On metric characterizations of the Radon–Nikodým and related properties of Banach spaces

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon–Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding.We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond.The paper also contains related characterizations of reflexivity and the infinite tree property.

A martingale characterization of Pólya-Lundberg processes

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

A martingale characterization of Pólya-Lundberg processes

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.