# Convergence to a Process with Independent Increments

• Jean Jacod
• Albert N. Shiryaev
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 288)

## Abstract

This chapter constitutes the second step on our way to general limit theorems. We consider a sequence (X n ) of semimartingales, with characteristics (B n, C n , v n ), and a limiting process X which is a PII with characteristics (B, C, v). Our main objective is to prove that the various conditions of Chapter VII still insure the (functional or finite-dimensional) convergence of (X n ) to X, although the X n ’s are no longer PII.

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1. The basic idea that underlies Theorem 1.9, in its present form, comes from Kabanov, Liptser and Shiryaev [121], and the formulation itself appears in Jacod, Klopotowski and Mémin [105]. But the first “general” convergence results for sums of dependent random variables are due to Bernstein [8] and Lévy [148], and the idea of considering conditional expectations and convergence in measure in the conditions for convergence of rowwise dependent triangular arrays originates in Dvoretsky [46].Google Scholar
2. Many authors have proved various versions of the theorems presented in Section 2, essentially (but not exclusively) for triangular arrays, and either for finite-dimensional convergence or for functional convergence (usually when the limiting process is a Wiener process, in which case the result is also called “invariance principle“). Let us quote for example Billingsley [11], Borovkov [16, 17], B. Brown [22], B. Brown and Eagleson [23], Durrett and Resnick [45], Gänssler, Strobel and Stute [58], P. Hall [83], Klopotowski [128, 129], McLeish [174, 175], Rootzen [209, 210], Rosén [212], Scott [220], etc... Books (partly) devoted to this subject include Ibragimov and Linnik [89], Hall and Heyde [84], and to a lesser extent Billingsley [12] and Ibragimov and Has’minski [88]. The forms 2.4 and 2.17 are taken from Jacod and Mémin [108], Liptser and Shiryaev [158, 160], Jacod, Klopotowski and Mémin [105] for the most general version. Theorem 2.20 is due to Jakubowski and Slominski [114].Google Scholar
3. The content of §§ 3a,b,c provides unification for a lot of results in the literature, especially concerning triangular arrays of martingale differences (McLeish [174, 175], Scott [220], B. Brown [22], etc.). It also contains the “necessary” part due to Gänssler and Hausier [57] and Rootzen [211] for triangular arrays, and to Liptser and Shiryaev [159, 161] in general (see also Rebolledo [205]).Google Scholar
4. Theorem 3.36 is essentially due to T. Brown [24], the method is taken from Kabanov, Liptser and Shiryaev [121]. Proposition 3.40 also is due to T. Brown [25]. Theorem 3.43 was first proved by Mémin. Theorem 3.54 is a particular case of a result due to Giné and Marcus [63] (a close look at [63] shows indeed that, although the authors do not speak about characteristics, the basic steps of the proof are the same as here): theorems of such type really belong to the theory of “central limit theorem in Banach spaces” (although D(R) is not even a topological vector space!). It should be emphazised that our approach does not seem to provide with a very powerful method to solve this type of problems.Google Scholar
5. Theorem 3.65 (continuous-time version) is borrowed to Touati [235]. The discrete-time version 3.74 is essentially due to Gordin and Lisic [67]. (see also Bhattacharya [9]). This type of theorems shows indeed that our convergence conditions sometimes cannot be applied directly: one has first to transform the semimartingales of interest into other semimartingales to which our theorems apply, plus some remainder terms which we can control.Google Scholar
6. The content of §3g (and §5e as well) is intended to give an idea of a vast subject, initiated by Rosenblatt [213], and pursued by many authors, e.g. Rozanov and Volkonski [218], Statulevicius [227], Serfling [221], Gordin [66], McLeish [173,174] (who introduced the concept of “mixingale” to unify martingales and mixing processes), etc... For more bibliographical information, and also for many other variants of the theorems, see the books [89] of Ibragimov and Linnik, and [84] of Hall and Heyde. 3.102a is due to Serfling [221], 3.102b is due to McLeish [173]. Results very similar to Theorems 3.79 or 3.97 may be found in Chikin [28] and in Dürr and Goldstein [44].Google Scholar
7. Theorem 4.1 comes from Jacod, Klopotowski and Mémin [105], and Theorem 4.10 from Kabanov, Liptser and Shiryaev [121]. §4c is due to Liptser and Shiryaev [163].Google Scholar
8. Convergence of triangular arrays to a mixture of infinitely divisible laws is a rather old subject: see the history in the book [84] of Hall and Heyde (see also Klopotowski [129]) and, from the statistical point of view, in the book [5] of Basawa and Scott. In the present functional setting, § 5a is takenGoogle Scholar
9. from Jacod, Klopotowski and Mémin [105], and § 5b is new (see also Grigelionis and Mikulevicius [78] and Rootzen [210]).Google Scholar
10. Stable convergence has been introduced by Renyi [207], but it also appears in various disguises in control theory (Schäl [219]), Markov processes (Baxter and Chacon [6]), stochastic differential equations (Jacod and Mémin [109]). Here we follow the exposition of Aldous and Eagleson [3]; see also Hall and Heyde [84]. Lemma 5.34 is due to Morando [186] (see also Dellacherie and Meyer [36]). The nesting condition 5.37 appears in McLeish [175] and Hall and Heyde [84] for the discrete-time case, in Feigin [52] for the continuous-time; Theorem 5.42 is due to Feigin [52]. Theorem 5.50 and Corollary 5.51 may be found in Aldous and Eagleson [3] and Durrett and Resnick [45]. The idea of Theorem 5.53 belongs to Renyi [206, 207], as well as the notion of mixing convergence (§ 5d).Google Scholar