The author establishes a uniform estimate for the mass function \(P(S_m =y)\) of an integer-valued random walk when \(y\to\infty\) and \((y-m\mu)/ \sqrt{m} \to \infty,\) where \(\mu\) is the mean of the step distribution. The assumptions are that the mass function \(p\) of the step distribution is regularly varying at \(\infty\) with \(-\kappa\), where \(\kappa >3\), and that \(\sum_{n=0}^{\infty} n^{\kappa'}p(-n) 2\). From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space-time Martin boundary of certain random walks.