,Y_{m}})=-\infty .$$ If $ \chi ({Y_{1},\dots ,Y_{m}})\neq -\infty  $, then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})=-\infty$$ implies that $$\chi ({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}})\leq -\infty +\chi ({Y_{1},\dots ,Y_{m}})=-\infty .$$

We shall write $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};l,k,\epsilon )$$ for $${\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l ,k ,\epsilon  )}$$ when the $ {Y_{1},\dots ,Y_{m}}$ are understood.

If $ p<m $, then $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|Y_{1},\dots ,Y_{p}) $, and a similar inequality holds for $ \chi _{R} $ and $ \chi ^{\omega } $.

We have the inclusion $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\subset {\Gamma _{R}({X_{1},\dots ,X_{n}}|y_{1},\dots ,y_{p} [Y_{1},\dots ,Y_{p} ];l ,k ,\epsilon  )}$$ for all $ ({y_{1},\dots ,y_{m}})\in M_{k} $. It follows that $$\begin{aligned}
\sup _{({y_{1},\dots ,y_{m}})}\lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\leq  &  & \\
\sup _{(y_{1},\dots ,y_{p})}\lambda {\Gamma _{R}({X_{1},\dots ,X_{n}}|y_{1},\dots ,y_{p} [Y_{1},\dots ,Y_{p} ];l ,k ,\epsilon  )} &  & \end{aligned}$$ which implies the desired inequality.

$ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}:{Y_{1},\dots ,Y_{m}}) $.

We clearly have $$\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,k,\epsilon )\subset \pi \Gamma _{r}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};l,k,\epsilon ),$$ where $ \pi  $ denotes the projection from $ (M_{k}^{{{\operatorname{sa}}}})^{n}\times (M^{{{\operatorname{sa}}}})^{m} $ onto $ (M_{k}^{{{\operatorname{sa}}}})^{n} $.

\[corr: max\]Let $ c^{2}=\frac{1}{n}\sum \tau (X_{i}^{2}) $. Then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \frac{n}{2}\log 2\pi ec^{2}.$$ 