Expectations on fractal sets

Abstract

Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals - being expectations on unit hypercubes - is extended to a class of fractal "string-generated Cantor sets" (SCSs) embedded in unit hypercubes of arbitrary dimension. Motivated by laboratory studies on the distribution of brain synapses, these SCSs were designed for dimensional freedom - a suitable choice of generating string allows for fine-tuning the fractal dimension of the corresponding set. We also establish closed forms for certain statistical moments on SCSs, develop a precision algorithm for high embedding dimensions, and report various numerical results. The underlying numerical quadrature issues are in themselves quite challenging. © 2013 Elsevier Inc. All rights reserved.