Cramer’s theorem is concerned with fluctuations of empirical averages of iid random variables. It states that, under an exponential moment condition, fluctuations are exponentially rare, and the level of rareness is defined through a function called ‘rate function’. However, this result fails to provide relevant information beyond the exponential moment condition, which is often violated in practice. Many works have been devoted to understanding the so-called ‘heavy-tail’ regime, but for some reason it seems there is no such a thing as a general, moment-condition oriented result for fluctuations of iid variables beyond the exponential case. This is the problem addressed in this talk.
I will first expose Cramer’s theorem for iid random variables and explain the main arguments of the lower and upper bound parts of the proof. In order to relax the exponential moment condition for the random variable at hand, I introduce the notion of subexponential moment condition and explain how it can be used to obtain an equivalent of Cramer’s result for subexponential variables. The theorem we obtain is surprisingly different from the original version: the large deviation principle is not at exponential scale, the rate function is non-convex and non-smooth, which is rather surprising for such a simple problem. If time allows I might touch a word about ongoing research on understanding the same kind of behavior for stochastic differential equations.