Abstract
We examine here two apparently disconnected sorts of problems. The relation between them essentially comes from the fact that, in order to solve both of them, we use the same tool, namely the Hellinger processes introduced in the previous chapter.
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Bibliographical Comments
The notions of contiguity and entire separation are due to LeCam [142]. An extensive account on contiguity and its statistical applications (especially for the independent and the Markov case) may be found in Roussas [217]). Some of the basic equivalences in Lemmas 1.6 and 1.13 appear in Hall and Loynes [85], Liptser, Pukelsheim and Shiryaev [156], Eagleson and Mémin [48], Hájek and Sidak [82], Jacod [102], Greenwood and Shiryaev [68].
In the discrete-time setting, the first general contiguity result (case of independent variables) is due to Oosterhof and Van Zwet [191], and the general criterion appeared in Liptser, Pukelsheim and Shiryaev [156] and Eagleson and Mémin [48] (the latter assumes local absolute continuity). The continuous-time problem was solved by Liptser and Shiryaev [162] (with a different method) and Jacod [102] (the criteria given here, though, are slightly different).
The relations between Hellinger integrals of various order and the variation metric can be found in Kraft [134] or Matusita [172]; see also Vajda [237]. Proposition 4.16 is just an exercise on multiplicative decompositions of nonnegative supermartingales; its corollary 4.19 was given by Kabanov, Liptser and Shiryaev [123], as well as the estimates of 4.21. The discrete-time version of Theorem 4.31 is in Vostrikova [241]. Essentially all the results of §4c are due to Kabanov, Liptser and Shiryaev [116,117,122], T. Brown [26], Valkeila [238], and Mémin [176]. The case of diffusion processes (§ 3d) has been investigated by Liese [151].
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© 2003 Springer-Verlag Berlin Heidelberg
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Jacod, J., Shiryaev, A.N. (2003). Contiguity, Entire Separation, Convergence in Variation. 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_5
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DOI: https://doi.org/10.1007/978-3-662-05265-5_5
Publisher Name: Springer, Berlin, Heidelberg
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