23
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ 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
  • 4
    $\begingroup$ 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
  • 2
    $\begingroup$ Related: stats.stackexchange.com/questions/34354/… $\endgroup$ – David R Aug 26 '16 at 22:43
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.