Strong limit theorems for osciallation moduli of PL-process and cumulative hazard process under truncation and censorship with applications (Q1292801)

Let \(F_n\) be the product-limit estimator of the survival distribution \(F\) for left truncated and right censored data. Define the PL-process as \[ \alpha_n(x)=\sqrt{n}\left(F_n(x)-F(x)\right), \] the oscillation moduli of \(\alpha_n\) as \[ \omega_n(a_n)=\sup\left\{\left|\alpha_n(t)-\alpha_n(s)\right|: |t-s|\leq a_n\right\}, \] and the Lipschitz-\(2^{-1}\) moduli of \(\alpha_n\) as \[ \tilde\omega_n(a_n)=\sup\left\{\left|\alpha_n(t+h)-\alpha_n(t)\right|h^{-1/2} : h\geq a_n\right\}, \] where \((a_n,n\geq 1)\) is a positive sequence. Strong limit theorems for \(\omega_n(a_n)\) and \(\tilde\omega_n(a_n)\) are given. Similar results for the standard estimator of the cumulative hazard function are presented.