Integro-local limit theorems including large deviations for sums of random vectors. II

This paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 43 (1998), pp. 1–12] and [A. A. Borovkov and A. A. Mogulskii, Siberian Math. J., 37 (1996), pp. 647–682]. Let $S(n)=\xi(1)+\cdots +\xi(n)$ be the sum of independent nondegenerate random vectors in $\mathbf{R}^d$ having the same distribution as a random vector $\xi$. It is assumed that $\varphi(\lambda)= \mathbf{E} \,e^{\langle\lambda,\xi\rangle}$ is finite in a vicinity of a point ${\lambda \in \mathbf{R}^d}$. We obtain asymptotic representations for the probability $\mathbf{P}\{S(n)\in \Delta (x)\}$ and the renewal function $H(\Delta (x))= \sum_{n=1}^{\infty}\mathbf{P}\{S(n)\in \Delta (x)\}$, where $\Delta(x)$ is a cube in $\mathbf{R}^d$ with a vertex at point $x$ and the edge length $\Delta$. In contrast to the above-mentioned papers, the obtained results are valid, in essence, either without any additional assumptions or under very weak restrictions.