Abstract and Applied Analysis

Volume 2013 (2013), Article ID 715054, 26 pages

http://dx.doi.org/10.1155/2013/715054

Research Article

Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Shunli Hao

Trade and Event Management, School of Economics, Beijing International Studies University, Beijing 100024, China

Received 15 July 2013; Accepted 18 September 2013

Academic Editor: Dumitru Motreanu

Copyright © 2013 Shunli Hao. 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.

We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

1. Introduction

The convergence rates in the law of large numbers have been considered by many authors. Let be a sequence of independent and identically distributed (i.i.d.) real valued random variables defined on a probability space with , and set . By the law of large numbers, for . Hsu and Robbins [1] introduced the notion of complete convergence and showed that

if ; Erdös [2, 3] proved that the converse also holds. Spitzer [4] showed that

whenever . Katz [5] and Baum and Katz [6] proved that, for and , or and ,

if and only if . Lai [7] studied the limiting case where and . Gafurov and Slastnikov [8] considered the case where and are replaced by more general sequences. Many authors have considered the generalization of the theorem of Baum and Katz [6] to arrays of independent (but not necessarily identically distributed) random variables; see for example Li et al. [9], Hu et al. [10–12], Kuczmaszewska [13], Sung et al. [14], and Kruglov et al. [15].

Let be a sequence of real-valued martingale differences defined on a probability space , adapted to a filtration , with . This means that for each (integer) , is -measurable and a.s. A natural question is whether the pre-mentioned theorem of Baum and Katz [6] is still valid for martingale differences . Lesigne and Volný [16] proved that, for , implies

(as usual, we write if and if the sequence is bounded) and that the exponent is the best possible, even for strictly stationary and ergodic sequences of martingale differences. Therefore, the theorem of Baum and Katz does not hold for martingale differences without additional conditions. (Stoica [17] claimed that the theorem of Baum and Katz still holds for in the case of martingale differences without additional assumption, but his claim is a contradiction with the conclusion of Lesigne and Volný [16], and his proof contains an error: when , we cannot choose satisfying (6) of [17].) Alsmeyer [18] proved that the theorem of Baum and Katz for and still holds for martingale differences if for some and with ,

where denotes the norm. This is a nice result; nevertheless, it is not always satisfied in applications; for example, (a) it does not apply to “nonhomogeneous” cases, such as martingales of the form , where and are i.i.d., as in this case the condition (5) (with ) is never satisfied; (b) in applications instead of a single martingale we often need to consider martingale arrays: for example, when we use the decomposition of a random sequence into martingale differences (such as in the study of directed polymers in a random environment), the summands usually depend on : , , where and for .

Our first main objective is to extend the theorem of Baum and Katz [6] to a large class of Banach valued martingale arrays. More precisely, under a simple moment condition on for some , we will find sufficient conditions for

for a large class of sequences of Banach valued martingale differences , , where , is a positive function, and . Of particular interest is the case where is a regular function:   , being slowly varying at ; that is, is a positive measurable function defined on such that for any . Our results improve and complete a result of Ghosal and Chandra [19] for martingale arrays and extend Alsmeyer's result [18] for martingales.

Our second main objective is to extend another important theorem of Baum and Katz [6] which states that for i.i.d. real valued random variables with and for each , if and only if for all . In fact, we prove that a similar result holds for a large class of Banach valued martingale arrays: under a simple moment condition on for some , we obtain sufficient conditions for

where and are defined as before, . The result is new and sharp even for independent but not identically distributed real valued random variables.

The consideration of a Banach valued martingale array (rather than a Banach valued single martingale) makes our results very adapted in the study of weighted sums of identically distributed Banach valued random variables. Many authors have contributed to this subject. Gut [20], Lanzinger and Stadtmüller [21] considered weighted sums of i.i.d. random variables. Li et al. [9], Wang et al. [22] studied weighted sums of independent random variables. Yu [23] considered weighted sums of martingale differences (see also the references therein). Ghosal and Chandra [19] considered weighted sums of arrays of martingale differences. As applications of our main results, we generalize or improve some of their results. For example, we prove a new theorem about the convergence rate for weighted sums of identically distributed Banach valued martingale differences.

As information, we mention that Baum-Katz type theorems in different dependent setups have been studied by many authors. For example, Li et al. [24] studied moving average processes; Shao [25, 26], Szewczak [27] considered mixing conditions; Baek and Park [28] studied negatively dependent random variables; Liang [29], Liang and Su [30], Kuczmaszewska [31], Kruglov [32], and Ko [33] studied negatively associated random variables.

