This expository article states and proves four, concrete, projective, central limit theorems. The results are known or suspected to be true by experts who are familiar with the more general central limit theorem for convex bodies, and related theory. Here we consider only four types of high dimensional geometric objects: spheres, balls, cubes, and boundaries of cubes. Each is capable of transfo...

In this paper we survey some recent progress on the Gaussian approximation for nonstationary dependent structures via martingale methods. First, present general theorems involving projective conditions triangular arrays of random variables and then various applications rho-mixing alpha-dependent arrays, stationary sequences in a time scenery, application to quenched FCLT, linear statistics with...

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Fréchet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quo...

We address the problem of the continuum limit for a system of Hausdorr lattices (namely lattices of isolated points) approximating a topological space M. The correct framework is that of projective systems. The projective limit is a universal space from which M can be recovered as a quotient. We dualize the construction to approximate the algebra C(M) of continuous functions on M. In a companio...

It is known that a Fréchet space F can be realized as a projective limit of a sequence of Banach spaces Ei. The space Kc(F) of all compact, convex subsets of a Fréchet space, F, is realized as a projective limit of the semilinear metric spacesKc(E). Using the notion of Hukuhara derivative for maps with values inKc(F), we prove the local and global existence theorems for an initial value problem...

For a Riemann domain Í2, a connected complex manifold where n (n = dimension) globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function / in Í2, the "Riemann surface" Slf, a maximal holomorphic extension Riemann domain for /, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an enve...

Evaluators and researchers often have to deal with situations in which conventional research tools are impossible to use, either because of the characteristics of a population or unclear research variables. This paper presents a technique that succeeds in overcoming this kind of problem--a projective technique, but one that differs from the usual approach to projective techniques. The approach ...