Empirical minimization

We investigate the behavior of the empirical minimization algorithm using various methods. We first analyze it by comparing the empirical, random, structure and the original one on the class, either in an additive sense, via the uniform law of large numbers, or in a multiplicative sense, using isomorphic coordinate projections. We then show that a direct analysis of the empirical minimization algorithm yields a significantly better bound, and that the estimates we obtain are essentially sharp. The method of proof we use is based on Talagrand's concentration inequality for empirical processes. Research partially supported by NSF under award DMS-0434393. Research partially supported by the Australian Research Council Discovery Porject DP0343616.     

On weak convergence of random fields

We suggest simple and easily verifiable, yet general, conditions under which multi-parameter stochastic processes converge weakly to a continuous stochastic process. Connections to, and extensions of, R. Dudley’s results play an important role in our considerations, and we therefore discuss them in detail. As an illustration of general results, we consider multi-parameter stochastic processes that can be decomposed into differences of two coordinate-wise non-decreasing processes, in which case the aforementioned conditions become even simpler. To illustrate how the herein developed general approach can be used in specific situations, we present a detailed analysis of a two-parameter sequential empirical process.     

Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity.      

On infinitesimal σ-fields generated by random processes

It is proved that the infinitesimal look-ahead and look-back σ-fields of a random process disagree at atmost countably many time instants.     

Random weighted projections,random quadratic forms and random eigenvectors

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms. (2) The infinity norm of most unit eigenvectors of a random ±1 matrix is of order . (3) An estimate on the threshold for the local semi‐circle law which is tight up to a factor. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 792–821, 2015     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Empirical processes indexed by smooth functions

In this paper we derive a general invariance principle for empirical processes indexed by smooth functions. The method is applied to prove bounds for the convergence of the empirical distributions which might be useful to verify asymptotic normality of smooth statistical functionals. As one further application we get the convergence of the so-called empirical characteristic function process.     

Superbranching processes and projections of random Cantor sets

We study sequences (X ₀, X ₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X _n (m, i) are distributed as X _n and have certain properties of independence. We prove that, under appropriate conditions, X _n ^1/n almost surely and in L ¹, where =sup E(X _n )^1/n . Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1] ^d . We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant     

Iterations of random and deterministic functions

Letf be a probability generating function on [0, 1]. The convergence of its iteratesf _n to fixed points is studied in this paper. Results include rates forf andf ^-1. Also iterates of independent identically distributed stable processes are studied and a trichotomy based on the order of the stability is established. Dedicated to the memory of Professor K G Ramanathan     

Oblique projections and frames

We characterize those frames on a Hilbert space which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension of . We show that all frames with infinite excess and frame bounds are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames.

     

Linear systems and determinantal random point fields

In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on L²(0,∞) from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems (−A,B,C) with one-dimensional input and output spaces, there exists a Hankel operator Γ with kernel ?(x)(s+t)=Ce−(2x+s+t)AB such that gx(z)=det(I+(z−1)ΓΓ†) is the generating function of a determinantal random point field on (0,∞). The inverse scattering transform for the Zakharov-Shabat system involves a Gelfand-Levitan integral equation such that the trace of the diagonal of the solution gives . When A?0 is a finite matrix and B=C†, there exists a determinantal random point field such that the largest point has a generalised logistic distribution.     

Scaling exponents of random walks in random sceneries

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n with . Here, the value of the exponent is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d[0;2[, the exponent is . For the simple walk on some specific graphs, whose volume grows like n^d for d[1;2[, the exponent is =1−d/4. We build a null recurrent walk, for which without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is . In that setting and when the scenery is i.i.d. by levels, the same result holds with .     

Perturbed projections and subgradient projections for the multiple-sets split feasibility problem

We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.     

Densities for infinitely divisible random processes

Let {ξ_j(t), t ∈ [0, T]} j = 1, 2 be infinitely divisible processes with distinct Poisson components and no Gaussian components. Let X be the set of all real-valued functions on [0, T] which are not identically zero, and $\text{B}$ be the σ-ring generated by the cylinder sets of ξ_j(t), j = 1, 2. Let μ_j be the measure on $\text{B}$ induced by ξ_j(t).Necessary and sufficient conditions on the projective limits of the Levy-Khinchine spectral measures of the processes are found to make μ₂ ? μ₁, and a representation for the density $\text{dμ}{}_2\text{dμ}{}_1$ is obtained.     

Symmetries on random arrays and set-indexed processes

A processX on the setÑ of all finite subsetsJ ofN is said to be spreadable, if for all subsequencesp=(p ₁,p ₂,...) ofN, wherepJ={p _j ;jJ}. Spreadable processes are characterized in this paper by a representation formula, similar to those obtained by Aldous and Hoover for exchangeable arrays of r.v.'s. Our representation is equivalent to the statement that a process onÑ is spreadable, iff it can be extended to an exchangeable process indexed by all finite sequences of distinct elements fromN. The latter result may be regarded as a multivariate extension of a theorem by Ryll-Nardzewski, stating that, for infinite sequences of r.v.'s, the notions of exchangeability and spreadability are equivalent.     

On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes

In this paper we define and analyze convergence of the geometric random walks, which are certain random walks on vector spaces over finite fields. We show that the behavior of such walks is given by certain random matroid processes. In particular, the mixing time is given by the expected stopping time, and the cutoff is equivalent to sharp threshold. We also discuss some random geometric random walks as well as some examples and symmetric cases.     

Optional and predictable projections of set-valued measurable processes

Abstract. In this paper,the optional and predictable projections of set-valued measurable pro-cesses are studied. The existence and uniqueness of optional and predictable projections of set-valued measurable processes are proved under proper circumstances.