  In this note we illustrate and develop further with mathematics and examples,
the work on successive standardization (or normalization) that is studied
earlier by the same authors in Olshen and Rajaratnam (2010) and Olshen and
Rajaratnam (2011). 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 Olshen and Rajaratnam (2010) 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. A basic result on convergence
given in Olshen and Rajaratnam (2010) is true, though the argument in Olshen
and Rajaratnam (2010) is faulty. The result is stated in the form of a theorem
here, and the argument for the theorem is correct. Moreover, many graphics
given in Olshen and Rajaratnam (2010) 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. Because we learned this set of
rules from Bradley Efron, we call it "Efron's algorithm". More importantly, the
rapidity of convergence is illustrated by numerical examples.
