Inference of the tail parameters of a distribution is a question of interest. Indeed, extreme events can have disastrous consequences and being able to estimate their probability of appearance allows us to manage them to an extent. It is however a difficult question because usual statistical theory do not work well in that case. Extreme value theory has been developed for this purpose. In particular, the Conditional Tail Moments (CTMs) are useful tools in risk quantification. For instance, the Expected Shortfall (ES), a particular case of CTM, is a risk measure widely used in finance. The estimation of CTMs and the proofs of convergence results for these estimators have been the purpose of my first year of PhD. In this talk, I will start with an introduction and motivation to Extreme Value Theory. I will then define the Conditional Tail Moment and give the mathematical framework in which estimation of extreme CTM is feasible. Finally, I would like to present some of the convergence results that I have been able to show so far.