It is known, that a standard multivariate Brownian bridge $ y(\mathbf u) $ is a centered Gaussian process with covariance function $$ \mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge v_j) - \prod_{j=1}^d u_j v_j $$

I am not sure about how to constuct such a multivariate Brownian bridge.

My first thought was to start somehow with a univariate Brownian bridge. I have found information about that and even a package in R that can do this, but only for the univariate Brownian bridge.

I found this, but as I understand it, what has been done there is not a standard multivariate Brownian bridge as defined above or e.g. in this paper.

I would appreciate any hints and support.

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    $\begingroup$ As I found out in Deheuvels paper link there is the following relationship between a Brownian Bridge $ B_t$ and a Brownian Sheet (or Wiener Sheet) $ W_t $: $$ B_t := W_t - \frac t T W_T $$ So I think the problem reduces to simulating a Brownian sheet. I will ask my questions about this in a seperate question. $\endgroup$ – andeliyeasi Apr 8 '14 at 12:34
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    $\begingroup$ correction, the relationship for more dimensions is $$ B_{\mathbf t} := W_{\mathbf t} - \prod_{j=1}^d t_j W_{(1,...,1)} $$ $\endgroup$ – andeliyeasi Apr 8 '14 at 12:46
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    $\begingroup$ Related: stats.stackexchange.com/questions/34354/… $\endgroup$ – David R Aug 26 '16 at 22:43

As you already pointed out in the comments, the question reduces to simulating a Brownian sheet. This can be done by generalizing simulation of Brownian motion in a straightforward way.

To simulating the Brownian motion, one can take an i.i.d. mean-0 variance-1 time series $W_i$, $i = 1, 2, \cdots$, and construct the normalized partial sum process $$ X_n(t) = \frac{1}{\sqrt{n}} \sum_{i = 1}^{[nt]} W_i. $$ As $n \rightarrow \infty$, $X_n$ convergence weakly (in the sense of Borel probability measures on a metric space) to the standard Brownian $B$ on the Skorohod space $D[0,1]$.

The i.i.d. with finite second moment case is the simplest way to simulate. The mathematical result (Functional Central Limit Theorem/Donsker's Theorem/Invariance Principle) holds in much greater generality.

Now to simulating the (say, two-dimensional) Brownian sheet, takes i.i.d. mean-0 variance-1 array $W_{ij}$, $i,j = 1, 2, \cdots$, and construct the normalized partial sum process $$ X_n(t_1, t_2) = \frac{1}{ n } \sum_{1 \leq i \leq [nt_1] , 1 \leq j \leq [nt_2]} W_{ij}. $$ As $n \rightarrow \infty$, $X_n$ convergence weakly to the standard Brownian sheet on the Skorohod space $D([0,1]^2)$ on the unit square.

(The proof is a standard weak convergence argument:

  1. Convergence of finite dimensional distribution follows from the Levy-Lindeberg CLT.

  2. Tightness on $D([0,1]^2)$ follows from a sufficient moment condition that holds trivially in the i.i.d. finite second moment case---see, e.g. Bickel and Wichura (1971). )

Then, by the continuous mapping theorem $$ X_n(t_1, t_2) - \prod_{j=1}^2 t_j X_n(t_1, t_2) $$ converges weakly to the two-dimensional Brownian bridge.

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