An information-spectrum approach to large deviation theorems

In this paper we show a some new look at large deviation theorems from the viewpoint of the information-spectrum (IS) methods, which has been first exploited in information theory, and also demonstrate a new basic formula for the large deviation rate function in general, which is a pair of the lower and upper IS rate functions. In particular, we are interested in establishing the general large deviation rate functions that can be derivable as the Fenchel-Legendre transform of the cumulant generating function. The final goal is to show a necessary and sufficient condition for the rate function to be of Cram\'er-G\"artner-Ellis type.