ON EQUIVALENT DEFINITIONS OF THE CORRELATION DIMENSION FOR A PROBABILITY MEASURE

摘要

In mathematical physics, one sometimes has to deal with averages of the type M mu(T) = 1/T-n integral(xiless than or equal to T)d xi<(mu)over cap>(xi)(2), T > 0 where <(mu)over cap> is the Fourier transform of some probability Borel measure mu. We show that the asymptotic behavior of M mu is governed by the usual (upper and lower) correlation dimension of the measure mu. [References: 7]