A bounded-density Gaussian bridge has a Borel multiscale law

The displayed kernel is symmetric and positive semidefinite because for any finite coefficients $a_j$, ${\sum }_{i,j}{a}_i{a}_{j}K(i,j)=P_0[{\sum }_j{a}_j({\psi }_j-P_0{\psi }_j){]}^2\ge 0$. Thus [F1] constructs a centered jointly Gaussian coordinate sequence $Z$ with this covariance. Each coordinate variance is at most $P_0{\psi }_{lk}^2\le D$.

For a centered normal of variance $v\le D$, completing the square in its normal density gives $\mathbb{E}{e}^{tZ}=e^{v{t}^2\mathrm{/}2}$; at $v=0$ the variable is zero almost surely. Exponential Markov with $t=u\mathrm{/}D$ therefore gives $\mathrm{Pr}(\mathrm{\mid }Z\mathrm{\mid }>u)\le 2e^{-u^2\mathrm{/}(2D)}$ when $D>0$, as it is for a probability density. Consequently $\mathrm{Pr}\{{\max }_k\mathrm{\mid }{Z}_{lk}\mathrm{\mid }>R\sqrt{l+1}\}\le 2C{2}^l{e}^{-R^2(l+1)\mathrm{/}(2D)}$. Choose $R$ with $R^2\mathrm{/}(2D)>2\log 2$. The sum over $l$ is finite, and the union bound over $l\ge J$ tends to zero. The probability that infinitely many of these events occur is therefore zero, since that event is contained in every such tail union.

Almost surely the normalized maxima ${\max }_k\mathrm{\mid }{Z}_{lk}\mathrm{\mid }\mathrm{/}{w}_l$ tend to zero, by step 2.1 and admissibility $w_l\mathrm{/}\sqrt{l+1}\to \mathrm{\infty }$; each of the finitely many remaining coordinates is finite. The event $E=\{Z\in M_0(w)\}$ is coordinate-measurable, since the little-o condition is expressed by countable unions and intersections of the inequalities ${\max }_k\mathrm{\mid }{Z}_{lk}\mathrm{\mid }\mathrm{/}{w}_l\le 1\mathrm{/}q$. It has probability one. Define $G=Z$ on $E$ and $G=0$ otherwise. Its finite projections are measurable finite-dimensional random elements and converge everywhere in the separable Banach space [F2]. Hence $G$ is Borel measurable: for every closed set $F$, the measurable functions $d(P_{J}G,F)$ converge to $d(G,F)$; inverse images of closed sets are thus measurable, and closed sets generate the Borel sigma-algebra.

Let $L$ be any continuous real linear functional on $M_0(w)$. Then $L(P_{J}G)\to L(G)$ almost surely, and each approximant is centered normal with some variance $v_J$. These variances are bounded. Indeed, almost-sure convergence makes the laws tight: choose $M$ so that $\mathrm{Pr}\{{\sup }_J\mathrm{\mid }L(P_{J}G)\mathrm{\mid }>M\}<1\mathrm{/}2$, using that the supremum is finite almost surely. If $v_J\to \mathrm{\infty }$ on a subsequence, the normal density bound would give $\mathrm{Pr}(\mathrm{\mid }L(P_{J}G)\mathrm{\mid }\le M)\le 2M\mathrm{/}\sqrt{2\pi v_J}\to 0$, a contradiction. Any convergent subsequence of $(v_J)$ with limit $v$ gives, by [F3] and [F5], $\mathbb{E}{e}^{itL(G)}=e^{-v{t}^2\mathrm{/}2}$. Taking $t=1$ shows that all subsequential variance limits coincide. Thus $v_J\to v\ge 0$, and [F3] identifies $L(G)$ as $N(0,v)$. Therefore $G$ is a centered Borel Gaussian random element.

Its coordinate covariance and finite Gaussian laws are unchanged by the null-set modification in step 3.1. Finally the Borel sigma-algebra of $M_0(w)$ equals the sigma-algebra generated by its countably many coordinates: the norm is their countable weighted supremum, so all open balls about a countable dense set are coordinate-measurable, and these balls generate the topology. Thus two Borel laws with the same finite-coordinate laws agree, by uniqueness of the cylinder law in [F1] restricted to this measurable space. This proves existence and uniqueness of the stated bridge law.