The Strong Law under Random Censorship

Let X₁, X₂, … be a sequence of i.i.d. random variables with d.f. F. We observe Zᵢ = min(Xᵢ,Yᵢ) and δᵢ = 1_ᵢ ≤ Yᵢ\, where Y₁, Y₂, … is a sequence of i.i.d. censoring random variables. Denote by F̂ₙ the Kaplan-Meier estimator of F. We show that for any F-integrable function φ, ∫φ dF̂ₙ converges almost surely and in the mean. The result may be applied to yield consistency of many estimators under random censorship.