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J. R. Statist. Soc. B, 72 (2010), 343–366

Assumptions 1 and 2 in the paper are not sufficient to ensure the validity of theorem 1. Assumption 2 on page 346 should be replaced by the following.

 Assumption 2.  Assume that RN=op(N−1/2) and inline image.

Then theorem 1 holds under assumption 1 and this assumption 2. The problem lies in the derivation of the limiting distribution of WN. Let

  • image

Note that

  • image

It is easy to see that I1N[RIGHTWARDS ARROW]D ΔVqΔ follows from assumption 1 and the continuous mapping theorem. Hence it suffices to show that I3N=op(1), as it implies that I2N=op(1) by the Cauchy–Schwarz inequality. It can be shown that the relationship I3N=op(1) is directly implied by assumption 2, but not by the original assumption 2 in the paper, because the following statement is in general false.

  • image

Consequently, to verify assumption 2 in remark 1 on page 346 for the smooth function model, we need to impose the assumption that inline image and the absolute summability condition on the jth-order cumulants of Yt, j=2,3,4. Furthermore, the assumptions on inline image and inline image in theorem 2 on page 351 should be changed to