*11*

*Projections*

*A projection of a random variable is defined as a closest element in a given set of functions. We can use projections to derive the asymptotic distribution of a sequence of variables by comparing these to projections of a simple form. Conditional expectations are special projections. The Hájek projection is a sum of independent variables; it is the leading term in the Hoeffding decomposition*.

**11.1 Projections**

A common method to derive the limit distribution of a sequence of statistics *T*_{n} is to show that it is asymptotically equivalent to a sequence *S*_{n} of which the limit behavior is known. The basis of this method is Slutsky’s lemma, which shows that the sequence *T*_{n} = *T*_{n} − *S*_{n} + *S*_{n} converges in distribution to *S* if both ...