Successive Normalization of Rectangular Arrays: Rates of Convergence

In this note we illustrate with examples and heuristic mathematics, figures that are given throughout the earlier paper by the same authors [1]. Thus, we deal with successive iterations applied to rectangular arrays of numbers, where to avoid technical difficulties an array has at least three rows and at least three columns. Without loss, an iteration begins with operations on columns: first subtract the mean of each column; then divide by its standard deviation. The iteration continues with the same two operations done successively for rows. These four operations applied in sequence completes one iteration. One then iterates again, and again, and again,.... In [1] it was argued that if arrays are made up of real numbers, then the set for which convergence of these successive iterations fails has Lebesgue measure 0. The limiting array has row and column means 0, row and column standard deviations 1. Moreover, many graphics given in [1] suggest that but for a set of entries of any array with Lebesgue measure 0, convergence is very rapid, eventually exponentially fast in the number of iterations. Here mathematical reason for this is suggested. More importantly, the rapidity of convergence is illustrated by numerical examples.