The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for some non-necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov (cf. [11]) in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.

The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

In this paper a new nonparametric estimate of conditional quantiles is proposed, that avoids the problem of crossing quantile curves [calculated for various p 2 (0;1)]: The method uses an initial estimate of the conditional distribution function in a flrst step and solves the problem of inversion and monotonization with respect to p 2 (0;1) simultaneously. It is demonstrated that the new estimates are asymptotically normal distributed and asymptotically flrst order equivalent to quantile estimates obtained by local constant or local linear smoothing of the conditional distribution function. The performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the proposed method are presented.

Non-crossing nonparametric estimates of quantile curves

Recent interest in modern regression modelling has focused on extending available (mean) regression models by describing more general properties of the response distribution. An alternative approac...

/pdf/expectile-and-quantile-regression-david-and-goliath-vebui2ev1l.pdf

Expectile and quantile regression—David and Goliath?

Dual divergence estimators and tests: Robustness results

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t? 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean.

/pdf/some-universal-limits-for-nonhomogeneous-birth-and-death-1nsivtzxo8.pdf

Some universal limits for nonhomogeneous birth and death processes

Given a scalar random variable Y and a random vector X defined on the same probability space, the conditional distribution of Y given X can be represented by either the conditional distribution function or the conditional quantile function. To these equivalent representations correspond two alternative approaches to estimation. One approach, distributional regression (DR), is based on direct estimation of the conditional distribution function; the other approach, quantile regression (QR), is instead based on direct estimation of the conditional quantile function. Indirect estimates of the conditional quantile function and the conditional distribution function may then be obtained by inverting the direct estimates obtained from either approach. Despite the growing attention to the DR approach, and the vast literature on the QR approach, the link between the two approaches has not been explored in detail. The aim of this paper is to fill-in this gap by providing a better understanding of the relative performance of the two approaches, both asymptotically and in finite samples, under the linear location model and certain types of heteroskedastic location-scale models.

/pdf/distributional-vs-quantile-regression-4olc51zd48.pdf

Distributional vs. Quantile Regression

https://hal.archives-ouvertes.fr/hal-00555139/document

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

http://www.eief.it/files/2010/07/leorato-peracchi-tanase-2010.pdf

Asymptotically efficient estimation of the conditional expected shortfall

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in Rn. The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.

The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.

Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R3 with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.

/pdf/minimal-cyclic-random-motion-in-rn-and-hyper-bessel-3lvmc24lht.pdf

Samantha Leorato

Papers

Some universal limits for nonhomogeneous birth and death processes

Distributional vs. Quantile Regression

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

Asymptotically efficient estimation of the conditional expected shortfall

Minimal cyclic random motion in Rn and hyper-bessel functions