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Asymptotic Statistics by A.W. van der Vaart

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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 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 = TnSn + Sn converges in distribution to S if both ...

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