Technical Report Series
For copies of the technical reports, or other enquiries, please contact mathstats@acadiau.ca.
Almudevar, A. (2001) A Stochastic Contraction Mapping Theorem
Abstract
A Stochastic Contraction Mapping Theorem Anthony Almudevar, Acadia University The submartingale convergence theorem is often described as a stochastic analogue to the bounded convergence theorem for monotone sequences of real numbers. We define an alternative process which is a stochastic analogue to an iterative contraction process. A general convergence theorem is presented, with an extension to multivariate processes. The convergence theorem functions in a manner similar to Kronecker's lemma. Given contraction properties imposed on the conditional expectations the convergence of a random sequence can be verified by establishing the convergence of a related random series. In the examples presented in this article these random series possess the martingale property. Examples of processes which satisfy this definition include the sample mean process of a sequence of martingale differences, multivariate linear least squares estimators and the Robbins-Monro stochastic approximation algorithm.
Revised: 23/01/2001
Lemire, D. (2002) Local Interpolation by High Resolution Subdivision Schemes
Abstract
Subdivision schemes are used to interpolate data samples locally. By using temporary placeholders on a dense grid, we improve one of the best known subdivision scheme (Deslauriers-Dubuc). Interpolated values require 2 steps to stabilize as they are first interpolated on a coarse scale through a tetradic filter and then on a finer scale using a dyadic filter. The interpolants are \( C^{1} \) and can be made to reproduce polynomials of degree 4 unlike regular subdivision schemes. These generalized interpolatory subdivision schemes have minimal support and no additional memory requirement.
Revised 25/02/2003