# zbMATH — the first resource for mathematics

On continuous conditional Gaussian martingales and stable convergence in law. (English) Zbl 0884.60038
Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 232-246 (1997).
Summary: We start with a stochastic basis $$(\Omega,{\mathcal F},\mathbb{F} = ({\mathcal F}_t)_{t\in [0,1]}, P)$$, the time interval being $$[0,1]$$, on which are defined a “basic” continuous local martingale $$M$$ and a sequence $$Z^n$$ of martingales or semimartingales, asymptotically “orthogonal to all martingales orthogonal to $$M$$”. Our aim is to give some conditions under which $$Z^n$$ converges “stable in law” to some limiting process which is defined on a suitable extension of $$(\Omega,{\mathcal F},\mathbb{F},P)$$. In the first section we study systematically some, more or less known, properties of extensions of filtered spaces and of $${\mathcal F}$$-conditional Gaussian martingales and so-called $$M$$-biased $${\mathcal F}$$-conditional Gaussian martingales. Then we explain our limit results. In Section 2 we give a fairly general result, and in Section 3 we specialize to the case when $$Z^n$$ is some “discrete-time” process adapted to the discretized filtration $$\mathbb{F}^n=({\mathcal F}^n_t)_{t\in[0, 1]}$$, where $${\mathcal F}^n_t= {\mathcal F}_{[nt]/n}$$. Finally, Section 4 is devoted to studying the limit of a sequence of $$M$$-biased $${\mathcal F}$$-conditional Gaussian martingales.
For the entire collection see [Zbl 0864.00069].

##### MSC:
 60G44 Martingales with continuous parameter 60E07 Infinitely divisible distributions; stable distributions
Full Text: