Journal of Probability and StatisticsVolume 2011 (2011), Article ID 580292, 34 pagesdoi:10.1155/2011/580292

Research Article

Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Andriy Yurachkivsky

Taras Shevchenko National University, 01601 Kyiv, Ukraine

Received 24 May 2011; Accepted 3 October 2011

Academic Editor: Tomasz J. Kozubowski

Copyright © 2011 Andriy Yurachkivsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let for each be an -valued locally square integrable martingale w.r.t. a filtration (probability spaces may be different for different ). It is assumed that the discontinuities of are in a sense asymptotically small as and the relation holds for all , row vectors , and bounded uniformly continuous functions . Under these two principal assumptions and a number of technical ones, it is proved that the 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes converge in distribution to some , then a sequence () converges in distribution to a continuous local martingale with initial value and quadratic characteristic , whose finite-dimensional distributions are explicitly expressed via those of .

1. Introduction

The theory of functional limit theorems for martingales may appear finalized in the monographs [1, 2]. This paper focuses at two points, where the classical results can be refined.

(1) The convergence in distribution to a local martingale with -conditional increments has been studied hitherto in the model, where the -algebra enters the setting along with the prelimit processes. This assumption is worse than restrictive—it is simply unnatural when one studies the convergence in distribution, not in probability. In the present paper, conditions ensuring asymptotic conditional independence of increments for a sequence of locally square integrable martingales are formulated in terms of quadratic characteristics of the prelimit processes (Theorem 4.5). Our approach to the proving of this property is based on the idea to combine the Stone-Weierstrass theorem (actually its slight modification—Lemma 2.2) with an elementary probabilistic result—Lemma 2.4, which issues in Corollaries 2.7 and 2.8. These corollaries, as well as Lemma 2.4 itself and the cognate Lemma 2.5, will be our tools.

(2) The main object of study in [1, 2] is semimartingale. So, some specific for local martingales facts are passed by. Thus, Theorem VI.6.1 and Corollary VI.6.7 in [2] assert that under appropriate assumptions about semimartingales , the relation where also is a semimartingale, entails the stronger one (below, the notation of convergence in law will be changed). For locally square integrable martingales, one can modify the problem as follows. Let relation (*) be fulfilled. What extra assumptions ensure that is a continuous local martingale and There is neither an answer nor even the question in [1, 2]. A simple set of sufficient conditions is provided by Corollary 5.2 (weaker but not so simple conditions are given by Corollary 5.5). Recalling that the quadratic variation of a continuous local martingale coincides with its quadratic characteristic, we see that the last two relations imply together asymptotic proximity of and . Actually, this conclusion requires even less conditions than in Corollary 5.2. They are listed in Corollary 5.3.

The main results of the paper are, in a sense, converse to Corollaries 5.2 and 5.5. They deal with the problem: what conditions should be adjoined to in order to ensure (**), where is a continuous local martingale with quadratic characteristic ? If the assumptions about the prelimit processes do not guarantee that performing as determines the distribution of , then results of this kind assert existence of convergent subsequences but not convergence of the whole sequence (Theorems 5.1 and 5.4). Combining Theorems 5.4 with 4.5, we obtain Theorem 5.6 asserting that the whole sequence converges to a continuous local martingale whose finite-dimensional distributions are explicitly expressed via those of its initial value and quadratic characteristic. The expression shows that the limiting process has conditionally independent increments—but this conclusion is nothing more than a comment to the theorem.

The proving of the main results needs a lot of preparation. Those technical results which do not deal with the notion of martingale are gathered in Section 2 (excluding Section 2.1), and the more specialized ones are placed in Section 3. The rationale in Sections 3 and 4 would be essentially simpler if we confined ourselves to quasicontinuous processes (for a locally square integrable martingale, this property is tantamount to continuity of its quadratic characteristic). To dispense with this restriction, we use a special technique sketched in Section 2.1.

All vectors are thought of, unless otherwise stated, as columns. The tensor square of will be otherwise written as . We use the Euclidean norm of vectors and the operator norm of matrices. The symbols , and signify: the space of -dimensional row vectors, the class of all symmetric square matrices of a fixed size (in our case—) with real entries, and its subclass of nonnegative (in the spectral sense) matrices, respectively.

By , we denote the space of complex-valued bounded continuous functions on a topological space . If and the dimension is determined by the context or does not matter, then we write simply Cb.

