Multi-Dimensional Local Limit Theorems for Large Deviations

In analogy to those theorems proved in 1, we will prove the following local theorems in the multi-dimensional case. We restrict ourselves to the case where the random vectors X^ (1), X^ (2), , have the common distribution (V) x = P\ Xₖ^{ (i) 1and|xᵢ | = o (n) as n, , j = 1, , m, then we obtain the relation (1). HereQk (x) is a multilinear form of degreek, whose coefficients are uniquely determined by the semi-invariants of the firstkorders of the distributionV (x). An analogous theorem is true for a lattice distribution of m-dimensional random vectors (Theorem 2).