The rest of the paper is organized as follows. In Section 2, we establish some maximal inequalities for Banach valued martingales. In Section 3, we show our main results on the convergence rates for Banach valued martingale arrays, which improve and complete Theorem 2 of Ghosal and Chandra [19]. In Section 4, we consider the important special case of triangular Banach valued martingale arrays, and obtain an extension of Theorem 1 and 2 of Alsmeyer [18]. We also generalize a result of Chow and Teicher (cf. [34, page 393]) about the complete convergence of sums of independent real valued random variables. In Section 5, we look for the convergence rates for the maxima of sequences of any Banach valued random variables, in order to obtain further equivalent conditions about the convergence rates for Banach valued martingales in the following section. In Section 6, we consider the convergence rates for Banach valued martingales. Our results extend Theorems 1–4 of Baum and Katz [6] for i.i.d. real valued random variables and generalize Theorems 1 and 2 of Alsmeyer [18]. As applications, in Section 7, we obtain new results on the convergence rates for weighted sums of Banach valued martingale differences, which extend Theorems 2 and 3 of Lanzinger and Stadtmüller [21] on weighted sums of the form . In Section 8, we consider more general weighted sums of Banach valued martingale differences, for which we extend Theorem 3.3 of Baxter et al. [35], Corollary 1 of Ghosal and Chandra [19], and Theorems 2.2–2.4 of Li et al. [9] and generalize Theorem 2 of Yu [23].

For notations, as usual, we write , and .

2. Maximal Inequalities for Banach Valued Martingales

In this section, we show new maximal inequalities for Banach valued martingales.

Let be a probability space and a real separable Banach space. For any real number , denote by the space of -valued random variables such that is finite. Let be an increasing sequence of sub--fields of . Let be an adapted sequence of -valued random variables defined on ; that is, for every , is measurable. We call it a sequence of -valued martingale differences if additionally a.s. and belongs to for any and a sequence of -valued supermartingale differences if additionally a.s. and belongs to for any . Following Pisier [36], we say that a Banach space is -smooth if there exists an equivalent norm such that

Set

and set

For , let

Accordingly, for an infinite -valued adapted sequence , we write

if the series converges, and

In the following, we consider relations among , , , and .

Our first theorem describes relations between and for an adapted sequence of -valued random variables .

Theorem 1. Let be an adapted sequence of -valued random variables. Then, for any , and ,

Our second theorem shows relations between and for a sequence of -valued martingale differences : that is, for each (integer) , is measurable and belongs to , and a.s.

Theorem 2. Let be a finite sequence of -valued martingale differences. For any , and , if is -smooth, then

where is a constant only depending on , , and .

Corollary 3. Let be a sequence of -valued martingale differences. Suppose that, for some ,

If is -smooth, then converges . and the inequalities (14) and (15) hold with replaced by .

We get Theorems 1 and 2 by a refinement of the method of Alsmeyer [18].

Proof of Theorem 1. The first inequality is obvious. We only consider the second one. Clearly,

Since is an adapted sequence of -valued random variables, by Markov's inequality (conditional on ),

Hence, by summing, we obtain

Therefore, the upper bound in (19) gives a lower bound of by (17), which implies the second inequality of (14).

Proof of Theorem 2. The first inequality is obvious, because if , then

We will prove the second inequality. For any , and ,

Define

where by convention . It is easily seen that are stopping times (cf. e.g., [34] for the definition) with respect to the filtration , where we take for all . As usual, we will write . Notice that for , if , then ; conversely, if there exists a positive integer such that , then . We proceed by three steps to estimate the second term of the right hand side of (21).

(a) We first prove that

which implies that

where

Assume that the first event in (23) takes place. Since , there exists such that . As , and it is clear that .

Let be the largest such that . Then, .

We will prove that . Suppose that . Then, by the definition of , so that

As , it follows that

where the last step holds because by the definition of and . As , by (26), (27), and the subadditivity of , we know that

This is a contradiction with , which proves that .

Therefore, for all . Thus, (23) holds.

(b) We next give an estimation of . For ,

Using successively Doob's inequality in a real separable Banach space [37, Theorem 3.1], the inequality in Assouad [38, Proposition 2] (conditional on ), and the subadditivity of the function , we obtain

Therefore, by (29),

Summing over , we obtain

(c) We finally give un upper bound for the term of the right hand side of (24), using (32). Set

Applying (32) for , we see that

Now, for any ,

where

Notice that

where (34) has been used for the last inequality. For , by (32), together with Markov's inequality and Jensen's inequality, we have

Therefore, by (35),

where

A simple calculation shows that

where . Therefore,

Together with (21) and (24), this proves (15).

Proof of Corollary 3. By Theorem 2 with and ,

By Theorem 1 and Markov's inequality,

Therefore,

which is equivalent to

Since is decreasing in , this implies that

which gives the desired conclusion.

3. Convergence Rates for Arrays of Banach Valued Martingale Differences

In this section, we consider the convergence rates in the law of large numbers for arrays of Banach valued martingale differences.

Let a probability space and be a real separable Banach space. For every , let be an increasing sequence of sub--fields of . For every , let be a sequence of -valued martingale differences defined on , adapted to the filtration : that is, for every , is measurable and belongs to , and a.s. Set for ,

if the series converges. We will call the double sequence an array of -valued martingale differences.