Our notation of classes of random processes follows [3]. In particular, and signify the class of all martingales with respect to a filtration (= flow of -algebras) and its subclass of uniformly integrable martingales. An -martingale will be called: square integrable if for all and uniformly square integrable if . The classes of such processes will be denoted and , respectively. The symbol will be suppressed if the filtration either is determined by the context or does not matter. If is a class of -adapted process, then by we denote the respective local class (see [2, Definition I.1.33], where the notation is used). Members of , and are called local martingales and locally (better local) square integrable martingales, respectively. All processes, except quadratic variations and quadratic characteristics, are implied -valued, where is chosen arbitrarily and fixed.

The integral will be written shortly (following [1, 2]) as if this integral is pathwise (i.e., is a process of locally bounded variation) or if it is stochastic. We use properties of stochastic integral and other basic facts of stochastic analysis without explanations, relegating the reader to [1–4]. The quadratic variation (see the definition in Section 2.3 [3] or Definition I.4.45 together with Theorem I.4.47 in [2]) of a random process and the quadratic characteristic of will be (and already were) denoted and , respectively. They take values in , which, of course, does not preclude to regard them as -valued random processes.

2. Some Technical Results

The Stone-Weierstrass theorem (see, e.g., [5]) concerns compact spaces only. In the following two, its minor generalizations (for real-valued and complex-valued functions, resp.) both the compactness assumption and the conclusion (that the approximation is uniform on the whole space) are weakened. They are proved likewise their celebrated prototype if one argues for the restrictions of continuous functions to some compact subset fixed beforehand.

Lemma 2.1. Let be an algebra of real-valued bounded continuous functions on a topological space . Suppose that separates points of and contains the module of each its member and the unity function. Then, for any real-valued bounded continuous function , compact set , and positive number , there exists a function such that and .

Lemma 2.2. Let be an algebra of complex-valued bounded continuous functions on a topological space . Suppose that separates points of , and contains the conjugate of each its member and the unity function. Then for any complex-valued bounded continuous function , compact set , and positive number there exists a function such that and .

We consider henceforth sequences of random processes or random variables given, maybe, on different probability spaces. So, for the th member of a sequence, and should be understood as and . In what follows, “u.i.” means “uniformly integrable”.

Lemma 2.3. In order that a sequence of random variables be u.i., it is necessary and sufficient that each its subsequence contain a u.i. subsequence.

Proof. Necessity is obvious; let us prove sufficiency.Suppose that a sequence is not u.i. Then, there exists such that for all Consequently, there exists an increasing sequence of natural numbers such that , where . Then, for any infinite set and , we have which means that the subsequence does not contain u.i. subsequences.

Lemma 2.4. Let for each be random variables given on a probability space , and a sub--algebra of . Suppose that for each , and for any the sequence is u.i. Then,

Proof. Denote . By the second assumption, the sequences ,, are stochastically bounded, which together with the first assumption yields for any . Hence, writing the identity and using both assumptions, we get (2.4) for . For arbitrary , this relation is proved by induction whose step coincides, up to notation, with the above argument.

The proof of the next statement is similar.

Lemma 2.5. Let and be random variables given on a probability space and a sub--algebra of . Suppose that and the sequences , and are u.i. Then,

Lemma 2.6. Let be an -valued random variable given on a probability space , and a sub--algebra of . Suppose that and the relation holds for all from some class of complex-valued bounded continuous functions on which separates points of the latter. Then, it holds for all .

Proof. Let denote the class of all complex-valued bounded continuous functions on satisfying (2.10). Obviously, it is linear. By Lemma 2.4, it contains the product of any two its members. So, is an algebra. By assumption, it separates points of . The other two conditions of Lemma 2.2 are satisfied trivially. Thus, that lemma asserts that for any and , there exists a function such that and . Then, By the choice of whence by the dominated convergence theorem Writing the identity we get from (2.11)–(2.13) which together with (2.9) and due to arbitrariness of proves (2.10).

Corollary 2.7. Let for each be -valued random variables given on a probability space and a sub--algebra of . Suppose that the relations hold for all and from some class of complex-valued bounded continuous functions on which separates points of the latter. Then, for all .

Proof. Denote . Condition (2.17) implies by Lemma 2.4 that relation (2.10) is valid for all of the kind , where . Obviously, such functions separate points of . Furthermore, condition (2.16) where runs over is tantamount to (2.9). It remains to refer to Lemma 2.6.

