Large deviations for a terminating compound renewal process

Let $(\xi(i),\eta(i)) \in \mathbf{R}^{d+1}$, $i \in \mathbf{N}$, be independent and identically distributed random vectors, let $\xi(i)\in \mathbf{R}^d$ be random vectors, let $\eta(i)$ be improper nonnegative random variables, and let $\mathbf{P}(\eta(i) = +\infty)\in(0,1)$. It is assumed that the distribution of the vector $(\xi(1),\eta(1))$ subject to $\{\eta(1)<+\infty\}$ satisfies the Cramér condition. By a terminating compound renewal process we mean the process $Z_T =\sum_{k=1}^{N_T}\xi(k)$, where $N_T = \max\{k \in \mathbf{N}\colon \eta(1)+\dots+\eta(k) \le T\}$ is the renewal process corresponding to improper random variables $\eta(1), \eta(2), \dotsc$. We find precise asymptotics of the probabilities $\mathbf{P}\bigl(Z_T\in I_{\Delta_T}(x)\bigr)$ and $\mathbf{P}(Z_T = x)$ in the nonlattice and strongly arithmetic cases, respectively; here $I_{\Delta_T}(x)=\{y\in\mathbf{R}^d\colon x_j\le y_j < x_j+\Delta_T$, $j=1,\dots,d\}$, and $\Delta_T$ is a positive function converging sufficiently slowly to zero.