The weak limit of a martingale of rank one (Q1080550)

Let \({\mathcal F}_ t^{\xi}=\sigma \{\xi (s):s\leq t\}\) be a filtration of a complete probability space (\(\Omega\),\({\mathcal F},P)\) and denote \(\bigvee^{k}_{i=1}{\mathcal F}_ t^{\xi_ i}\) the minimal \(\sigma\)- algebra which contains all \({\mathcal F}_ t^{\xi_ i}\), \(i=1,2,...,k\). A martingale \(\xi\) (t) is said to be of rank k if there exist independent Wiener processes \(w_ i(t)\), \(i=1,2,...,k\) such that \({\mathcal F}_ t^{\xi}=\bigvee^{k}_{i=1}{\mathcal F}_ t^{\xi_ i}.\) It is proved that there exist a sequence of martingales of rank one weakly converging to that of rank \(\infty\).