a normal limit regardless of the tail heaviness of the loss variable. Hence an interval with asymptotically correct confidence level can be obtained easily either by the normal approximation method via estimating the asymptotic variance or by a bootstrap method. A simulation study and real data analysis confirm the effectiveness of the proposed new inference procedure for estimating the H–G risk measure.

Reliability index for non-normal distributions of limit state functions

Reliability analysis is a probabilistic approach to determine a safety level of a system. Reliability is defined as a probability of a system (or a structure, in structural engineering) to functionally perform under given conditions. In the 1960s, Basler defined the reliability index as a measure to elucidate the safety level of the system, which until today is a commonly used parameter. However, the reliability index has been formulated based on the pivotal assumption which assumed that the considered limit state function is normally distributed. Nevertheless, it is not guaranteed that the limit state function of systems follow as normal distributions; therefore, there is a need to define a new reliability index for no-normal distributions. The main contribution of this paper is to define a sophisticated reliability index for limit state functions which their distributions are non-normal. To do so, the new definition of reliability index is introduced for non-normal limit state functions according to the probability functions which are calculated based on the convolution theory. Eventually, as the state of the art, this paper introduces a simplified method to calculate the reliability index for non-normal distributions. The simplified method is developed to generate non-normal limit state in terms of normal distributions using series of Gaussian functions.

Asymptotics of Mean-Field O(N) Models

We study mean-field classical $N$-vector models, for integers $N\ge 2$. We use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in \cite{KM} but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.

Limit theorems in the imitative monomer-dimer mean-field model via Stein’s method

We consider the imitative monomer-dimer model on the complete graph introduced in the work of Alberici et al. [J. Math. Phys. 55, 063301-1–063301-27 (2014)]. It was shown that this model is described by the monomer density and has a phase transition along certain coexistence curve, where the monomer and dimer phases coexist. More recently, it was understood [D. Alberici et al., Commun. Math. Phys. (published online, 2016)] that the monomer density exhibits the central limit theorem away from the coexistence curve and enjoys a non-normal limit theorem at criticality with normalized exponent 3/4. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein’s method of exchangeable pairs. Our approach recovers what were established in the work of Alberici et al. [Commun. Math. Phys. (published online, 2016)] and furthermore allows to obtain the conditional central limit theorems along the coexistence curve. In all these results, the Berry-Esseen inequalities for the Kolmogorov-Smirnov distance are given.

Convergence of trimmed Lévy processes to trimmed stable random variables at [formula omitted

Let (r,s)Xt be the Lévy process Xt with r largest jumps and s smallest jumps up till time t deleted and let (r)X˜t be Xt with r largest jumps in modulus up till time t deleted. We show that ((r,s)Xt−at)/bt or ((r)X˜t−at)/bt converges to a proper nondegenerate nonnormal limit distribution as t↓0 if and only if (Xt−at)/bt converges as t↓0 to an α-stable random variable, with 0<α<2, where at and bt>0 are nonstochastic functions in t. Together with the asymptotic normality case treated in Fan (2015) [7], this completes the domain of attraction problem for trimmed Lévy processes at 0.

Edgeworth–Cornish–Fisher–Hill–Davis expansions for normal and non-normal limits via Bell polynomials

Cornish and Fisher gave expansions for the distribution and quantiles of asymptotically normal random variables whose cumulants behaved like those of a sample mean. This was extended by Hill and Davis to the case, where the asymptotic distribution need not be normal. Their results are cumbersome as they involve partition theory. We overcome this using Bell polynomials. The three basic expansions (for the distribution and its derivatives, for the inverse of the quantile, and for the quantile) involve three sets of polynomials. We give new ways of obtaining these from each other. The Edgeworth expansions for the distribution and density rest on the Charlier expansion. We give an elegant form of these as linear combinations of generalized Hermite polynomials, using Bell polynomials.

Pólya Urns Via the Contraction Method

We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

Limit theorems for iteration stable tessellations

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^{d}$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

Asymptotics of the Mean-Field Heisenberg Model

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Stein's method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.