Corollary 2.8. Let for each be an -valued random process given on a probability space , a sub--algebra of , and an -measurable -valued random variable. Suppose that the relations and (2.16) hold for , all and any bounded uniformly continuous function on . Then, for any and the relation is valid.

Recall that for any where is the unit sphere in .

Lemma 2.9. For any symmetric matrices and ,

Proof. It suffices to note that the left-hand side of the equality does not exceed .

Let be -valued random processes with trajectories in the Skorokhod space (= càdlàg processes on ). We write if the induced by the processes measures on the Borel -algebra in weakly converge to the measure induced by . If herein is continuous, then we write . We say that a sequence is relatively compact (r.c.) in (in ) if each its subsequence contains, in turn, a subsequence converging in the respective sense. The weak convergence of finite-dimensional distributions of random processes, in particular the convergence in distribution of random variables, will be denoted . Likewise means equality of distributions.

Denote , Proposition VI.3.26 (items (i), (ii)) [2] together with VI.3.9 [2] asserts that a sequence of càdlàg random processes is r.c. in if and only if for all positive and Hence, two consequences are immediate.

Corollary 2.10. Let and be sequences of -valued and -valued, respectively, càdlàg processes such that is r.c. in and for any Then, the sequence is also r.c. in C.

Corollary 2.11. Let and be r.c. in C sequences of càdlàg processes taking values in and , respectively. Suppose also that for each and are given on a common probability space. Then the sequence of -valued processes is also r.c. in C.

Lemma 2.12. Let , , and be sequences of càdlàg random processes such that for any positive and for each the sequence is r.c. in . Then, there exists a random process such that .

Proof. Let be a bounded metric in metrizing Skorokhod's -convergence (see, e.g., [2, VI.1.26]). Then, condition (2.26) with arbitrary and implies that Hence, by the triangle inequality, we have Let be a uniformly continuous with respect to bounded functional on . Denote , . Then, and for any which together with (2.29) yields By condition (2.27), which jointly with (2.31) proves fundamentality and, therefore, convergence of the sequence . Now, the desired conclusion emerges from relative compactness of in .

Corollary 2.13. Let the conditions of Lemma 2.12 be fulfilled. Then, , where is the existing by Lemma 2.12 random process such that .

Proof. Repeating the derivation of (2.31) from (2.29), we derive from (2.28) the relation It remains to write .

Corollary 2.14. Let , , and be sequences of càdlàg random processes such that for any and (2.26) holds; for each relation (2.27) is valid; the sequence is r.c. in C. Then, there exists a random process such that and .

Below, is the symbol of the locally uniform (i.e., uniform in every interval) convergence.

Lemma 2.15. Let be càdlàg random processes such that . Then, for any -continuous functional on D.

Proof. Lemma VI.1.33 and Corollary VI.1.43 in [2] assert completeness and separability of the metric space , where is the metric used in the proof of Lemma 2.12. Then, it follows from the assumptions of the lemma by Skorokhod's theorem [6] that there exist given on a common probability space càdlàg random processes such that (so that is continuous), and a.s. By the choice of , the last relation is tantamount to a.s. Hence, and from continuity of , we get by Proposition VI.1.17 [2] a.s. and, therefore, by the choice of a.s. It remains to note that .

2.1. Forestopping of Random Processes

Let be a filtration on some probability space, an -adapted random process, and a stopping time with respect to . We put and denote ,, . Obviously, provided exists. In case is -predictable, the operation was called in [7] the forestopping. The following three statements were proved in [7].

Lemma 2.16. Let a random process and a stopping time be -predictable. Then, the process is -predictable.

Theorem 2.17. Let be an -martingale and an -predictable stopping time. Then, is a -martingale. If is uniformly integrable, then so is .

Lemma 2.18. Let be an -valued right-continuous -predictable random process and a closed set in . Then, the stopping time is -predictable.

The operation of forestopping was used prior to [7] by Barlow [8] who took the assertion of Theorem 2.17 (which he did not even formulate) for granted.

We will need some subtler properties of this operation.

Lemma 2.19. Let be a starting from zero locally square integrable martingale with respect to , a positive number, and an -predictable stopping time such that Then, .

