choose ${\alpha}={\frac{1}{2}}-\delta_4$ where $\delta_4>0$ is sufficiently small, then by taking $\delta_1$ in (\[eq2.33\]) to be small we obtain: $$\label{eq2.34} \max_{1\leq r\leq r_0} n^{-r/2} \sup_{-\infty<z<\infty} \biggl|\sum
_{\nu\in{{\mathcal N}}_\ell} \nu_\ell^r \psi
\biggl(z-\frac{e_2 n_1}{ e_1 n_2} \nu \biggr) \biggr| =O \bigl(n^{(1/2) \{1-(1/\eta)\}+\delta_3} \bigr),$$ provided that $$\label{eq2.35} |\rho_{12}-\rho_0|=O \bigl(n^{-(1/2) \{1+(1/\eta)+\delta_5\}}
\bigr) ,$$ where $\delta_5>0$ can be made as small as we like simply by choosing $\delta_4$ small. Now, (\[eq2.35\]) follows from (\[eq2.15\]). It therefore follows from (\[eq2.34\]), and the definition of $K_n(x)$ at (\[eq2.10\]), that if (\[eq2.15\]) holds for some $\delta>0$ then $$\label{eq2.36} \sup_{-\infty<x<\infty} (n_1n_2)^{-1/2}\bigl|K_n(x)\bigr|
=O \bigl(n^{\delta-(1/2) \{1+(1/\eta)\}} \bigr)$$ for all $\delta>0$. Results (\[eq2.14\]) and (\[eq2.36\]) imply (\[eq2.16\]), as had to be shown.

Expansions relating to the bootstrap {#sec2.7}
------------------------------------

In this section we show that, despite the potential for problems arising from discreteness, the bootstrap (including the double bootstrap) applied to inference based on the distribution of $\{S-E(S)\}
/({\operatorname{var}}S)^{1/2}$, generally (when (\[eq2.15\]) holds and $\rho_0$ is of type 1) produces confidence regions and hypothesis tests with the same orders of magnitude of coverage or level accuracy, up to terms of size $n^{\delta-1}$ for all $\delta>0$, as it would in the case of smooth sampling distributions. This result is of practical importance, since standard 