The asymptotic behavior of the iterative proportional fitting procedure (IPF procedure) is analyzed comprehensively. Given a nonnegative matrix as well as row and column marginals the IPF procedure generates a sequence of matrices, called the IPF sequence, by alternately fitting rows and columns to match their respective marginals. We prove that the IPF sequence has at most two accumulation points. They originate as the limits of the even-step subsequence, and of the odd-step subsequence. The wellknown IPF convergence criteria are then retrieved easily. Our proof is based on Csiszár's and Tusnády's (1984) results on the interplay of the I-divergence geometry and alternating minimization procedures.

This paper proves continuity of f-projections and the continuous dependence of the limit matrix of the iterative proportional fitting procedure (IPF procedure) on the given matrix as well as the given marginals under certain regularity constraints. For discrete spaces, the concept of f-projections of finite measures on a compact and convex set is introduced and continuity of f-projections is proven. This result is applied to the IPF procedure. Given a nonnegative matrix as well as row and column marginals the IPF procedure generates a sequence of matrices, called the IPF sequence, by alternately fitting rows and columns to match their respective marginals. If the IPF sequence converges, the application of the previous result yields the continuous dependence of the limit matrix on the given matrix. By generalized convex programming and under some constraints, it is shown that the limit matrix of the IPF sequence continuously depends not only on the given matrix but also on the marginals.Keywords Continuity of f-projections · Iterative proportional fitting · Limit matrix · Dependence on input matrix and marginals · f-divergence · f-projection · I-divergence · I-projection Mathematics Subject Classification (2000) 68W40 · 62B10 · 62H17 · 90C25 *

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