  Large deviations in the context of first-passage percolation was first
studied in the early 1980s by Grimmett and Kesten, and has since been revisited
in a variety of studies. However, none of these studies provides a precise
relation between the existence of moments of polynomial order and the decay of
probability tails. Such a relation is derived in this paper, and is used to
strengthen the conclusion of the shape theorem. In contrast to its
one-dimensional counterpart - the Hsu-Robbins-Erd\H{o}s strong law - this
strengthening is obtained without imposing a higher-order moment condition.
