Improving the rate of convergence in the multidimensional central limit theorem for sums of locally dependent random vectors

Estimates of the rate of convergence in multidimensional limit theorems for sums of dependent random vectors are considered in many papers. The types of dependence in a sequence of random vectors can be different, for example, $m$-dependent and locally dependent sequences of random vectors. It is important that these estimates are implicit. They do not specify how the estimate depends on the dimension of random vectors.
In this connection, in one of the author's previous papers, an explicit estimate for the distance between a multidimensional normal distribution and the distribution of the sum of locally dependent random vectors was obtained. In this paper, we improve this estimate. Also, it is proved that for centered and normalized sums of independent random vectors, the order of this estimate is equal to $d^{9/2}n^{-1/2}\ln n,$ where $d$ is dimension and $n$ is number of vectors.
Results of this paper have applications for discrete mathematical objects. For example, in the paper we consider a fixed regular graph. Each vertex is independently assigned one of the colors with a certain probability. A condition for the normal approximation of the number of edges incident to vertices of the same color is obtained.