Limit Theorems for Stochastic Processes pp 456-520 | Cite as

# Convergence to a Process with Independent Increments

Chapter

## 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.

### Keywords

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### Bibliographical Comments

- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- from Jacod, Klopotowski and Mémin [105], and § 5b is new (see also Grigelionis and Mikulevicius [78] and Rootzen [210]).Google Scholar
- 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

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