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Information Center for Mathematical Science

PAC

Information Center for Mathematical Science

PAC

Convolution powers of complex functions on Z
Author Persi Diaconis (Stanford University)
Homepage Url http://www-stat.stanford.edu/~cgates/PERSI/year.html
Coauthors Laurent Saloff-Coste
Abstract Repeated convolution of a probability measure on Z leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thom´ee which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thom´ee). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on Z. If M is an hermitian contraction, an estimate for the off-diagonal entries of the powers Mn k of Mk = I −(I −M)k is obtained. This generalizes the Carne–Varopoulos Markov chain estimate.
Abstract Url http://www-stat.stanford.edu/~cgates/PERSI/papers/scho1.pdf