Proof. Predictability of implies by Theorem 2.1.13 [3] that there exists a sequence of stopping times such that By the choice of there exists a sequence of stopping times such that Then, Herein, obviously, From (2.41) we have by Doob's inequality Noting that: (1) for any , (2) , we may rewrite the last inequality in the form Writing we get from (2.37) and (2.38) , which together with (2.45) results in . Then, from (2.42) and (2.43), we have by Fatou's theorem The assumption yields Relations (2.38) and (2.39) imply that which together with (2.47) yields by Fatou's theorem . It remains to note that in view of (2.34).

Lemma 2.20. Let be a locally square integrable martingale with respect to such that and for any Let, further, be a positive number and a predictable time satisfying condition (2.37). Then, .

Proof. In view of (2.50) it suffices to show that . In other words, we may consider that . Then condition (2.51) and the evident inequality imply by Lemma 2.19 that for any It remains to prove that for all , Taking a sequence with properties (2.40) and (2.41), we write To deduce (2.54) from this inequality and (2.40), it suffices to note that so that (2.53) provides uniform integrability of the sequence .

Corollary 2.21. Under the conditions of Lemma 2.20  .

Theorem 2.22. Let be a locally square integrable martingale with respect to satisfying conditions (2.50) and (2.51), a positive number, and a predictable time satisfying condition (2.37). Then, .

Proof. Denote . It suffices to show that is a -martingale. To deduce this fact from Theorem 2.17, we note that, firstly, by construction and Lemma 2.20, and, secondly, by construction of both processes and because of (2.35).

3. Martingale Preliminaries

The next statement is obvious.

Lemma 3.1. Let be a sequence of martingales such that and for any the sequence is uniformly integrable. Then, is a martingale.

Lemma 3.2. Let be a sequence of martingales such that (3.1) holds and Then, .

Proof. By condition (3.2) and the definition of quadratic characteristic, there exists a constant such that for all and . Hence, and from (3.1), we have by Fatou's theorem (applicable due to the above-mentioned Skorokhod's principle of common probability space) .

Corollary 3.3. Let a sequence of square integrable martingales satisfy conditions (3.1) and (3.2). Then, is a uniformly integrable martingale.

Lemma 3.4. Let be a local martingale and be an -valued random process. Suppose that they are given on a common probability space and . Then for any a.s.

Proof. By assumption, for all and . Hence, recalling the definition of quadratic variation, we get .

We shall identify indistinguishable processes, writing simply if . Theorem 2.3.5 [3] asserts that for a continuous local martingale . Hence, and from Lemma 3.4, we have

Corollary 3.5. Let be a continuous local martingale and an -valued random process. Suppose that they are given on a common probability space and . Then, .

Proof. Lemma 3.4 and formula (3.4) yield . Continuity of both processes enables us to substitute by .

Lemma 3.6. Let be a locally square integrable martingale. Then, is an increasing process.

Proof. For any , the process is a numeral locally square integrable martingale and, therefore, the process increases. It remains to note that and to recall formula (2.21).

Lemma 3.7. Let and be locally square integrable martingales with respect to a common filtration. Then,

Proof. For (then ), this is the Kunita-Watanabe inequality [3, page 118]. In the general case, we take an arbitrary vector and write hereupon the required conclusion ensues from (2.21) and Lemma 2.9.

Lemma 3.8. Let and be locally square integrable martingales with respect to some common filtration. Then, for any

Proof. Writing the identities we deduce from Lemma 3.7 that It remains to note that the right-hand side increases in by Lemma 3.6.

For a function we denote .

Let us introduce the conditions:(RC) The sequence is r.c. in .(UI1) The sequence is u.i.(UI2) For any the sequence is u.i.(UI3) The sequence is u.i.

Lemma 3.9. Let be a sequence of local square integrable martingales satisfying the conditions: (RC), and, for each , the condition Then, is r.c. in C.

Proof. It follows from (RC) and (3.10) by Rebolledo's theorem [2, VI.4.13] that is r.c. in . Hereon, the desired conclusion follows from Proposition VI.3.26 (items (i) and (iii)) [2] with account of VI.3.9 [2].

Combining Lemma 3.9 with Corollary 2.11, we get

Corollary 3.10. Under the assumptions of Lemma 3.9, the sequence of compound processes is r.c. in C.

Some statements below deal with random processes on , not on . In this case, the time variable is denoted by and C means instead of .

Lemma 3.11. Let be a r.c. in C and satisfying condition (UI1) sequence of martingales on . Then, for any , the sequence is u.i.

