Let Z_1,Z_2,... be independent identically distributed lattice random vectors in R^k with mean zero and nonsingular covariance matrix V. We construct for S_n=n^{-1/2}( Z_1+...+Z_n ) an approximation of the form
P( S_n^TV^{-1}S_n < x ) - G_k(x) = J_1 + O(n^{-1})
that holds uniformly for all real x, provided E|Z_1|^4 < \infty, where G_k denotes the distribution function of a chi-square random variable with k degrees of freedom. The term J_1 is of order O(n^{-1}) when k \ge 5. Moreover, we prove non-uniform bounds for the difference P( S_n^TV^{-1}S_n < x ) - G_k(x) which are small when x is large. Furthermore, we discuss applications of the general results to the family of goodness-of-fit statistics.