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In this paper, we prove $L_p,\ p\geq 2$ and almost sure convergence of tailindex estimator mentioned in \cite{grama2008} under random censoring andseveral assumptions. $p$th moment of the error of the estimator is proved to beof order $O\left(\frac{1}{\log^{m\kappa/2}n}\right)$ with given assumptions. Wealso perform several finite sample simulations to quantify performance of thisestimator. Finite sample results show that the proposed estimator is effectivein finding underlying tail index even when censor rate is high.

$L_p$ and almost sure convergence of estimation on heavy tail index under random censoring