Séminaire de Probabilités XLIX pp 75-117 | Cite as

# Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

## Abstract

The iterative proportional fitting procedure (IPFP), introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Thus, given a rectangular non-negative matrix *X*_{0} and two positive marginals *a* and *b*, the algorithm generates a sequence of matrices (*X*_{n})_{n≥0} 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*_{1}*X*_{0}*D*_{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})_{n≥0} 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})_{n≥0} 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*_{1}*X*_{0}*D*_{2}.

In the opposite case (when there exists no matrix with marginals *a* and *b* whose support is included in the support of *X*_{0}), the sequence (*X*_{n})_{n≥0} diverges but both subsequences (*X*_{2n})_{n≥0} and (*X*_{2n+1})_{n≥0} 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 ⋯*M*_{2}*M*_{1} of stochastic 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-stochastic 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}⋯*M*_{1})_{n≥0}, but also its finite variation.

## Keywords

Infinite products of stochastic matrices Contingency matrices Distributions with given marginals Iterative proportional fitting Relative entropy I-divergence## Subject Classifications

15B51 62H17 62B10 68W40## Notes

### Acknowledgements

We thank A. Coquio, D. Piau, G. Geenens, F. Pukelsheim and the referee for their careful reading and their useful remarks.

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