Abstract
In this chapter, we lay down the last cornerstone that is needed to derive functional limit theorems for processes. Namely, we consider the space 𝔻(ℝd) of all càdlàg functions: ℝ+→ℝd; we need to provide this space with a topology, such that: (1) the space is Polish (so we can apply classical limit theorems on Polish spaces); (2) the Borel σ-field is exactly the σ-field generated by all evaluation maps (because the “law” of a process is precisely a measure on this o-field).
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Bibliographical Comments
The first examples of weak convergence are due to Kolmogorov [130], Erdös and Kac [51], Donsker [40] and Maruyama [171].
The basic facts of the chapter, weak convergence and properties of the Skorokhod (J 1) topology, originate in the works of Prokhorov [200] and Skorokhod [223], and they also appear in Billingsley [12]. In these references, the authors consider processes indexed by [0,1], but in many instances it is more natural to consider processes indexed by (ℝ+. For this purpose, the Skorokhod topology was extended by Stone [230] and Lindvall [154], and here we essentially follow Lindvall’s method. The metric δ’ of Remark 1.27 has been described by Skorokhod [223]; Kolmogorov [131] showed that the space D with the associated topology is topologically complete, and the metric δ of 1.26 for which it is complete was exhibited by Prokhorov [199].
It should be emphazised that Prokhorov’s Theorem 3.5 has two parts:
(1) all relatively compact sequences of measures are tight,
(2) all tight sequences are relatively compact.
For (2) we only need a metric separable space, and of course (2) is the most useful of the two statements. However, (1) requires completeness, and we also use (1) in this book (in Section 6 for example).
The results of Section 2 are essentially “well-known”, and scattered through the literature. See e.g. Billingsley [12], Aldous [2], Whitt [245] , Pagès [192]. §2b is taken from Jacod and Mémin [107].
Aldous’ criterion was introduced in [1]. Theorem 4.13 is due to Rebolledo [202], and 4.18 is a modernized version of results in Liptser and Shiryaev [158] and Jacod and Mémin [107] (see also Lebedev [141]; other results belonging to the same circle of ideas can be found in Billingsley [13] and Grigelionis [73]).
Section 5 is based upon Jacod, Mémin and Métivier [111], with an amelioration due to Pagès [193] (condition C5). Section 6 has its origin in Liptser and Shiryaev [159], and the general case comes from Jacod [100].
The condition P-UT has been introduced, under the (slightly misleading) name UT, by Jakubowski, Mémin and Pagès in [113] in order to obtain a stability result for stochastic integrals (Theorem 6.22). As said in 6.2, This condition is strongly related with the Bichteler-Dellacherie-Mokobodski characterization of semimartingales. The various criteria given in § 6a can be found in various papers by Kurtz and Protter [277], [278], [279] and Mémin and Slominski [282], and also Strieker [293]. Theorem 6.26 has its origin in Liptser and Shiryaev [159] and the general case comes from Jacod [100] (as well as Remark 6.28), except that the P-UT condition is replaced by a condition expressed in terms of the characteristics (and which turns out to be equivalent to P-UT). Proposition 7.3 is taken from Slominski [291], while Proposition 7.5 comes from Jacod and Protter [272].
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© 2003 Springer-Verlag Berlin Heidelberg
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Jacod, J., Shiryaev, A.N. (2003). Skorokhod Topology and Convergence of Processes. 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_6
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DOI: https://doi.org/10.1007/978-3-662-05265-5_6
Publisher Name: Springer, Berlin, Heidelberg
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