A CHARACTERIZATION OF MONOTONE AND REGULAR DIVERGENCES

J. M. CORCUERA^{1} AND F. GIUMMOLÈ^{2}

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 Čencov, using the invariant properties of Amari’s α-connections.

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