A multivariate central limit theorem for continuous local martingales
Let \(M\) be a \(d\)-dimensional continuous local martingale with \(K_t\langle M\rangle_tK^d_t \to^P\eta \eta^T\) for some non-random invertible \(d\times d\)-matrices \(K_t\) such that \(\|K_t\|\to 0\) as \(t\to\infty\) and some possibly random \(d\times d\)-matrix \(\eta\). The main result: Under these conditions, \((K_tM_t)\) converges for \(t\to\infty\) stably to a ``variance mixed'' normal distribution with mixing distribution equal to those of \(\eta\eta^T\) in the sense that \((K_tM_t)\) converges stably to \(V\) where \(V\) has the characteristic function \(u\to E\exp (-{1\over 2}u^T \eta\eta^Tu)\). This theorem forms a generalization of a similar result of the reviewer and \textit{M. Sørensen} [Bernoulli 5, No. 3, 483--493 (1999; Zbl 0943.60016)] for (in general discontinuous) square integrable martingales.
scientific article; zbMATH DE number 1546527