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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

MSC:

60F17 Functional limit theorems; invariance principles
60J65 Brownian motion

Citations:

Zbl 0593.00014
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