We prove a linear version of Dawson-G{ \"a}rtner theorem: weak large deviation principles and the equality --s = p* between the negentropy and the Fenchel-Legendre transform of the pressure are preserved through linear projective limits. As a result, the equality --s = p* holds in great generality for empirical means of independent and identically distributed random variables (Cram{ \'e}r's theory), e.g. in any measurable normed space, and even in any projective limit of such spaces. Eventually, we give an original example where --s is different from p* and discuss the dual equality.
Categories: math.PR