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⁾ < xₖ ;k = 1, ⋯ ,m\ ,j = 1,2, ⋯. It will be assumed that all mixed second moments of the vectors X⁽ˡ⁾ are finite, and \[ \| {σ ʲᵏ } \| = Δ ≠ 0.\]Theorem 1. LetEX⁽ⁱ⁾ = o. Let the following conditions be satisfied: (A) There exists an no such that the density of the distribution of∑ j = 1n₀ X⁽ʲ⁾is bounded. (B) There exists a positive numberαsuch that for allm-dimensional vectorsb,|vᵢ | < αthe integral∫Rₘ e^ b,x dV(x)converges. Then for sufficiently largenthe inversion formula is valid, and if|xᵢ | > 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).