Proof. The obvious equality allows us to consider that . Then, condition (UI1) together with Doob's inequality yields whence By assumption, for any infinite set , there exist an infinite subset and a random process such that Condition (UI1) implies that is a square integrable martingale and for any , From (3.14) and (3.13), we have by Corollary VI.6.7 [2] Hence, and from (3.15), recalling that for any -valued one has [3, Theorem ], we get Comparing this relation with (3.16), we conclude that the sequence is uniformly integrable. Hence, in view of arbitrariness of , we deduce by Lemma 2.3 uniform integrability of .

Lemma 3.12. Let be a sequence of martingales on satisfying condition (UI1) and (UI2). Suppose that there exists an -valued random process such that Then, firstly, where is the null matrix, is a continuous martingale, and, secondly,

Proof. For the same reason as in the proof of Lemma 3.11, we may consider that . Then, as was shown above, condition (UI1) implies (3.13). Combining the latter with (a part of (3.18)), we get by Corollary VI.6.7 [2] that From (3.18) and (3.22), we get by Corollary 2.11 that for any infinite set , there exist an infinite subset and an -valued random process such that (Of course .)Denote . This is a martingale by Lemma 10.4 in [4]. Relation (3.23) implies that For any , the sequence is, by Lemma 3.11 and condition (UI2), u.i. So, relation (3.24) implies by Lemma 3.1 that is a martingale. Also, it implies its continuity. Relation (3.23) shows that the processes and increase and start from zero. So, starts from zero and has finite variation in . These four properties of imply together that for all . Thus, any subsequence contains, in turn, a subsequence such that . This proves (3.19).From (3.22) and (3.19), we have . And this is, in view of (3.4), tantamount to Comparing this relation with (3.18), we arrive at (3.20).

Remark 3.13. The second conclusion of Lemma 3.12 implies by Corollary 3.5 that .

Corollary 3.14. Let a sequence of martingales on satisfy conditions (RC), (3.11), (UI1), and (UI2). Then, relation (3.19) holds.

Proof. By Corollary 3.10 for any infinite set , there exist an infinite set and an -valued random process such that relation (3.18) holds as . Then, by Lemma 3.12, so does (as ) (3.19). Due to arbitrariness of this relation holds when ranges over , too.

Corollary 3.15. Let be a sequence of martingales on satisfying conditions (RC) and for all , conditions (3.11), (UI1), (UI2). Then, relation (3.19) holds.

Lemma 3.16. Let a sequence of martingales on satisfy conditions (RC) and, for any , (3.11) and (UI3). Then, relation (3.19) holds.

Proof. Denote . Obviously, By Corollary 2.21   is a square integrable martingale with respect to . By Theorem 2.22, Condition (RC) implies relative compactness of the sequence . By construction . So, condition (UI3) implies that for any and the sequence is u.i. Thus, Corollary 3.15 asserts that for any Equalities (3.27) and (2.36) yield the relation which together with (3.26), (3.28) and (RC) entails (3.19).

Corollary 3.15 and Lemma 3.16 are only the steps towards the final result about asymptotic proximity of quadratic variations and quadratic characteristics—Corollary 5.3.

Corollary 3.17. Let a sequence of martingales on satisfy conditions (RC) and for any , (3.11) and (UI3). Suppose also that there exists an -valued random process such that relation (3.18) is valid. Then, is a continuous martingale, and (3.20) holds.

Proof. Lemma 3.16 asserts (3.19). The implications ((3.21) and (UI1) (3.22)); ((3.22) and (3.19) (3.25)), were established in the proof of Lemma 3.12.

4. Sequences of Martingales with Asymptotically Conditionally Independent Increments

Lemma 4.1. Let for each be an -valued square integrable martingale on with respect to a flow and a sub--algebra of . Suppose that conditions (UI1) and (RC) are fulfilled for , and there exists a nonrandom number such that for all Then,

Proof. Conditions (RC) , (4.1), and (4.2) entail, by Lemma 3.9, relative compactness of in .Denote , , , , Condition (RC) implies that In view of (4.1) . Then, by Itô's formula Hence, recalling that , we get By the definition of and by condition (4.3), Consequently, and . The right-hand side of the last equality being less than , the sequence is u.i., and so is by Lemma 3.11 whose conditions (those not postulated) we have verified. This together with (4.9) and (4.3) implies uniform integrability of . Now, (4.8) and inequality (4.9) show that has this property, too.By construction and Lemma 10.4 in [4], is a martingale. Then, it follows from (4.9) that , which together with (4.10) and (4.8) yields So, it suffices to show that Obviously, for any real and Hence, from (4.5), (4.9), we get Now, (4.12) ensues from (4.6), (4.3), (4.2), and stochastic boundedness of the sequence .

