Strings: mathematical theory and statistical examples

Abstract

Strings , in the sense of the present paper, are sequences of multiarrays with two types of indices; tensorial and structural, and they are characterized by a transformation law that generalizes those for tensors, affine connections and derivatives of scalars. The original definition Barndorff-Nielsen ( Proc. R. Soc. Lond. A 406, 127-137 (1986)) is here extended and a systematic study of the mathematics of strings is undertaken. In particular, a convolutive multiplication of strings is introduced and is used in the discussion of intertwining , a type of operation that produces tensors from strings and strings from tensors and connection strings , a special kind of string. It appears that tensors and connection strings have a role, respectively, as ‘coordinates ’ and ‘coordinate frames ’ in the calculus of strings. A definition of ‘covariant differentiation of strings’ is proposed and is related to convolutive multiplication and to intertwining. The general theory is illustrated by various examples from the context of statistical inference. Finally, a brief comparison is made between strings and the somewhat related concept of extensors.