Normal limits, nonnormal limits, and the bootstrap for quantiles of dependent data

We will show under very weak conditions on differentiability and dependence that the central limit theorem for quantiles holds and that the block bootstrap is weakly consistent. Under slightly stronger conditions, the bootstrap is strongly consistent. Without the differentiability condition, quantiles might have a nonnormal asymptotic distribution and the bootstrap might fail.

Fluctuations of eigenvalues for random Toeplitz and related matrices

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real random variables such that $$ \mathbb{E}[a_{j}]=0, \mathbb{E} [|a_{j}|^{2}]=1 \mathrm{for}\, j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty \mathrm{for} k\geq 3.$$ Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree at least 2. Without this assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hermitian Toeplitz and symmetric Hankel matrices. Since Toeplitz matrices are quite different from Wigner and Wishart matrices, our results enrich this topic.

Confidence sets for split points in decision trees

We investigate the problem of finding confidence sets for split points in decision trees (CART). Our main results establish the asymptotic distribution of the least squares estimators and some associated residual sum of squares statistics in a binary decision tree approximation to a smooth regression curve. Cube-root asymptotics with nonnormal limit distributions are involved. We study various confidence sets for the split point, one calibrated using the subsampling bootstrap, and others calibrated using plug-in estimates of some nuisance parameters. The performance of the confidence sets is assessed in a simulation study. A motivation for developing such confidence sets comes from the problem of phosphorus pollution in the Everglades. Ecologists have suggested that split points provide a phosphorus threshold at which biological imbalance occurs, and the lower endpoint of the confidence set may be interpreted as a level that is protective of the ecosystem. This is illustrated using data from a Duke University Wetlands Center phosphorus dosing study in the Everglades.

Estimating optimal step-function approximations to instantaneous hazard rates

We investigate the problem of estimating the best binary decision tree approximation to the baseline hazard function in the Cox proportional hazards model. Our motivation is to find an effective way of condensing key functional information in the baseline hazard into a small number of estimable parameters. The parameters consist of a threshold and two hazard levels, one to the left of the threshold and one to the right, defined in terms of the best L 2 approximation to the nonparametric baseline hazard function. Estimators of these parameters are introduced and shown to converge at cube-root rate to a non-normal limit distribution. Two alternate ways of constructing confidence intervals for the threshold are compared. Results from a simulation study and an example concerning a threshold for the age of onset of schizophrenia in a large cohort study are discussed.

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett. 36, 401–413) as a robustified L ∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett. 36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O P ( n −1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.

Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

We characterize all limit laws of the quicksort-type random variables defined recursively by ${\cal L}(X_n)= {\cal L}(X_{I_n}+X^*_{n-1-I_n}+T_n)$ when the toll function Tn varies and satisfies general conditions, where (Xn), (Xn*), (In, Tn) are independent, In is uniformly distributed over {0, . . .,n-1}, and ${\cal L}(X_n)={\cal L}(X_n^\ast)$. When the toll function Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly $\limsup_{n\rightarrow\infty}\log E(T_n)/\log n\le 1/2$), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on a broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc.

The sample ACF of a simple bilinear process

We consider a simple bilinear process X t = aX t−1 + bX t−1 Z t−1 + Z t , where ( Z t ) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207–248) that X t has a distribution with regularly varying tails of index α>0 provided the equation E| a+ bZ 1| u =1 has the solution u= α. We study the limit behaviour of the sample autocorrelations and autocovariances of this heavy-tailed non-linear process. Of particular interest is the case when α<4. If α∈(0,2) we prove that the sample autocorrelations converge to non-degenerate limits. If α∈(2,4) we prove joint weak convergence of the sample autocorrelations and autocovariances to non-normal limits.

Convergence in Distribution for Best-Fit Decreasing

Consider independent random variables $X_1 , \ldots ,X_n $ uniformly distributed over $[0,1]$, and denote by $B_n $ the number of bins needed to pack items of these sizes using the best-fit decreasing algorithm. We prove that the random variable $n^{{{ - 1} / 2}} $ converges in distribution to a nonnormal limit. The method consists of showing that the patterns created by the algorithm exhibit some kind of convergence.