The structure of isomorphisms of discrete memoryless correlated sources

Summary For the class of discrete memoryless stationary correlated sources with non-singular dependence and maximal correlation less than 1, it is shown that two such sources can be isomorphic only if there exists a single letter isomorphism between them, and they can have only trivial isomorphisms.     

Isomorphisms of certain correlated probability spaces and of discrete memoryless correlated sources

Summary In an earlier paper [5], we defined a sufficient set of invariants for the isomorphy of discrete memoryless correlated sources with maximal correlation <1. Using the structure of isomorphisms of certain correlated probability spaces, we give here a sufficient set of invariants also for the case of maximal correlation equal to 1. We show, in particular, that two discrete memoryless stationary correlated sources with maximal correlation 1 may be isomorphic in a non-trivial way.     

On variable length coding with side information for discrete memoryless sources

In this paper we consider the zero error variable length coding problem for discrete memoryless sources with side information at the decoder only. Using concepts of graph theory, we obtain a characterization of the minimum achievable rater for this coding problem. Furthermore, computable upper and lower bounds for the minimum achievable rater is given and a condition under which the upper and lower bounds are equal is found. Finally, this result is applied to some examples in which the minimum achievable rater can be evaluated.     

On the asymptotic normality of sequences of weak dependent random variables

The aim of this paper is to investigate the asymptotic normality for strong mixing sequences of random variables in the absense of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences. The class of applications include filters of certain Gaussian sequences.Supported in part by an NSF grant, cost-sharing from the University of Cincinnati, and a Taft research grant.     

Limits of nonlinear weak asymptotic methods

We propose a mathematical limit of $L^1$-stable weak asymptotic methods. A family of $L^1$-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.     

On asymptotic solutions of the problems of moving sources of diffusion and oscillations

Asymptotic expansions are proposed for the solutions of the problems of moving sources of diffusion and high-frequency oscillations in the presence of a flow. The intensity of each of the sources is a given arbitrary function of time.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 6–13, 1989.     

The weak asymptotic equivalence and the generalized inverse

In this paper, we discuss the relationship between the weak asymptotic equivalence relation and the generalized inverse in the class \({\mathcal{A}} \) of all nondecreasing and unbounded functions, defined and positive on a half-axis [a,+∞) (a > 0). In the main theorem, we prove a proper characterization of the functional class \({{ORV} \cap \mathcal{A}} \), where ORV is the class of all \({\mathcal{O}} \)-regularly varying functions (in the sense of Karamata).     

Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.     

A weak condition of globally asymptotic stability for neural networks

In this work we consider a general class of continuous activation functions which may be neither bounded nor differentiable; however, many sigmoidal functions are included as special cases. With this class of activation functions we give a result on asymptotic stability for neural networks under a weak condition of nonnegative definiteness. Then we show that differentiability is a condition for its exponential stability.     

Higher-order spectral analysis and weak asymptotic stability of convex processes

This paper deals with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process. We provide necessary and sufficient conditions for weak asymptotic stability, and obtain sharp estimates for the asymptotic null-controllability set. These estimates involve not only standard, but also higher-order spectral information on the convex process and its adjoint.     

On asymptotic contractions

A constructive generalization of the contraction mapping principle to the case of mappings with arbitrary rate of contraction is given.     

On Bombieri's asymptotic sieve

We improve the error term in the Bombieri asymptotic sieve when the summation is restricted to integers having at most two prime factors. This results in a refined bilinear decomposition for the characteristic function of the primes that enables us to get a best possible estimate for the trigonometric polynomial over primes.     

On asymptotic Assouad-Nagata dimension

For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdimX of X coincides with the covering dimension dim(νLX) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X×R)=AN-asdimX+1. We note that the similar equality for Gromov's asymptotic dimension asdim generally fails to hold [A. Dranishnikov, Cohomological approach to asymptotic dimension, Preprint, 2006].Additionally we construct an injective map from the asymptotic cone without the basepoint to the sublinear Higson corona.     

On Kirk's asymptotic contractions

Recently Kirk introduced the notion of asymptotic contractions on a metric space and using ultrapower techniques he obtained an asymptotic version of the Boyd-Wong fixed point theorem. In this paper we extend this result and moreover, we give a constructive proof of it. Furthermore, we obtain a complete characterization of asymptotic contractions on a compact metric space. As a by-product we establish a separation theorem for upper semicontinuous functions satisfying some limit condition.     

On Bombieri's asymptotic sieve

If a sequence of non-negative real numbers has ``best possible' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum for . By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.

     

Reducing asymptotic bias of weak instrumental estimation using independently repeated cross-sectional information

In this paper, we consider the instrumental variable estimation (the two-stage least squares estimator and the limited information maximum likelihood estimator) using weak instruments in a repeated measurements or a panel data model. We show that independently repeated cross-sectional data can reduce the asymptotic bias of the instrumental variable estimation when instruments are weakly correlated with endogenous variables. When the number of repeated measurements tends to infinity, we can achieve consistent instrumental variable estimation with weak instruments.