content_image/317555.jpg content_image/317556.jpg content_image/317559.jpg content_image/317557.jpg
Again, G ∗ n will satisfy (7) under mild conditions. On the other hand, because 

we would expect 

<img src='content_image/317555.jpg'>

<img src='content_image/317556.jpg'>

Unlike Q , the process Q ∗ is random in general. Indeed, Q ∗ coincides with Q if and only if G is of the form G ( s ) = s ′ ˙ G for some zero-mean Gaussian vector ˙ G . As a consequence, the bootstrap analogue of (4) fails unless G ( s ) = s ′ ˙ G . Because S = V − 1 0 ˙ G when G ( s ) = s ′ ˙ G , the limiting distribution of r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) is Gaussian whenever the bootstrap analogue of (4) holds. In perfect qualitative agreement with Fang and Santos (2016, Corollary 3.1), we therefore find that Gaussianity of the limiting distribution of r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) is a “heuristically necessary” condition for consistency of the nonparametric bootstrap. 

The rate of convergence of ˆ θ$_{n}$ is also a “red herring” in our main constructive results. In particular, the reshaped function ˜ m$_{n}$ employed in the construction of ˜ θ ∗ n makes no use of the fact that ˆ θ$_{n}$ was assumed to be cube root consistent. Moreover, because 

<img src='content_image/317559.jpg'>

the factor 3 √$_{n}$ has been included in the statement of the bootstrap consistency result (9) itself solely to facilitate its interpretation. As a consequence, also in the more general setting of the current discussion one would expect that if ˜ V$_{n}$ → P V 0 , then our bootstrap-based approximation will be consistent in the sense that 

<img src='content_image/317557.jpg'>

