Limit distributions of U-statistics resampled by symmetric stable laws

Summary If (Y _i) and (V _i) are independent random sequences such thatY _i are i.i.d. random variables belonging to the normal domain of attraction of a symmetric -stable law, 0<<2, andV _i are i.i.d. random variables, then the limit distributions of U-statistics , coincide with the probability laws of multiple stochastic integralsX ^d f = ... f (t ₁, ... ,t _d)dX(t _d) with respect to a symmetric -stable processX(t).The research was originated during author's visit at ORIE, Cornell University     

Limit theorems for some polynomial statistics of the poisson process

The central limit theorem and the theorem on large deviations for the functionals of the Poisson random process are proved. The formulas for cumulants of multiple stochastic integrals (m.s.i.) with respect to the Poisson process are obtained. The m.s.i. may be considered as anU-statistics arising in queueing theory as well as a generalization of the well-known Poisson shot-noise process, having wide applications.     

Central limit theorems for multiple stochastic integrals and Malliavin calculus

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.     

U-statistics in danach spaces

U-statistics in Banach spaces are considered and thoroughly investigated. The martingale structure, estimates of moments, the law of large numbers, the central limit theorem, the invariance principle, estimates of the rate of convergence, and large deviations are established     

Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

On a continuous time stochastic approximation problem

This paper is concerned with a continuous time stochastic approximation/optimization problem. The algorithm is given by a pair of differential-integral equations. Our main effort is to derive the asymptotic properties of the algorithm. It is shown that ast , a suitably normalized sequence of the estimation error,t(¯x _tr–) is equivalent to a scaled sequence of the random noise process, namely, (1/t) ₀ ^tr _sds. Consequently, the asymptotic normality is obtained via a functional invariance theorem, and the asymptotic covariance matrix is shown to be the optimal one. As a result, the algorithm is asymptotically efficient.Supported in part by the National Science Foundation, and in part by Wayne State University.Supported in part by Wayne State University through a research assistantship.     

Unbiased estimates for gradients of stochastic network performance measures

Three classes of stochastic networks and their performance measures are considered. These performance measures are defined as the expected value of some random variables and cannot normally be obtained analytically as functions of network parameters in a closed form. We give similar representations for the random variables to provide a useful way of analytical study of these functions and their gradients. The representations are used to obtain sufficient conditions for the gradient estimates to be unbiased. The conditions are rather general and usually met in simulation study of the stochastic networks. Applications of the results are discussed and some practical algorithms of calculating unbiased estimates of the gradients are also presented.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

Limit distributions of least squares estimators in linear regression models with vague concepts

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have lately been introduced (cf. [V. Krätschmer, Induktive Statistik auf Basis unscharfer Meßkonzepte am Beispiel linearer Regressionsmodelle, unpublished postdoctoral thesis, Faculty of Law and Economics of the University of Saarland, Saarbrücken, 2001; V. Krätschmer, Least squares estimation in linear regression models with vague concepts, Fuzzy Sets and Systems, accepted for publication]) to improve the empirical meaningfulness of the relationships between the involved items by a more sensitive attention to the problems of data measurement, in particular, the fundamental problem of adequacy. The parameters of such models are still real numbers, and a method of estimation can be applied which extends directly the ordinary least squares method. In another recent contribution (cf. [V. Krätschmer, Strong consistency of least squares estimation in linear regression models with vague concepts, J. Multivar. Anal., accepted for publication]) strong consistency and -consistency of this generalized least squares estimation have been shown. The aim of the paper is to complete these results by an investigation of the limit distributions of the estimators. It turns out that the classical results can be transferred, in some cases even asymptotic normality holds.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

On a “lack of memory” property

For two independent nonnegative random variablesX andY we say thatX is ageless relative toY if the conditional probability P[X> Y+x|X>Y] is defined and is equal to P[X>x] for allx>0. Suppose thatX is ageless relative to a nonlatticeY with P[Y=0]<P [Y<X]. We show that the only suchX is the exponential variable. As a corollary it follows that exponential variable is the only one which possesses the ageless property relative to a continuous variable. Research partially supported by NRC of Canada grants #A8057 and #T0500. Work partially completed while on leave at Division of Math. Stat., C.S.I.R.O., Australia.     

Limit theorems for stochastic measures of the accuracy of density estimators

Stochastic measures of the distance between a density f and its estimate f_n have been used to compare the accuracy of density estimators in Monte Carlo trials. The practice in the past has been to select a measure largely on the basis of its ease of computation, using only heuristic arguments to explain the large sample behaviour of the measure. Steele [11] has shown that these arguments can lead to incorrect conclusions. In the present paper we obtain limit theorems for the stochastic processes derived from stochastic measures, thereby explaining the large sample behaviour of the measures.     

On the asymptotics of numbers of observations in random regions determined by order statistics

In this paper, we consider random variables counting numbers of observations that fall into regions determined by extreme order statistics and Borel sets. We study multivariate asymptotic behavior of these random variables and express their joint limiting law in terms of independent multinomial and negative multinomial laws. First, we give our results for samples with deterministic size; next we explain how to generalize them to the case of randomly indexed samples.     

A general correlation inequality and the Almost Sure Local Limit Theorem for random sequences in the domain of attraction of a stable law

In the present paper we obtain a new correlation inequality and use it for the purpose of extending the theory of the Almost Sure Local Limit Theorem to the case of lattice random sequences in the domain of attraction of a stable law. In particular, we prove ASLLT in the case of the normal domain of attraction of α$\alpha$-stable law, α∈(1,2)$\alpha \in (1,2)$.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.     

High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus

We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy.