# How to create a multivariate Brownian Bridge?

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.

• 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. – andeliyeasi Apr 8 '14 at 12:34
• correction, the relationship for more dimensions is $$B_{\mathbf t} := W_{\mathbf t} - \prod_{j=1}^d t_j W_{(1,...,1)}$$ – andeliyeasi Apr 8 '14 at 12:46
• – 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.