Nikolai Leonenko

Bio: Nikolai Leonenko is an academic researcher from Cardiff University. The author has contributed to research in topics: Random field & Estimator. The author has an hindex of 32, co-authored 269 publications receiving 4206 citations. Previous affiliations of Nikolai Leonenko include Steklov Mathematical Institute & Taras Shevchenko National University of Kyiv.

Statistical Analysis of Random Fields

Abstract: 1. Elements of the Theory of Random Fields.- 1.1 Basic concepts and notation.- 1.2 Homogeneous and isotropic random fields.- 1.3 Spectral properties of higher order moments of random fields.- 1.4 Some properties of the uniform distribution.- 1.5 Variances of integrals of random fields.- 1.6 Weak dependence conditions for random fields.- 1.7 A central limit theorem.- 1.8 Moment inequalities.- 1.9 Invariance principle.- 2. Limit Theorems for Functionals of Gaussian Fields.- 2.1 Variances of integrals of local Gaussian functionals.- 2.2 Reduction conditions for strongly dependent random fields.- 2.3 Central limit theorem for non-linear transformations of Gaussian fields.- 2.4 Approximation for distribution of geometric functional of Gaussian fields.- 2.5 Reduction conditions for weighted functionals.- 2.6 Reduction conditions for functionals depending on a parameter.- 2.7 Reduction conditions for measures of excess over a moving level.- 2.8 Reduction conditions for characteristics of the excess over a radial surface.- 2.9 Multiple stochastic integrals.- 2.10 Conditions for attraction of functionals of homogeneous isotropic Gaussian fields to semi-stable processes.- 3. Estimation of Mathematical Expectation.- 3.1 Asymptotic properties of the least squares estimators for linear regression coefficients.- 3.2 Consistency of the least squares estimate under non-linear parametrization.- 3.3 Asymptotic expansion of least squares estimators.- 3.4 Asymptotic normality and convergence of moments for least squares estimators.- 3.5 Consistency of the least moduli estimators.- 3.6 Asymptotic normality of the least moduli estimators.- 4. Estimation of the Correlation Function.- 4.1 Definition of estimators.- 4.2 Consistency.- 4.3 Asymptotic normality.- 4.4 Asymptotic normality. The case of a homogeneous isotropic field.- 4.5 Estimation by means of several independent sample functions.- 4.6 Confidence intervals.- References.- Comments.

A class of Rényi information estimators for multidimensional densities

Abstract: A class of estimators of the Renyi and Tsallis entropies of an unknown distribution f in R^m is presented. These estimators are based on the k-th nearest-neighbor distances computed from a sample of N i.i.d. vectors with distribution f. We show that entropies of any order q, including Shannon's entropy, can be estimated consistently with minimal assumptions on f. Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each.

Limit Theorems for Random Fields with Singular Spectrum

Abstract: 1 Second-Order Analysis of Random Fields 2 Limit Theorems for Non-Linear Transformations of Random Fields 3 Asymptotic Distributions of Geometric Functionals of Random Fields 4 Limit Theorems for Solutions of the Burgers' Equation with Random Data 5 Statistical Problems for Random Fields with Singular Spectrum Comments Bibliography Index

Spectral Analysis of Fractional Kinetic Equations with Random Data

Abstract: We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.

A new class of random vector entropy estimators and its applications in testing statistical hypotheses

Abstract: This paper proposes a new class of estimators of an unknown entropy of random vector Its asymptotic unbiasedness and consistency are proved Further, this class of estimators is used to build both goodness-of-fit and independence tests based on sample entropy A simulation study indicates that the test involving the proposed entropy estimate has higher power than other well-known competitors under heavy tailed alternatives which are frequently used in many financial applications

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: 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

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.