Fernique integrability for a separable Banach Gaussian law

On $(B\times B,\mathcal{B}(B)\otimes \mathcal{B}(B),\mu \otimes \mu )$ let $W(x,y)=x$ and $W^{\mathrm{\prime }}(x,y)=y$ be the coordinate copies. The product-measure rectangle formula makes them independent. A countable dense subset of $B$ and rational-radius balls give a countable base, and products of base sets give one for $B\times B$, so [F5] identifies the product sigma-algebra with $\mathcal{B}(B\times B)$. The coordinate projections are continuous, hence Borel; the maps $(x,y)\mapsto (x+y)\mathrm{/}\sqrt{2}$ and $(x,y)\mapsto (x-y)\mathrm{/}\sqrt{2}$ are continuous as well, so [F5] makes $U=(W+W^{\mathrm{\prime }})\mathrm{/}\sqrt{2}$ and $V=(W-W^{\mathrm{\prime }})\mathrm{/}\sqrt{2}$ Borel random elements. For any finite list of functionals used on either $U$ or $V$, the corresponding vectors from $W$ and $W^{\mathrm{\prime }}$ are independent centered Gaussian vectors, so [F6] makes their stack multivariate Gaussian. The finite coordinate vector of $(U,V)$ is a linear image of that stack, hence is also centered multivariate Gaussian.

Centering and independence give, for all $\mathrm{\ell },m\in B^{\ast }$, $E[\mathrm{\ell }(U)m(V)]=\frac{1}{2}(E[\mathrm{\ell }(W)m(W)]-E[\mathrm{\ell }(W)m(W^{\mathrm{\prime }})]+E[\mathrm{\ell }(W^{\mathrm{\prime }})m(W)]-E[\mathrm{\ell }(W^{\mathrm{\prime }})m(W^{\mathrm{\prime }})])=0$. Independence and centering give $E[\mathrm{\ell }(U)m(U)]=E[\mathrm{\ell }(W)m(W)]$ and $E[\mathrm{\ell }(V)m(V)]=E[\mathrm{\ell }(W)m(W)]$. Thus each finite coordinate vector of $U$ or $V$ is centered Gaussian with the same covariance as the corresponding vector of $W$, so [F2] gives matching finite laws.

By [A2], the HB and countable-choice hypotheses of [F1] and [F4] hold. The zero mixed covariances from step 2.1 let [F1] make $U,V$ independent, while [F4] upgrades their matching finite norming-coordinate laws to equality of Borel laws with $W$. Thus $U,V$ are independent copies of $W$.

Let $p(r)=P(\mathrm{\parallel }W\mathrm{\parallel }>r)$. By [F3] and $\mathrm{\parallel }W\mathrm{\parallel }<\mathrm{\infty }$ almost surely, $p(r)\to 0$ as $r\to \mathrm{\infty }$; choose $s>0$ with $q:=1-p(s)>0$. For $r>s$, the event $\{\mathrm{\parallel }W\mathrm{\parallel }>r,\mathrm{\parallel }{W}^{\mathrm{\prime }}\mathrm{\parallel }\le s\}$ has probability $p(r)q$. On it, $\sqrt{2}U=W+W^{\mathrm{\prime }}$ and $\sqrt{2}V=W-W^{\mathrm{\prime }}$ imply $\mathrm{\parallel }U\mathrm{\parallel }>(r-s)\mathrm{/}\sqrt{2}$ and $\mathrm{\parallel }V\mathrm{\parallel }>(r-s)\mathrm{/}\sqrt{2}$, by the reverse triangle inequality. Since $U,V$ are independent copies by step 3.1, $p(r)q\le p((r-s)\mathrm{/}\sqrt{2}{)}^2\mathrm{.}$

Choose $r_0>s$ with $p(r_0)<q$, and define $r_{n+1}=\sqrt{2}{r}_n+s$. If $p(r_0)=0$, then $W$ is bounded almost surely and the theorem follows immediately. Otherwise put $\theta =p(r_0)\mathrm{/}q\in (0,1)$. Iterating step 4.1 gives $p(r_n)\le q{\theta }^{2^n}$. The explicit recurrence solution is $r_n=(r_0+c)2^{n\mathrm{/}2}-c$, where $c=s\mathrm{/}(\sqrt{2}-1)$. Hence for all sufficiently large $n$, $r_n$ is bounded below by a positive multiple of $2^{n\mathrm{/}2}$ and $r_{n+1}\le 2r_n$. If $r_n<t\le r_{n+1}$, then $p(t)\le p(r_n)\le q{e}^{-\mathrm{\mid }\log \theta \mathrm{\mid }{2}^n}\le C{e}^{-b{t}^2}$ for constants $C,b>0$ independent of $n$. Thus $P(\mathrm{\parallel }W\mathrm{\parallel }>t)\le C{e}^{-b{t}^2}$ for all sufficiently large $t$.

Choose $0<a<b\mathrm{/}3$. Partition according to $\mathrm{\parallel }W\mathrm{\parallel }\le r_N$ and the intervals $r_n<\mathrm{\parallel }W\mathrm{\parallel }\le r_{n+1}$ for sufficiently large fixed $N$. On the $n$th interval, $e^{a\mathrm{\parallel }W{\mathrm{\parallel }}^2}\le e^{a{r}_{n+1}^2}$, while step 5.1 bounds its probability by $C{e}^{-b{r}_n^2}$. Since $r_{n+1}^2\mathrm{/}{r}_n^2\to 2$, the resulting series is bounded for large $n$ by a constant times $e^{-(b-3a)r_n^2}$ and converges. The initial bounded interval contributes finitely. Therefore $E{e}^{a\mathrm{\parallel }W{\mathrm{\parallel }}^2}<\mathrm{\infty }$, and then $\mathrm{\parallel }W{\mathrm{\parallel }}^2\le c_a{e}^{a\mathrm{\parallel }W{\mathrm{\parallel }}^2}$ for a finite constant $c_a$ proves the second claim.