Projections

A common method to derive the limit distribution of a sequence of statistics Tn is to show that it is asymptotically equivalent to a sequence Sn of which the limit behavior is known. The basis of this method is Slutsky's lemma, which shows that the sequence Tn = Tn–Sn + Sn converges in distribution to S if both Tn–Sn and S.

How do we find a suitable sequence Sn? First, the variables Sn must be of a simple form, because the limit properties of the sequence Sn must be known. Second, Sn must be close enough. One solution is to search for the closest Sn of a certain predetermined form. In this chapter, “closest” is taken as closest in square expectation.

Let T and be random variables (defined on the same probability space) with finite second-moments. A random variable S is called a proi-edion of and minimizes

Often S is a linear space in the sense that isfor every, whenever In this case S is the projection of if and only if is orthogonal to for the inner product This is the content of the following theorem.

Theorem. Let S be a linear space of random variables with finite second moments. Then S is the projection of Tonto S if and only if Sand

Every two projections of Tonto S are almost surely equal.