Summary.
This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ∞(Ψ)-valued continuous-time stochastic processes t⇝X t n=(X t n ,ψ|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X t n ,ψ is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived.
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Received: 6 May 1996 / In revised form: 4 February 1997
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Nishiyama, Y. Some central limit theorems for ℓ∞-valued semimartingales and their applications. Probab Theory Relat Fields 108, 459–494 (1997). https://doi.org/10.1007/s004400050117
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DOI: https://doi.org/10.1007/s004400050117
- Mathematics Subject Classification (1991): 60F05
- 60F17
- 62E20