AISM 50, 433-450

## A characterization of monotone and regular divergences

### J.M. Corcuera1 and F. Giummole2

1Departament d'Estadística, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
2Dipartimento di Scienze Statistiche, Università degli Studi di Padova, Via S. Francesco, 33, 35121 Padova, Italy

(Received April 15, 1996; revised June 23, 1997)

Abstract.    In this paper we characterize the local structure of monotone and regular divergences, which include $f$-divergences as a particular case, by giving their Taylor expansion up to fourth order. We extend a previous result obtained by Cencov, using the invariant properties of Amari's $\alpha$-connections.

Key words and phrases:    Differential geometry, divergence, embedding invariance, Markov embedding, \alpha-connection.

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