At first the paper investigates the asymptotic behavior of the finite-time ruin probability with constant interest rate and subexponentially tailed claim sizes, which extends the result recently established by Tang^[14] for the classical risk model to the delayed renewal risk model; then, within the intersection class of the claim sizes with subexponential tails and the claim sizes with dominatedly varying tails, the paper discusses the finite-time ruin probability with claims arriving according to an arbitrary counting process.