arXiv2018
The iterative proportional fitting procedure, introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Given a rectangular non-negative matrix $X_0$ and two positive marginals $a$ and $b$, the algorithm generates a sequence of matrices $(X_n)$ starting at $X_0$, supposed to converge to a biproportional fitting, that is, to a matrix $Y$ whose marginals are $a$ and $b$ and of the form $Y=D_1X_0D_2$, for some diagonal matrices $D_1$ and $D_2$ with positive diagonal entries. When a biproportional fitting does exist, it is unique and the sequence $(X_n)$ converges to it at an at least geometric rate. More generally, when there exists some matrix with marginal $a$ and $b$ and with support included in the support of $X_0$, the sequence $(X_n)$ converges to the unique matrix whose marginals are $a$ and $b$ and which can be written as a limit of matrices of the form $D_1X_0D_2$. In the opposite case, the sequence $(X_n)$ diverges but both subsequences $(X_{2n})$ and $(X_{2n+1})$ converge. In the present paper, we use a new method to prove again these results and determine the two limit-points in the case of divergence. Our proof relies on a new convergence theorem for backward infinite products $\cdots M_2M_1$ of stochatic matrices $M_n$, with diagonal entries $M_n(i,i)$ bounded away from $0$ and with bounded ratios $M_n(j,i)/M_n(i,j)$. This theorem generalizes Lorenz' stabilization theorem. We also provide an alternative proof of Touric and Nedić's theorem on backward infinite products of doubly-stochatic matrices, with diagonal entries bounded away from $0$. In both situations, we improve slightly the conclusion, since we establish not only the convergence of the sequence $(M_n \cdots M_1)$, but also its finite variation.