For a set of i.i.d. random variables indexed by $Z^d_+, d \geqslant 1$, the positive integer $d$-dimensional lattice points, convergence rates for moderate deviations are derived, i.e., the rate of convergence to zero of, for example, certain tail probabilities of the partial sums, are determined. As an application we obtain results on the integrability of last exit times (in a certain sense) and the number of boundary crossings of the partial sums.
Allan Gut. "Convergence Rates for Probabilities of Moderate Deviations for Sums of Random Variables with Multidimensional Indices." Ann. Probab. 8 (2) 298 - 313, April, 1980. https://doi.org/10.1214/aop/1176994778