A　kernel　distribution　estimator　(KDE)　is　obtained　based　on　residuals　of　innovation　distribution　in　ARCH　time　series.　The　deviation　between　KDE　and　the　innovation　distribution　function　is　shown　to　converge　to　a　Gaussian　process.　Based　on　this　convergence,　a　smooth　simultaneous　confidence　band　is　constructed　for　the　innovation　distribution　and　an　invariant　procedure　proposed　for　testing　the　symmetry　of　innovation　distribution　function.　Quantiles　are　further　estimated　from　the　KDE,　and　multi-step-ahead　prediction　intervals　(PIs)　of　future　observations　are　constructed　using　the　estimated　quantiles,　which　achieve　asymptotically　the　nominal　prediction　level.　The　multi-step-ahead　PI　is　constructed　for　the　S&P　500　daily　returns　series　with　satisfactory　performance,　which　corroborates　the　asymptotic　theory.　©　The　Author(s)　under　exclusive　licence　to　Sociedad　de　Estadística　e　Investigación　Operativa　2024.