Invariance Principle Under Self-Normalization for Nonidentically Distributed Random Variables

Let V _n ^–1 _n be the adaptive process of self-normalized partial sums S _k of independent random variables X _i , defined by linear interpolation between the points (V _k ²/V _n ²,S _k /V _n ), kn, where V _k ²= _ik X _i ². We prove that if the X _k 's are symmetric, V _n ^–1 _n converges weakly to the Brownian motion W in each Hölder space supporting W if and only if V _n ^–1 max _kn |X _k |=o _P (1). We give some partial extension to the non symmetric case.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

Nonparametric estimation of distributions with categorical and continuous data

In this paper we consider the problem of estimating an unknown joint distribution which is defined over mixed discrete and continuous variables. A nonparametric kernel approach is proposed with smoothing parameters obtained from the cross-validated minimization of the estimator's integrated squared error. We derive the rate of convergence of the cross-validated smoothing parameters to their ‘benchmark’ optimal values, and we also establish the asymptotic normality of the resulting nonparametric kernel density estimator. Monte Carlo simulations illustrate that the proposed estimator performs substantially better than the conventional nonparametric frequency estimator in a range of settings. The simulations also demonstrate that the proposed approach does not suffer from known limitations of the likelihood cross-validation method which breaks down with commonly used kernels when the continuous variables are drawn from fat-tailed distributions. An empirical application demonstrates that the proposed method can yield superior predictions relative to commonly used parametric models.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.     

On Solvability of Functional Equations Relating to Dynamical Systems with Two Generators

In this paper, some solvability problems for functional equations of the form

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.     

The Coordinate of the Singular Point of Generating Functions of Clusters in the High-Temperature Dynamics of Spin Lattice Systems with Axially Symmetric Interaction

We investigate generating functions for equipped trees composed of double bonds of two sorts on a hypercubic lattice of dimension d with built-in fragments. Rules for constructing these clusters are chosen to ensure the estimate for coefficients of power series in time for the longitudinal and transverse autocorrelation functions of the spin system with axially symmetric interaction. We derive a system of two equations for the tree-generating functions and an equation for the generating functions of chains leading from the root to a fragment in a tree using the Bethe approximation and under the condition that mainly bonds of one sort are taken into account. For the face-centered hypercubic lattice, we find the first terms of the 1/d expansion for the coordinate of the singular point of the generating function in both the anisotropic and the isotropic cases taking fragments in the forms of a triangle from four bonds and a four-fold bound pair into account. The obtained result is written in terms of ratios of lattice sums and is generalized to nuclear spin systems with dipole–dipole interaction. The theoretical value of the singular-point coordinate agrees well with the experimental value calculated from the tail of the absorption spectrum of the nuclear magnetic resonance in a barium fluoride monocrystal.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

Permanence for non-autonomous food chain systems with delay

In this paper we first establish a new general criterion for the permanence of Kolmogorov-type systems of nonautonomous functional differential equations. Then, as applications of this criterion we study the permanence of a class of n-species general nonautonomous food chain systems with delay and new sufficient condition are established.