Again, G ∗ n will satisfy (7) under mild conditions. On the other hand, because we would expect 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 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 Q ∗ $_{n}$( s ) = Q$_{n}$ ( ˆ s$_{n}$ + s ) + G$_{n}$ ( ˆ s$_{n}$ + s ) − Q$_{n}$ ( ˆ s$_{n}$ ) − G$_{n}$ ( ˆ s$_{n}$ ) , Q ∗ $_{n}$( s ) ⇝ Q ( S + s ) + G ( S + s ) − Q ( S ) − G ( S ) = Q ( s ) − S $^{′}$V 0 s + G ( S + s ) − G ( S ) = Q $^{∗}$( s ) . sup t ∈ R d ∣ P $^{∗}$[ r$_{n}$ ( ˜ θ ∗ n − ˆ θ$_{n}$ ) ≤ t ] − P [ r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) ≤ t ] ∣ → P 0 . 13 sup t ∈ R d ∣ P $^{∗}$[ r ( ˜ θ ∗ n − ˆ θ$_{n}$ ) ≤ t ] − P [ r ( ˆ θ$_{n}$ − θ 0 ) ≤ t ] ∣ = sup t ∈ R d ∣ P $^{∗}$[ ˜ θ ∗ n − ˆ θ$_{n}$ ≤ t ] − P [ ˆ θ$_{n}$ − θ 0 ≤ t ] ∣ , ∀ r > 0 ,