Morrow, Gregory J.; Silverstein, Martin L. Two parameter extension of an observation of Poincaré. (English) Zbl 0609.60042 Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 396-418 (1986). [For the entire collection see Zbl 0593.00014.] Let \(B^ n_ u\) (u\(\geq 0)\) be a standard n-dimensional Brownian motion starting at the origin, \(T^ n(t)=\inf (u>0:| B^ n(u)|^ 2=e^ t)\), \(y^ n(t)=e^{-t/2}B^ n(T^ n(t))\), \(y^ n(t,j)=j\)-th component of \(y^ n(t)\), \(t\in [-T,T]\), \(T>0\). Define the following càdlàg process \(X^ n_ t\) with values in the space of continuous functions on [0,1] by \(X^ n_ t(0)=0\), \(X^ n_ t(k/n)=\sum^{k}_{j=1}y^ n(t,j)\), \(1\leq k\leq n\), interpolating for \(k/n<s<(k+1)/n\). The authors prove that \(X^ n\) converges weakly as \(n\to \infty\) to the infinite dimensional Ornstein-Uhlenbeck process of Malliavin. Reviewer: R.Mikulevičius Cited in 1 Document MSC: 60F17 Functional limit theorems; invariance principles 60J65 Brownian motion Keywords:infinite dimensional Ornstein-Uhlenbeck process Citations:Zbl 0593.00014 PDFBibTeX XML Full Text: Numdam EuDML