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For times $t_1,\dots ,t_n<1$ and real coefficients $a_i$, the linear combination of coordinates isThe Gaussianity of deterministic Brownian stochastic integrals makes this a centered normal random variable. Indeed step-function approximations give linear combinations of independent normal increments, and the Itô isometry gives convergence in $L^2$; their characteristic functions converge to that of a normal variable with the limiting variance. Since every linear combination is normal, the coordinate vector is a Gaussian random vector. Appending the deterministic coordinate $X_1=0$ preserves that property, so $X$ is a centered Gaussian process on $[0,1]$.

For $0\le s\le t<1$, the covariance version of the Itô isometry givesAt $t=1$ the covariance is zero because $X_1=0$, agreeing with the formula. Thus the Brownian bridge covariance kernel isIts finite-dimensional distributions identify it as a Brownian bridge; continuity at the terminal endpoint is established next.

Let $(B_t{)}_{0\le t\le 1}$ be standard Brownian motion with its continuous version, and defineEvery finite vector is a linear transform of a Gaussian Brownian vector, so $Z$ is a centred Gaussian process. Using $\mathbb{E}[B_s{B}_t]=\min (s,t)$,For $s\le t$, this is $s(1-t)$, the prescribed Brownian bridge covariance kernel. Its paths are continuous on the probability-one event of Brownian continuity, because $t\mapsto t{B}_1$ is continuous on every sample. The construction also gives $Z_0=Z_1=0$. Thus it realizes a Brownian bridge with almost surely continuous paths, and part (i) identifies its process law.

The kernel is the Brownian bridge covariance kernel. If $C_X\varphi =\lambda \varphi$, differentiating the integral equation twice givesThe normalized solutions and eigenvalues are therefore

The stated kernel is the Brownian bridge covariance kernel. Its eigenvalue equation isThe right-hand side vanishes at $s=0$ and $s=1$, and differentiating it twice givesThus the normalized eigenfunctions and eigenvalues areThe Karhunen–Loève expansion is consequentlywith convergence in $L^2(\Omega ;L^2[0,1])$, where $\mathbb{E}{Z}_k=0$ and $\mathbb{E}[Z_j{Z}_k]=1_{\{j=k\}}$. Covariance alone does not imply that the $Z_k$ are independent or normal; they are independent standard normal variables when $X$ is Gaussian.

The Brownian bridge covariance kernel is invariant under reversing both times, and its mean is zero. The two centered Gaussian processes therefore have equal finite-dimensional laws. Equality on countably many rational coordinates transfers the event of continuity at zero to the reversed endpoint. Interior continuity then extends the rational-time limit to all times. This proves the terminal continuity of the stochastic integral representation of a Brownian bridge.