Subgaussian rates of convergence of means in directed first passage percolation

We consider directed first passage percolation on the integer lattice, with time constant $\mu$ and passage time $a_{0n}$ from the origin to $(n,0,\mathrm{.}\mathrm{.}\mathrm{.},0)$. It is shown that under certain conditions on the passage time distribution, $E{a}_{0n}-n\mu =O(n^{1\mathrm{/}2}(\log \log n)\mathrm{/}\log n)$.