Maximal inequalities via bracketing with adaptive truncation

The paper provides a recursive interpretation for the technique known as bracketing with adaptive truncation. By way of illustration, a simple bound is derived for the expected value of the supremum of an empirical process, thereby leading to a simpler derivation of a functional central limit theorem due to Ossiander. The recursive method is also abstracted into a framework that consists of only a small number of assumptions about processes and functionals indexed by sets of functions. In particular, the details of the underlying probability model are condensed into a single inequality involving finite sets of functions. A functional central limit theorem of Doukhan, Massart and Rio, for empirical processes defined by absolutely regular sequences, motivates the generalization.     

Asymptotics of weighted empirical processes of linear fields with long-range dependence

We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform-topology to a degenerated process of the form fZ, where Z is a standard normal random variable and f is the marginal probability density of the underlying random field.     

Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower and upper bounds for the sums of subexponential claims in this model are given. The proof method is based on the results of the total claim amount under subexponential class.     

Asymptotic behavior of the empirical conditional value-at-risk

We study asymptotic behavior of the empirical conditional value-at-risk (CVaR). In particular, the Berry–Essen bound, the law of iterated logarithm, the moderate deviation principle and the large deviation principle for the empirical CVaR are obtained. We also give some numerical examples.     

Lower classes of the Riemann–Liouville process

Let be a Riemann–Liouville process with index H>0. We characterize the lower classes of its sup-norm statistic by a unique integral test and thus measure the influence of the non-stationarity of increments.     

Une inégalité de Bennett pour les maxima de processus empiriquesA Bennet type inequality for maxima of empirical processes

We give new constants in Talagrand's concentration inequality for maxima of empirical processes. Our approach is based on the Herbst method. The improvement we get concerns the constant in the variance factor, which is the one conjectured by Massart.     

Subsampling tests for variance changes in the presence of autoregressive parameter shifts

In this paper, we consider the problem of testing for variance changes in the linear autoregressive processes including AR(p) processes when there are autoregressive parameter shifts. In performing a test, we employ the conventional residual CUSUM of squares test (RCUSQ) statistic. The RCUSQ test is based on the subsampling method introduced by Jach and Kokoszka (2004) [16] to eliminate the influence caused by autoregressive parameter shifts. It is shown that under regularity conditions, the test statistic behaves asymptotically the function of a standard Brownian bridge. We establish the asymptotic validity of this method and assess its performance both theoretically and numerically.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Scaling analysis of multiple-try MCMC methods

Multiple-try methods are extensions of the Metropolis algorithm in which the next state of the Markov chain is selected among a pool of proposals. These techniques have witnessed a recent surge of interest because they lend themselves easily to parallel implementations. We consider extended versions of these methods in which some dependence structure is introduced in the proposal set, extending earlier work by Craiu and Lemieux (2007).We show that the speed of the algorithm increases with the number of candidates in the proposal pool and that the increase in speed is favored by the introduction of dependence among the proposals. A novel version of the hit-and-run algorithm with multiple proposals appears to be very successful.     

Perturbed and non-perturbed Brownian taboo processes

In this paper we study the Brownian taboo process, which is a version of Brownian motion conditioned to stay within a finite interval, and the α-perturbed Brownian taboo process, which is an analogous version of an α-perturbed Brownian motion.We are particularly interested in the asymptotic behaviour of the supremum of the taboo process, and our main results give integral tests for upper and lower functions of the supremum as t→∞. In the Brownian case these include extensions of recent results in Lambert [4], but are proved in a quite different way.     

On quantitative versions of theorems due to F.E. Browder and R. Wittmann

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.     

Remarks on t-transformations of measures and convolutions

A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators” which act on the Interacting Fock Spaces.     

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models.The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.     

Large deviations for increasing sequences on the plane

We prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among Poisson points on the plane. The rate function for lower tail deviations is derived from a 1977 result of Logan and Shepp about Young diagrams of random permutations. For the upper tail we use a coupling with Hammersley's particle process and convex-analytic techniques. Along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric Hammersley's process. Received: 22 July 1997 / Revised version: 23 March 1998     

Correction added in proof

Summary We extend Sanov's theorem on i.i.d. large deviations to independent but not identically distributed random variables, and study the generalization of relative entropy that appears as the rate function.     

A remark on the law of the logarithm for weighted sums of random variables with multidimensional indices

In this work, a sharp upper bound on the law of the logarithm for the weighted sums of random variables with multidimensional indices is obtained. The main result improves the result in [Li, Rao and Wang, 1995. On strong law of large numbers and the law of the logarithm for weighted sums of independent random variables with multidimensional indices. J. Multivariate Anal. 52, 181–198], partly.     

The pentagram map and Y-patterns

The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its “shortest” diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

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.     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.