The weak approximation of the empirical characteristic function process when parameters are estimated (Q1185786)

Let \(X_ 1,\dots,X_ n\) be i.i.d. real random variables with characteristic function \(\phi (t)\). Define the empirical characteristic function of \(X_ 1,\dots,X_ n\) by \(\phi_ n (t) = n^{-1} \sum_{j=1}^ n \exp (it X_ j)\). Let \(n^{1/2} \bigl( \phi_ n (t) - \phi(t)\bigr)\) be the empirical characteristic function process. \textit{S. Csörgö} [Ann. Probab. 9, 130-144 (1981; Zbl 0453.60025)] has shown that the empirical characteristic function process may be strongly approximated by a complex-valued Gaussian process. The present paper similarly approximates the empirical characteristic function process with estimated parameters under a sequence of local alternatives. In this setting the characteristic function has the assumed form \(\{ \phi (t;\beta_ 0,\theta); \theta \in\Theta\}\) where \(\beta_ 0\) is a \(p\)- dimensional vector of specified parameters and \(\Theta\) is a \(q\)- dimensional vector of unknown parameters in some suitable parameter space \(\Theta\).