Triangular Gaussian coordinates: metric and moment control

For any complex matrix $A$ and vector $u$, put $a_i=(\overline{A_{ij}}{)}_j$. In the vertex convention, $⟨a_i,u⟩={\sum }_j{A}_{ij}{u}_j$, so [F2] gives $\mathrm{\parallel }Au{\mathrm{\parallel }}_2^2={\sum }_i\mathrm{\mid }⟨a_i,u⟩{\mathrm{\mid }}^2\le {\sum }_i({\sum }_j\mathrm{\mid }{A}_{ij}{\mathrm{\mid }}^2)\mathrm{\parallel }u{\mathrm{\parallel }}_2^2=\mathrm{\parallel }A{\mathrm{\parallel }}_F^2\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. Thus the matrix action is bounded, and its operator norm in [F4] satisfies $\mathrm{\parallel }A\mathrm{\parallel }\le \mathrm{\parallel }A{\mathrm{\parallel }}_F$. Also $\mathrm{\parallel }T(z)-T(w){\mathrm{\parallel }}_F^2={\sum }_i(z_{ii}-w_{ii}{)}^2+2{\sum }_{i<j}(z_{ij}-w_{ij}{)}^2\le 2\mathrm{\parallel }z-w{\mathrm{\parallel }}_2^2$. Taking nonnegative square roots proves the stated metric estimates.

The operator-norm triangle inequality in [F4], applied in both directions, gives $\mathrm{\mid }\mathrm{\parallel }T(z)\mathrm{\parallel }-\mathrm{\parallel }T(w)\mathrm{\parallel }\mathrm{\mid }\le \mathrm{\parallel }T(z)-T(w)\mathrm{\parallel }\le \sqrt{2}\mathrm{\parallel }z-w{\mathrm{\parallel }}_2$. Hence $h:{\mathbb{R}}^N\to \mathbb{R}$, $h(z)=\mathrm{\parallel }T(z)\mathrm{\parallel }$, is continuous.

Choice supplies the countable-choice hypothesis in [F5] by restriction to countable families. The Gaussian coordinates are measurable real random variables on their common probability space by [F1], so [F5] makes $Z$ measurable into $({\mathbb{R}}^N,\mathcal{B}({\mathbb{R}}^N))$, where $N\ge 1$. Step 2.1 and [F5] make $h$ Borel measurable, so $h\circ Z=\mathrm{\parallel }Y\mathrm{\parallel }$ is measurable. For any fixed $k\ge 1$, the map $p_k(t)=t^{2k}$ is continuous: factorization gives $\mathrm{\mid }{s}^{2k}-t^{2k}\mathrm{\mid }\le 2k\max (\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }t\mathrm{\mid }{)}^{2k-1}\mathrm{\mid }s-t\mathrm{\mid }$, which tends to zero as $s\to t$. Applying [F5] to $p_k$ and then composing with $h\circ Z$ or an individual Gaussian coordinate proves measurability of $\mathrm{\parallel }Y{\mathrm{\parallel }}^{2k}$ and $\mathrm{\mid }{Z}_{ij}{\mathrm{\mid }}^{2k}$. Their finite sums and nonnegative scalar multiples are measurable by [F3].

By step 1.1, $\mathrm{\parallel }Y{\mathrm{\parallel }}^{2k}\le 2^k({\sum }_{i\le j}{Z}_{ij}^2{)}^k\le 2^k{N}^k{\sum }_{i\le j}\mathrm{\mid }{Z}_{ij}{\mathrm{\mid }}^{2k}$: the sum of the $N$ coordinate squares is at most $N$ times their maximum, and the $k$th power of that maximum is at most the sum of their $k$th powers. Every summand has finite expectation by [F1]. Using the measurability from step 3.1, finite linearity and monotonicity in [F3] give $\mathbb{E}\mathrm{\parallel }Y{\mathrm{\parallel }}^{2k}\le 2^k{N}^k{\sum }_{i\le j}\mathbb{E}\mathrm{\mid }{Z}_{ij}{\mathrm{\mid }}^{2k}<\mathrm{\infty }$. Finally $Y_{ii}=Z_{ii}$ has mean zero and second moment one by [F1], hence variance one, while $Y_{ij}=Y_{ji}=Z_{ij}$ for $i<j$ by construction.