Abstract
Let us roughly describe the problems which will retain our attention in this last chapter.
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Bibliographical Comments
The motivations of this chapter are statistical, and originate in the work of LeCam [142], who was the first to give a precise meaning to “asymptotically Gaussian experiments“. LeCam discussed this notion and conditions to achieve it in various papers (see e.g. [144]) and his book [145]; see also Hajek [81] or Ibragimov and Has’minski [88], while Kutoyants [140] provides a discussion which concerns the case of continuous-time stochastic processes.
Of course, in statistical applications, one is concerned with general parametric models rather than looking at “simple hypotheses”: for each n there is a family (math) of measures, and thus a family (math) of likelihood processes with respect to some reference measure Q n. Then one looks at the limit (math), either as finite-dimensional in (θ, t), or as functional in 0 for a given value of t, or functional in (θ, t).
The main result of asymptotic normality 1.12 is new in this form, but it has been proved in the discrete-time case (i.e. when each pre-limiting filtration F” is a discrete-time filtration) by Greenwood and Shiryaev [68], and in continuous-time by Vostrikova [242] under a mild additional assumption: this paper also contains the finite-dimensional (in 0) convergence result refered to above, while the functional convergence in (0, t) is proved in Vostrikova [243]. § 1d is new.
The statistical models called here “exponential families of stochastic processes”, which naturally extend the classical exponential statistical models, have been considered by various authors: LeCam [143], Stefanov [228,229], and also Sörensen [226] (see a complete bibliography in this last paper). They encompass a lot of particular cases (as models for branching processes) and are used mainly for sequential analysis (optimal stopping, etc.). Proposition 2.5 is just an exercise; Theorem 2.12 is new, but Mémin [178] has proved a closely related result (where the conditions are expressed in terms of the processes Z n themselves), and his method is different (and simpler, but it only gives the functional convergence). See also Taraskin [234] for a result closely related to 2.12.
§ 3a contains simple variations about LeCam’s third Lemma (see e.g. Grige-lionis and Mikulevicius [79] for results in this direction). § 3b is new, but the same results when the processes X n are discrete-time are due to Greenwood and Shiryaev [68].
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© 2003 Springer-Verlag Berlin Heidelberg
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Jacod, J., Shiryaev, A.N. (2003). Limit Theorems, Density Processes and Contiguity. In: Limit Theorems for Stochastic Processes. Grundlehren der mathematischen Wissenschaften, vol 288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05265-5_10
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DOI: https://doi.org/10.1007/978-3-662-05265-5_10
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