The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the vali...

Safety cases are now regularly used to communicate the argument about the achievement of acceptable levels of safety for safety critical systems. Increasingly, safety standards such as Defence Standard 00-56 require the scope of the safety case to cover not only the development of the system, but also operating and maintenance through life, including decommissioning. This involves two dimension...

We study the behaviour of Yang-Mills theory under the inclusion of gravity. In the weakgravity limit, the running gauge coupling receives no contribution from the gravitational sector, if all symmetries are preserved. This holds true with and without cosmological constant. We also show that asymptotic freedom persists in general field-theory-based gravity scenarios including gravitational shiel...

Sadao Ikeda University of North Carolina Institute of Statistics~oSeries No. 455 Decemb.er,J..965 Con'tents Summary and 5.ntroduction 1. Definition of four types of asymptotic equivalence 2. Pr~erties of four types of asymptotic equivalence 3. Asymptotic independence of a set of random variables 4. Some classes of transformations of random variables pteserving the asymptotic equivalence in the ...

We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented. 1. Asymptotic summation Consider a power series ∞ ∑

On arithmetic and asymptotic properties of up-down numbers. We prove that the set of all up-down numbers C(σ) can be expressed by a single universal polynomial Φ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Φ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ), for fix...

The first problem studied in the paper is relations between the Kolmogorov n-width of a compact and its projections. The second problem is: to what extent the width can change if we enlarge the space. 2000 Mathematics Subject Classification: 46B20, 41A46, 41A65

We consider the second-order asymptotic properties of‎  ‎the bootstrap of L_1 regression estimators by looking at‎ ‎the difference between the L_1 estimator and ‎its first-order approximation‎, ‎where the latter‎ ‎is the minimizer of a quadratic approximation to the‎ ‎L_1 objective function‎. ‎It is shown that the bootstrap ‎distribution of the normed difference does not converge‎ ‎(eit...

Let {Xn ,n=>1} be a strictly stationary sequence of negatively associated random variables, with common distribution function F. In this paper, we consider the estimation of the two-dimensional distribution function of (X1, Xk+1) for fixed $K /in N$ based on kernel type estimators. We introduce asymptotic normality and properties and moments. From these we derive the optimal bandwidth...