\mathbb U(M,\Sigma,f_{\tilde{z}})$, then it is easy to see that $\tilde{M}(\tilde{M}'\tilde{M})^{-1/2} \in \mathbb G(f_{\tilde{z}})$. Because conditioning on $P_{\tilde{M}}\tilde{z}$ is equivalent to conditioning on $(\tilde{M}'\tilde{M})^{-1/2} \tilde{M}'\tilde{z}$, it follows that ${{\mathbb P}}( |{{\mathbb E}}[e\|x]| > t)$ is bounded from above by with $t$ replaced by $t / \|(I_d-P_{\tilde{M}})\tilde{\theta}\|$ and that ${{\mathbb P}}( |{{\mathbb E}}[e^2\|x] - s^2| > t)$ is bounded by with $t$ replaced by $t/\|(I_d-P_{\tilde{M}})\tilde{\theta}\|^2$.

The consideration in the preceding paragraph suggests that the effect of misspecification in , where ${{\mathbb E}}[e\|x]$ may be non-zero and $\operatorname{Var}[e\|x]$ may be non-constant, may be negligible in an asymptotic setting where $p/\log d$ becomes small, provided that $f_{\tilde{z}} \in \mathcal F_{d,20}$ and that $R \in \mathbb U(M,\Sigma,f_{\tilde{z}})$. This idea is formalized in the following two results, which show that the distribution of certain statistics is unaffected asymptotically if the error $e$ is replaced by a substitute error $e^\ast$ that has mean zero and constant variance conditional on $x$. The following results are stated for sequences where the data-generating model - and the working model are allowed to depend on $n$, that is, a ‘triangular array’ setting where all parameters depend on $n$.

\[lemma:LinQuad\] Fix finite positive constants $D$ and $E$. For every $n\in{{\mathbb N}}$, let 