  In this paper we consider first passage percolation on the square lattice
\(\mathbb{Z}^d\) with edge passage times that are independent and have
uniformly bounded second moment, but not necessarily identically distributed.
For integer \(n \geq 1,\) let \(T_n\) be the minimum passage time between the
origin and the point \((n,0,\ldots,0).\) We prove that
\(\frac{1}{n}(T_n-\mathbb{E}T_n)\) converges to zero almost surely and in
\(L^2\) as \(n~\rightarrow~\infty.\) The convergence is nontrivial in the sense
that \(\frac{T_n}{n}\) is asymptotically bounded away from zero and infinity
almost surely. We first define a truncated version \(\hat{T}^{(n)}_n\) that is
asymptotically equivalent to~\(T_n.\) We then use a finite box modification of
the martingale method of Kesten~(1993) to estimate the variance of
\(\hat{T}^{(n)}_n.\) Finally, we use a subsequence argument to obtain almost
sure convergence for \(\frac{1}{n}(\hat{T}^{(n)}_n -
\mathbb{E}\hat{T}^{(n)}_n).\) The corresponding result for \(T_n\) is then
obtained using asymptotic equivalence of \(T_n\) and \(\hat{T}^{(n)}_n.\) For
identically distributed passage times, our method alternately obtains almost
sure convergence of~\(\frac{T_n}{n}\) to a positive constant~\(\mu_F,\) without
invoking the subadditive ergodic theorem.
