An asymptotic formula of the Hellinger transform for multivariate point processes (Q1873264)

Let \((\Omega, \mathbb{F}, P)\) be a filtered measurable space, and let \((E, \mathbb{E})\) be a Blackwell space. A multivariate (marked) point process with mark space \(E\) is defined as a sequence \((T_n, X_n)_{n\geq 1}\), where \(T_n\) are \(F\)-stopping times such that \(0< T_1\), \(T_n < T_{n+1}\) if \(T_n<\infty\), \(T_n = T_{n+1}\) if \(T_n = \infty\) and \(X_n\) are \(\mathbb{F}_{T_n}\)-measurable random variables with values in \((E, \mathbb{E})\). The author and \textit{Yu. N. Lin'kov} [cf. ``Asymptotic statistical methods for stochastic processes'' (2001; Zbl 0959.62068)] used the Hellinger transform in the investigation of properties of optimal estimates and statistical criteria. The author establishes a formula of the rate function for probability measures corresponding to stochastic integrals for \((T_n, X_n)_{n\geq 1}\) and finds an asymptotic formula of the Hellinger transform for such processes.