A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R?u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R?(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.     

Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Hüsler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.     

Asymptotic Properties of <Emphasis Type="Italic">QML</Emphasis> Estimation of Multivariate Periodic <Emphasis Type="Italic">CCC</Emphasis> − <Emphasis Type="Italic">GARCH</Emphasis> Models

In this paper, we explore some probabilistic and statistical properties of constant conditional correlation (CCC) multivariate periodic GARCH models (CCC ? PGARCH for short). These models which encompass some interesting classes having (locally) long memory property, play an outstanding role in modelling multivariate financial time series exhibiting certain heteroskedasticity. So, we give in the first part some basic structural properties of such models as conditions ensuring the existence of the strict stationary and geometric ergodic solution (in periodic sense). As a result, it is shown that the moments of some positive order for strictly stationary solution of CCC ? PGARCH models are finite.Upon this finding, we focus in the second part on the quasi-maximum likelihood (QML) estimator for estimating the unknown parameters involved in the models. So we establish strong consistency and asymptotic normality (CAN) of CCC ? PGARCH models.     

Kulish–Sklyanin-type models: Integrability and reductions

We start with a Riemann–Hilbert problem (RHP) related to BD.I-type symmetric spaces SO(2r + 1)/S(O(2r ? 2s+1) ? O(2s)), s ≥ 1. We consider two RHPs: the first is formulated on the real axis R in the complex-λ plane; the second, on R ? iR. The first RHP for s = 1 allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrödinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.     

Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ?_φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θ?_φ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θ?_φ(x) - Θ_φ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.     

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Given a class \(\mathcal{F(\theta)}\) of differential equations with arbitrary element θ, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member \(f\in\mathcal{F(\theta)}\) the structure of its Lie symmetry group G _f , conditional symmetry Q _f and conservation law \(\mathop {\rm CL}\nolimits _{f}\) under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u _tt =(g(x)H(u)u _x ) _x +h(x)K(u)u _x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

On reliability analysis of consecutive k-out-of-n systems with arbitrarily dependent components

In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n?r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T_r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable.     

Mass distributions of two-dimensional extreme-value copulas and related results

Working with Markov kernels (conditional distributions) and right-hand derivatives D ⁺ A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) C _A distribute mass. Underlining the usefulness of working directly with D ⁺ A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of C _A concentrates its mass on the graphs of some increasing homeomorphisms f _t , we determine which EVC assigns maximum mass to the union of the graphs of \(f_{t_{1}},\ldots ,f_{t_{N}}\), derive the absolutely continuous component of an arbitrary EVC C _A and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs C _A which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel \(K_{C_{A}}(x,\cdot )\) of C _A have full support [0,1] for every x∈[0,1].     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

Conjugate variables in analytic number theory. Phase space and Lagrangian manifolds

For an arithmetic semigroup (G, ?), we define entropy as a function on a naturally defined continuous semigroup ? containing G. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (G, ?).     

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z _n: C[0, t] → R ⁿ⁺¹ by \({Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)\), where a ∈ C[0, t], h ∈ L ₂[0, t], and 0 < t ₁ <... < t _n ≤ t is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z _n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t].     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models

This paper establishes consistency and asymptotic normality of the generalized quasi-maximum likelihood estimate (GQMLE) for a general class of periodic conditionally heteroskedastic time series models (PCH). In this class of models, the volatility is expressed as a measurable function of the infinite past of the observed process with periodically time-varying parameters, while the innovation is an independent and periodically distributed sequence. In contrast with the aperiodic case, the proposed GQMLE is rather based on S instrumental density functions where S is the period of the model while the corresponding asymptotic variance is in a “sandwich” form. Application to the periodic asymmetric power GARCH model is given. Moreover, we also discuss how to apply the GQMLE to the prediction of power problem in a one-step framework and to PCH models with complex periodic patterns such as high frequency seasonality and non-integer seasonality.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

Tail dimension reduction for extreme quantile estimation

In a regression context where a response variable Y ∈ ? is recorded with a covariate X ∈ ? ^p , two situations can occur simultaneously: (a) we are interested in the tail of the conditional distribution and not on the central part of the distribution and (b) the number p of regressors is large. To our knowledge, these two situations have only been considered separately in the literature. The aim of this paper is to propose a new dimension reduction approach adapted to the tail of the distribution in order to propose an efficient conditional extreme quantile estimator when the dimension p is large. The results are illustrated on simulated data and on a real dataset.     

Nonasymptotic approach to Bayesian semiparametric inference

The classical semiparametric Bernstein–von Mises (BvM) results is reconsidered in a non-classical setup allowing finite samples and model misspecication. We obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the target parameter and in the dimension of sieve approximation of the nuisance parameter. This helps to identify the so called critical dimension p_n of the sieve approximation of the full parameter for which the BvM result is applicable. If the bias induced by sieve approximation is small and dimension of sieve approximation is smaller then critical dimension than the BvM result is valid. In the important i.i.d. and regression cases, we show that the condition “p_n²q/n is small”, where q is the dimension of the target parameter and n is the sample size, leads to the BvM result under general assumptions on the model.     

Bernoulli Trials of Fixed Parity,Random and Randomly Oriented Graphs

Suppose you can colour n biased coins with n colours, all coins having the same bias. It is forbidden to colour both sides of a coin with the same colour, but all other colourings are allowed. Let X be the number of different colours after a toss of the coins. We present a method to obtain an upper bound on a median of X. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs of fixed graphs. It turns out that there are parity restrictions on the random variables that are under consideration. Hence, in order to present our result, we introduce a class of Bernoulli random variables whose total number of successes is of fixed parity and are closely related to Poisson trials conditional on the event that their outcomes have fixed parity.     

Approximate solution of a parabolic equation with the use of a rational approximation to the operator exponential

For the abstract parabolic equation \(\dot x = Bx + bv\left( t \right)\) with an unbounded self-adjoint operator B, where b is a vector and v(t) is a scalar function, we suggest a solution method based on the evaluation of some rational function of the operator B. We obtain a priori estimates of the approximation error for the output function y(t) = <x(t), l>, where l is a given vector. The results of a numerical experiment for the inhomogeneous heat equation are presented.