In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process X_n=-μ +∑_j=-∞^∞ arphi_n-jε_j, where {ε,ε_n ;-∞0 is a constant and the coefficients {\varphi_i; -∞<i<∞} satisfy 0<∑_j=-∞^∞|j arphi_j | <∞. Under conditions the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{sup_n≥0( -n μ +∑_j=-∞^∞ε_jβ_nj)> x } is discussed. Then the result is applied to ultimate ruin probability.