How to get Central Limit Theorems for global errors of estimates

The asymptotic behavior of global errors of functional estimates plays a key role in hypothesis testing and confidence interval building. Whereas for pointwise errors asymptotic normality often easily follows from standard Central Limit Theorems, global errors asymptotics involve some additional techniques such as strong approximation, martingale theory and Poissonization. We review these techniques in the framework of density estimation from independent identically distributed random variables, i.e., the context for which they were introduced. This will avoid the mathematical difficulties associated with more complex statistical situations in which these tools have proved to be useful.