Lemma 4.2. Let for each be an -valued starting from zero locally square integrable martingale with respect to some flow and a sub--algebra of . Suppose that condition (RC) is fulfilled for ; for any ; there exists a nonrandom function such that for all and . Then, for any , relation (4.4) holds.

Proof. Let us denote, only in this proof, , , (so that and . The evident inequality and condition (4.15) show us that for any positive and , the sequence is u.i.By assumption, there exists a sequence of stopping times such that a.s. and for each . Then, for any , and , , Writing , we deduce from (4.17) and (4.15) uniform integrability of the sequence . So, letting in (4.18), we get that is, is a martingale. It is square integrable because of (4.17) and (4.15). Thus, for any and the sequence satisfies all the conditions of Lemma 4.1 which, therefore, asserts that Here, in because of (4.16), so Obviously, . From (4.16), we have by the Lenglart-Rebolledo inequality The last three relations imply that which together with (4.20) yields (4.4).

Lemma 4.3. Let for each   be -valued starting from zero locally square integrable martingales with respect to a flow and a sub--algebra of . Suppose that for all , , , there exists a nonrandom function such that for all , and ; for each , the sequence is r.c. in C. Then, for any relation (4.4) holds.

Proof. Denote . By Lemma 4.2   as . So, in view of (4.26), it suffices to prove that for any , Conditions (4.25) and (4.26) imply by Lemma 3.8 that for all positive and Furthermore, (4.25) together with the Lenglart-Rebolledo inequality and the assumed equalities yields which jointly with the previous relation and condition (4.25) entails (4.27).

Lemma 4.4. Let for each be an -valued starting from zero locally square integrable martingale with respect to a flow and a sub--algebra of . Suppose that conditions (RC) , (4.24) (for all and ) and (4.25) (for all and ) are fulfilled; for all and , for any and bounded uniformly continuous function , Then, for any ,

Proof. () Let us fix and denote , so that . If there exists a nonrandom constant such that for all and , then all the conditions of Lemma 4.3 are fulfilled, and therefore, Also, under this assumption , where , So, substituting to (4.31), we obtain (2.7), whence by Lemma 2.5, relation (2.8) follows. Juxtaposing it with (4.33), we get . Dividing both sides of this relation by , we arrive at (4.32).() Let us waive the extra assumption.Denote , and likewise with instead of . Lemma 2.18 asserts predictability of . By construction and condition (4.33), . Thus, Theorem 2.22 asserts that and are square integrable martingales and . Consequently, for any In view of (4.35) and (4.34), whence Here, in condition (4.6) is fulfilled because of (RC).Let be a bounded uniformly continuous function. Then, by condition (4.30); on the strength of (4.39), (4.6) and uniform continuity of . From the second relation, we get, since is bounded, These three relations together yield Thus, the sequences and satisfy all the conditions of the lemma plus the above extra assumption. Then, according to item as . Hence, and from (RC), relation (4.32) emerges by the same argument as (4.4) was derived from (4.20) and (4.22).

Theorem 4.5. Let for each be locally square integrable martingales with respect to a flow . Suppose that the sequence is r.c. in C and for all and bounded uniformly continuous functions Then, (1) for any , , , and (2) under the extra assumption that the sequence is stochastically bounded, for all , , , and .

Proof. The relative compactness condition implies that for any , the sequence is stochastically bounded. Then, it follows from (4.47) by Corollary 2.8 that for any , and If, moreover, the sequence is stochastically bounded, then the same corollary asserts that, in the notation of formula (4.49), Let us fix and denote (likewise with a superscript), . Then, So we have the implications: (4.44) (4.24); (4.45) (4.25). Setting in (4.50) , , , (), and taking to account (4.52), we get (4.31) with . Equality (4.52) shows that the sequence is r.c. in , since has this property. The similar equality for and condition (4.46) imply (4.30). Thus, Lemma 4.4 asserts that for any relation (4.32) with holds. Putting , we convert it to Denote the left-hand side of this relation by . Inequality allows to rewrite it in the form . Consequently, for any Hence, and from (4.50) (), we have for Now, (4.48) emerges from Lemma 2.4.Relation (4.49) follows from (4.48) and (4.51) by Lemma 2.5.

