An elementary proof of the lower bound of Cramér’s Theorem in RdRd

Let {ξi}{ξi} be a family of i.i.d. random vectors in Rd(d≥2), and μnμn be the distributions in RdRd of the empirical means ξ1+⋯+ξnn. Without imposing any conditions, a new and elementary approach is presented in this note for showing the lower bound of the large deviation principle for {μn}{μn}, lim infn→∞1nlnμn(O)≥−infx∈OΛ⋆(x),open O⊆Rd, for some rate function Λ⋆(⋅)Λ⋆(⋅) which is defined in this note.