A Gaussian wavelet series at the Besov threshold

Let $H_0=L^2([0,1])$. By [F1] under countable choice [A2], it is a real Hilbert space. The complete basis [F10] has a countable index set, so its finite rational linear combinations form a countable set. Its finite real linear span is dense by [F10], and approximating each finite list of real coefficients by rationals makes the rational span dense as well. Hence $H_0$ is separable by [F11]. For the first application of [F3], take $H_1=H_0$ with the identity inclusion; the supplied wavelet family is a complete orthonormal basis in both spaces, so $w_j=1$ for every index. Enumerate the finite scaling family first and the detail wavelets by increasing $\mathrm{\ell }$ and then $k$, set ${\lambda }_i=0$ on the scaling family, and set ${\lambda }_{\mathrm{\ell }k}=2^{-2\mathrm{\ell }(s+1\mathrm{/}2)}$ and retain $w_j=1$ on all indices. Then ${\sum }_j{\lambda }_j{w}_j={\sum }_{\mathrm{\ell }\ge 0}{2}^{\mathrm{\ell }}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}={\sum }_{\mathrm{\ell }\ge 0}{2}^{-2s\mathrm{\ell }}<\mathrm{\infty }$ because $s>0$. Apply [F3] under AC and countable choice [A1]–[A2] to obtain the Borel centered Gaussian limit $W$ in $H_0$. The identity on $D=H_0$ is measurable and $\mathbb{P}(W\in D)=1$, so [F4] makes its law a prior on $D$.

Enumerate the detail coefficients by increasing level and then by $k$, write the corresponding standard normals as $({\eta }_n{)}_{n\ge 1}$, and set $Z_n={\eta }_n^2$. By [F9] under countable choice [A2], the square map is measurable. Since each ${\eta }_n$ has the same standard normal law, the $Z_n$ have the same pushforward law under this map; [F6] preserves independence, so [F7] makes $(Z_n)$ IID. By [F5] under AC [A1], their mean is $1$ and their variance is finite; [F8] gives $R_N\mathrm{/}N\to 1$ almost surely for $R_N={\sum }_{n=1}^N{Z}_n$. For every $\mathrm{\ell }\ge 0$, set $A_{\mathrm{\ell }}=2^{-\mathrm{\ell }}{\sum }_{k<2^{\mathrm{\ell }}}{\xi }_{\mathrm{\ell }k}^2$. For $\mathrm{\ell }\ge 1$, put $N_{\mathrm{\ell }}=2^{\mathrm{\ell }}-1$, so $A_{\mathrm{\ell }}=(R_{N_{\mathrm{\ell }+1}}-R_{N_{\mathrm{\ell }}})\mathrm{/}{2}^{\mathrm{\ell }}$. Since $N_{\mathrm{\ell }+1}\mathrm{/}{2}^{\mathrm{\ell }}\to 2$ and $N_{\mathrm{\ell }}\mathrm{/}{2}^{\mathrm{\ell }}\to 1$ as $\mathrm{\ell }\to \mathrm{\infty }$, the strong-law limits give $A_{\mathrm{\ell }}\to 2-1=1$ almost surely for the same variables used in step 1.1.

Fix $0\le r<s$ and take $H_1=B_{2,2}^r([0,1])$. By [F2], it is a separable real Hilbert space continuously included in $H_0$, and the same wavelet family is complete and orthogonal in $H_1$, with squared basis norms $1$ on the finite scaling family and $w_{\mathrm{\ell }k}=2^{2r\mathrm{\ell }}$ on detail levels. The weighted eigenvalue sum is ${\sum }_{\mathrm{\ell }\ge 0}{2}^{\mathrm{\ell }}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}{2}^{2r\mathrm{\ell }}={\sum }_{\mathrm{\ell }\ge 0}{2}^{-2(s-r)\mathrm{\ell }}<\mathrm{\infty }$. Apply [F3] under AC and countable choice [A1]–[A2] to the same coefficients as in step 1.1; the partial wavelet sums converge almost surely in $H_1$. Their $H_0$ limit is $W$ by step 1.1, and continuity of the inclusion makes the $H_1$ limit the same element. Hence $W\in B_{2,2}^r([0,1])$ almost surely for this fixed $r$.

For each detail wavelet, its coordinate functional is continuous on $H_0$; after the term enters the partial sums, the coefficient is $2^{-\mathrm{\ell }(s+1\mathrm{/}2)}{\xi }_{\mathrm{\ell }k}$, so convergence in step 1.1 gives the same coefficient for $W$. The scaling coefficients vanish by construction. Therefore $\mathrm{\parallel }W{\mathrm{\parallel }}_{B_{2,2}^s}^2={\sum }_{\mathrm{\ell }\ge 0}{2}^{2\mathrm{\ell }s}{\sum }_{k<2^{\mathrm{\ell }}}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}{\xi }_{\mathrm{\ell }k}^2={\sum }_{\mathrm{\ell }\ge 0}{A}_{\mathrm{\ell }}$. By step 2.1, $A_{\mathrm{\ell }}\to 1$ almost surely, so the nonnegative series diverges and $W\mathrm{\notin }{B}_{2,2}^s([0,1])$ almost surely.