## 1 Introduction
Preliminaries**  
We begin precise definitions:  
\[\bar{X}_{i\cdot}^{(0)}=\frac{1}{J}\,\sum_{j=1}^{J}\,X_{ij}\]  
\[(S_{i}^{(0)})^{2}=\frac{1}{J}\,\sum_{j=1}^{J}\,(X_{ij}-\bar{X}_{i\cdot}^{(0)})^ {2}=(\frac{1}{J}\,\sum_{j=1}^{J}\,X_{ij}^{2}-J(\bar{X}_{i\cdot}^{(0)})^{2})\]  
\[\mathbf{X}^{(1)}=\{X_{ij}^{(1)}\},\text{ where }X_{ij}^{(1)}=\frac{X_{ij}-\bar{X}_{i \cdot}^{(0)}}{S_{i}^{(0)}}\]  
By analogy, set  
\[\mathbf{X}^{(2)}=\{X_{ij}^{(2)}\}\]  
where  
\[X_{ij}^{(2)}=\frac{X_{ij}^{(1)}-\bar{X}_{\cdot\cdot j}^{(1)}}{S_{j}^{(1)}}\text { and }\bar{X}_{\cdot\cdot}^{(1)}=\frac{1}{I}\,\sum_{i=1}^{I}\,X_{ij}^{(1)}\]  
Now,  
\[(S_{j}^{(1)})^{2}=\frac{1}{I}\,\sum_{i=1}^{I}(X_{ij}^{(1)}-\bar{X}_{\cdot\cdot }^{(1)})^{2}\]  
and analogously define \((S_{j}^{(n)})^{2}\) for \(n=2,3,...\). By induction on dimension applied to regular conditional distributions, \((S_{i}^{(0)})^{2}>0\) a.s.  
As explained in [1] the \(2\times 2\) case illustrates the need to work with dimensions greater than or equal to \(3\). Consider the following arbitrary \(2\times 2\) matrix  
\[\mathbf{X}= \begin{bmatrix}a&b\\ c&d\end{bmatrix} \]  
If \(a<b\) and \(c<d\), then  
\[\mathbf{X}^{(1)}= \begin{bmatrix}-1&1\\ -1&1\end{bmatrix} \]  
so \((S_{j}^{(1)})^{2}=0\)If \(a,b,c,d\) are, say, _iid_ and have symmetric distributions, then  
\[P((S_{j}^{(1)})^{2}=0)=\frac{1}{2}\]  
A moment's reflection shows that if \(\mathbf{X}\) is \(I\times 2\), with \(I\) odd, then after each row normalization, each column has an odd number of entries, each entry being \(+1\) or \(-1\). However, each row has exactly one \(+1\) and one \(-1\). Thus \((S_{i}^{(n)}-S_{j}^{(n+1)})^{2}\to 0\) is impossible. With \(\min(I,J)\geq 3\), both tend to 1 a.s. as \(n\
earrow\infty\). The fact that \(\{X_{ij}\}\) iid implies in particular that \(\mathbf{X}\) is row and column exchangeable. Thus, if  
\[\mathbf{X}=[\mathbf{x}_{1,c},...\mathbf{x}_{J,c}]\]  
where each column is \(I\times 1\), and \(\pi\) is a permutation of the integers \(\{1,...,J\}\), then \(\mathbf{X}\sim[\mathbf{x}_{\pi(1),c},...,\mathbf{x}_{\pi(J),c}]\). The analogous holds for rows, where  
\[\mathbf{X}= \begin{bmatrix}\mathbf{x}_{1,r}\\ \vdots\\ \mathbf{x}_{I,r}\end{bmatrix} \]  
where \(\mathbf{x}_{i,r}\) is \(1\times J\). **3. Background and Motivation**  
A first step in the argument of [1] is to note that Lebesgue measure on the Borel subsets of \(\mathbb{R}^{IJ}\) is mutually absolutely continuous with respect to IJ product Gaussian measure, each coordinate being standard Gaussian. Thereby, the distinction between measure and topology is blurred; arguments of [1] as corrected here. Having translated a problem concerning Lebesgue measure to one concerning Gaussian measure, one cannot help note from graphs in [1] Figures 1-3,6 and 7 that suggest with entries of \(\mathbf{X}\) chosen independently from a common absolutely continuous distribution, and as led to these figures, almost surely ultimately, convergence is at least exponentially fast. In these graphs, the ordinate is always the difference of logarithms (the base does not matter) of squared Frobenius norm of the difference between current iteration and the immediately previous iteration; always abscissa indexes iteration. One purpose of this paper is to demonstrate the ultimately almost sure rapidity of convergence. Readers will note we assume that coordinates are independent and identically distributed, with standard normal distributions. This assumption is used explicitly though unnecessarily in [1], but only implicitly here, and then only to the extent that arguments here depend on those in [1]. Obviously, this Gaussian assumption is sufficient. It is pretty obviously not necessary. For the sake of motivation we illustrate the patterns of convergence from successive normalizations on \(5\times 5\) matrices. We first describe the details of the successive normalization in a manner very similar to [1]. Consider a simple \(5\times 5\) matrix with entries generated from a normal distribution with a given mean and standard deviation.