The　kernel　distribution　estimator　(KDE)　is　proposed　based　on　residuals　of　the　innovation　distribution　in　the　autoregressive　moving-average　(ARMA)　time　series.　The　deviation　between　KDE　and　the　innovation　distribution　function　is　shown　to　converge　to　the　Brownian　bridge,　leading　to　the　construction　of　a　Kolmogorov–Smirnov　smooth　simultaneous　confidence　band　for　the　innovation　distribution　function.　Additionally,　an　empirical　cumulative　distribution　function　(CDF)　based　on　prediction　residuals　is　introduced　for　the　multi-step-ahead　prediction　error　distribution　function.　This　empirical　process　weakly　converges　to　a　Gaussian　process　with　a　specific　covariance　function.　Furthermore,　a　quantile　estimator　is　derived　from　the　empirical　CDF　of　prediction　residuals,　and　multi-step-ahead　prediction　intervals　(PIs)　for　future　observations　are　established　using　these　estimated　quantiles.　The　PIs　achieve　the　nominal　prediction　level　asymptotically　for　the　finite　variance　ARMA　model.　Simulation　studies　and　analysis　of　crude　oil　price　data　confirm　the　validity　of　the　asymptotic　theory.　©　2024　American　Statistical　Association　and　Taylor　&　Francis.