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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
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