Remark 4.6. Relation (4.49) implies that every partial limit (with respect to the weak convergence in law) of a sequence is a process with conditionally independent increments.

The following result can facilitate the verification of condition (4.47).

Lemma 4.7. Let for each be an -valued or -valued random process adapted to a flow on a probability space . Suppose that there exists a sequence of scalar random processes such that, for any and is an -measurable positive random variable; for all , Then, for all for and bounded uniformly continuous functions on (or on ),

Proof. Denote . Condition (4.57) implies that for any , and sequence whose th member is a sub--algebra of . Hence, and from the evident inequality we get by the choice of Setting here at first and then , subtracting the second relation from the first and recalling that the random variable is, by the assumptions about and , -measurable, we arrive at (4.58).

Example 4.8. Let and , where is an -adapted random process (so that is -adapted). Writing we see that condition (4.57) will be fulfilled with if we demand that tend in probability to some limit as .

5. The Convergence Theorems

Theorem 5.1. Let be a sequence of local square integrable martingales satisfying conditions (RC), (3.10), and, for each , the condition Then, for any infinite set there exist an infinite set and a continuous local martingale such that

Proof. () Denote , , (so that ), regarding and as -valued processes.Conditions (RC), (3.10) and (5.1) imply by Corollary 3.10 that the sequence is r.c. in . Then, by Corollaries 2.10 and 2.11, for any the sequence of compound processes is r.c. in , too. Hence, using the diagonal method, we deduce that for any infinite set , there exist an infinite set and random processes such that for all where The distribution of the right-hand side of (5.4) may depend on , so the minute notation would be something like . We suppress, “for technical reasons”, the superscript , keeping, however, it in mind.() By the definition of , which together with (5.1) shows that for any and , the sequence is uniformly integrable. Then, it follows from (5.3)–(5.5) by Corollary 3.17 and Remark 3.13 that is a continuous martingale and () Writing and recalling that is r.c. in , we arrive at (2.26).() Note that the processes are given, in view of (5.4), on a common probability space. Let us show that where is the metric in defined by From (5.4), we have by Lemma 2.15 for all natural and . Then, Alexandrov's theorem asserts that for any , which together with the definitions of , and yields, for , Hence, and from the evident inequality where is an arbitrary nonnegative random variable, we get for , By the Lenglart-Rebolledo inequality, for any . Relation (5.4) implies, by Alexandrov's theorem, that which together with (5.10)–(5.16) yields Hence, letting , then finally , we obtain (5.9).() Obviously, metrizes the -convergence and the metric space is complete. Relation (5.9) means that the sequence of -valued random elements is fundamental in probability. Then, by the Riesz theorem, each of its subsequences contains a subsequence converging w.p.1. The limits of every two convergent subsequences coincide w.p.1 because of (5.9). So, there exists a -valued random element (= continuous random process) such that And this is a fortified form of the relation In particular, the sequence is r.c. in (which can be proved directly, but such proof does not guarantee that partial limits are given on the same probability space that the prelimit processes are).() Relation (5.4) together with the conclusions of items () and () shows that all the conditions of Corollary 2.14 (with the range of restricted to ) are fulfilled (and even overfulfilled: relation (5.20) proved above without recourse to Corollary 2.14 contains both an assumption and a conclusion of the latter). So, Corollary 2.14 asserts, in addition to (5.20), that This pair of relations can be rewritten, in view of (5.3) and (5.5), in the form where is a synonym of . We wish to stress again that, firstly, all the processes in (5.22) are given on a common probability space and, secondly, they depend on the choice of .() Let us show that is a local martingale.Denote , and , . Equalities (5.19), (5.10), and (5.5) yield whence On the strength of (5.7), By the construction of the processes and for any and , the sequence increases. Then, due to (5.4) so does . Hence, we have with account of (5.19), (5.10), and (5.5) for all and . Comparing this with (5.26), we see that But is a continuous increasing process, so . Now, it follows from (5.25) and (5.28) by Corollary 3.3 that is a uniformly integrable martingale. Thus, the sequence localizes .() Relation (a part of (5.22)) where the prelimit processes are, according to item (), continuous martingales implies by Corollary VI.6.7 [2] that Comparing this with (5.22), we get with account of (3.4) , hereupon Corollary 3.5 asserts that .

