Multidimensional large deviation local limit theorems

An earlier paper by the author ([4], 97–114) established large deviation local limit theorems for arbitrary sequences of real valued random variables. This work showed clearly the connection between the Cramér series and large deviation rates. In this article we present large deviation local limit theorems for arbitrary multidimensional random variables based solely on conditions imposed on their moment generating functions. These results generalize the theorems of [12], 100–106) for sums of independent and identically distributed random vectors.