Multi-Dimensional Limit Theorems for Large Deviations and Their Application to the $\chi ^2 $ Distribution

Going out from multi-dimensional local limit theorems for large deviations (see [3], Theorem 1, as well as Theorem 2 of the present note), two integral limit theorems are proved (Theorems 4 and 5). In the proof a generalization of the method is used, by which A. Ya. Khinchin derived the first integral theorem for large deviations in the case of Bernoulli schemes [7]. Theorem 1 is a consequence of these theorems applied to the distribution of the $\chi ^2 $ statistics.