Corollary 5.2. Let be a sequence of local square integrable martingales satisfying conditions (RC), (3.21), and, for all , (5.1). Then, is a continuous local martingale and relation (3.25) holds.

Proof. Let be an arbitrary infinite set of natural numbers. Then, Theorem 5.1 whose condition (3.10) is covered by (3.21) asserts existence of an infinite set and a continuous local martingale such that (5.2) holds. By assumption, the distribution of and, consequently, of does not depend on , which allows to delete the superscript in (5.2). Hence, taking to account arbitrariness of , we conclude that (5.2) holds for .

Corollary 5.3. Let a sequence of locally square integrable martingales satisfy conditions (RC) and, for all , (5.1). Then, relation (3.19) holds.

Proof. It was shown in items and of the proof of Theorem 5.1 that for each , the sequence satisfies all the conditions of Lemma 3.16 which, therefore, asserts that as . Hence, by the same argument as in item , relation (3.19) follows.

Theorem 5.4. Let for each be locally square integrable martingales with respect to a common filtration. Suppose that for all , and conditions (4.44) and (4.45) are fulfilled, and Then, for any infinite set , there exist an infinite set and a continuous local martingale such that

Note that relation (5.34) is, up to notation, a duplicate of (5.2). So, the superscript on the right-hand side is tacitly implied (but suppressed because the conditions of the theorem contain another superscript).

Proof. Conditions (5.31) and (5.32) imply that for each , the sequence is r.c. in . Then, it follows from (4.44) and (5.30) by Lemma 3.9 that the sequence is r.c. in . So, there exist an infinite set and a random process such that Consequently, if we denote by the set whose th member is that of , then for each , And this together with (4.44) and relative compactness of implies by Corollary 5.2 that is a continuous local martingale and Then, it follows from (5.30)–(5.32) that and therefore, the sequences , and are r.c. in .Conditions  (5.33) and (4.45) imply by the Lenglart-Rebolledo inequality that for all positive and Conditions(5.31) and (4.45) imply by Lemma 3.8 that which together with the previous relation yields (2.26). Then, Corollary 2.14 asserts existence of a random process such that The ensuing relation , continuity (due to (5.37)) of all and relative compactness of imply by Corollary 5.2 that is a continuous local martingale and . Comparing this with (5.41), we get , which converts (5.42) to (5.34).

Repeating the deduction of Corollary 5.2 from Theorem 5.1, we get from Theorem 5.4 the following conclusion.

Corollary 5.5. Let for each be locally square integrable martingales with respect to a common filtration. Suppose that, they have the same initial value; conditions (4.44), (4.45), (5.31), and (5.32) are fulfilled for all and ; there exists a random process such that . Then, is a continuous local martingale and .

Theorem 5.6. Let for each be locally square integrable martingales with respect to a flow . Suppose that, conditions (4.44)–(4.47) and (5.30)–(5.33) are fulfilled for all , and bounded uniformly continuous functions ; there exist an -valued random variable and an -valued random process such that Then, (1) for any and bounded continuous function (2) there exists a continuous local martingale with initial value and quadratic characteristic such that and for any and bounded continuous function .

Proof. Since the assumptions of this theorem contain those of Theorem 5.4, the conclusion of the latter is valid. It implies, in particular, that the sequence is r.c. in . So, firstly, the assumptions of Theorem 4.5 are also fulfilled (and therefore the conclusions are valid), and, secondly, the relation holds.If , then Theorem 4.5 asserts relation (4.49) which together with (5.43) yields, by the dominated convergence theorem, (5.44). Relation (5.46) and continuity (due to (5.43)) of enable us to let in (5.44), thus waiving the interim assumption .Combining (4.49) with the conclusion of Theorem 5.4 and with the dominated convergence theorem, we see that for any infinite set , there exist an infinite set and a continuous local martingale such that for all and bounded continuous function as . The comparison of (5.43) and (5.34) shows that the right-hand side of (5.47) equals and, therefore, does not depend on the choice of and . So, (5.47) holds as ranges over , too. This together with (5.44) proves the second statement under the extra assumption which can be waived exactly as above.

Corollary 5.7. Let the conditions of Theorem 5.6 be fulfilled. Then, has conditionally with respect to independent increments.

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