In the following, we give a sufficient condition for the convergence of -valued martingale arrays. For , let

Theorem 4. Assume that for some , as ,

If is -smooth, then for all , as ,

Proof. Notice that, by Corollary 3, the condition (50) implies the a.s. convergence of . Equation (51) comes from (50) as

(52) follows from (51) and Theorem 1; (53) is a consequence of (52) and Theorem 2; (54) is implied by (53) and Corollary 3.

We are interested in the convergence rates of the probabilities and . We will describe their rates of convergence by comparing them with an auxilary function and by considering the convergence of the related series.

We begin with some relations among , , , and .

Lemma 5. Let be a positive function. Suppose that for some and some integer ,

If is -smooth, then the following conditions are equivalent:

Proof. Equation (57) are equivalent by Theorem 2.

Lemma 6. Let be a positive function. Suppose that for some and ,

Then, the following conditions are equivalent:

Proof. The conclusion comes directly from Theorem 1.

Lemma 7. Let be a positive function. Suppose that (58) holds for some and . Then,

is implied by

Proof. The equivalence is an immediate consequence of Corollary 3.

Theorem 8. Let be a positive function. Suppose that for some and ,

If is -smooth, then one has the following implications (63)⇔(64)⇔(65)⇒(66):

Remark 9. The condition (62) holds if for some and ,

Proof of Theorem 8. Notice that when (62) holds for and some , then for ,

Therefore, we can assume that (62) holds for some and . Since as , , we can choose an integer large enough such that . Therefore, the condition (56) holds with . By Lemma 6, (63) and (64) are equivalent; by Lemma 5, (64) and (65) are equivalent; by Lemma 7, (65) implies (66).

Theorem 10. Let be a positive function. Suppose that for some and ,

If is -smooth, then one has the following implications (70)⇔(71)⇔(72)⇒(73):

Notice that, by (69), for each with , so that is well defined (cf. Corollary 3). When , we use the convention that the associated term containing as a factor is defined by 0.

When , , and is a sequence of real-valued martingale differences, the implication “(70)⇒(72)” reduces to Theorem 2 of Ghosal and Chandra [19]. (Although the condition does not appear in Theorem 2 of [19], it is implicitly used in its proof.) So, our result improves and completes that of Ghosal and Chandra [19] in the sense that we prove the equivalence between (70) and (72) (not just the implication “(70)⇒(72)”) under much weaker conditions.

Remark 11. Theorem 10 also holds if is replaced by for some . In fact, the case can be reduced to the case by considering the subsequences () of , which are still sequences of -valued martingale differences.

Corollary 12. Suppose that (67) holds for some , and . Then one has the implications (70)⇔(71)⇔(72)⇒(73).

Proof of Theorem 10. As in the proof of Theorem 8, we can assume that . Since as , , we can choose an integer large enough such that . Let be large enough such that for all . Then,

By Theorem 1, (70) and (71) are equivalent; by Theorem 2, (71) and (72) are equivalent; since (69) implies for each with , by Corollary 3, (72) implies (73).

Proof of Corollary 12. Choose , then by (67), we have

So, the condition (69) holds for , and the conclusion follows from Theorem 10.

4. Convergence Rates for Triangular Arrays of Banach Valued Martingale Differences

In this section, we consider the convergence rates in the law of large numbers for triangular arrays of Banach valued martingale differences.

Let be a probability space and a real separable Banach space. For every , let be an increasing sequence of sub--fields of . For each , let be a sequence of -valued martingale differences defined on , adapted to the filtration : that is, for every and every , is measurable and belongs to , and a.s. Set for ,

We will call the double sequence a triangular array of -valued martingale differences. In the following, we first give a sufficient condition for the convergence of triangular arrays of -valued martingale. For and , let

Theorem 13. Let . Assume that for some , as ,

If is -smooth, then for all , as ,

Proof. It suffices to apply Theorem 4 for the array of -valued martingale differences defined by

We are interested in the convergence rates of the probabilities and . We will describe their rates of convergence by comparing them with an auxilary function and by considering the convergence of the related series.

We begin with some relations among , , , and .

Theorem 14. Let and be a positive function. Suppose that for some and ,

If is -smooth, then one has the following implications (82)⇔(83)⇔(84)⇒(85):

Corollary 15. Let and . Let be a function slowly varying at and . Suppose that for some , with ,

If is -smooth, then one has the implications (82)⇔(83)⇔(84)⇒(85).

Remark 16. It is obvious that (86) holds with if for some constant , all and ,

Proof of Theorem 14. It suffices to apply Theorem 8 for the array of -valued martingale differences defined by (80).

Proof of Corollary 15. Since , we have

As , we can choose . For this ,

Thus, the condition (81) holds, so that the conclusion follows from Theorem 14.

Theorem 17. Let and be a positive function. Suppose that for some and ,

If is -smooth, then one has the following implications (91)⇔(92)⇔(93)⇒(94):

Corollary 18. Let and . Let be a function slowly varying at and . Suppose that (86) holds for some and with . If is -smooth, then one has the implications (91)⇔(92)⇔(93)⇒(94).

For a single real-valued martingale, when and , Corollary 18 reduces to Alsmeyer's result in [18]. We notice that the consideration of a triangular array makes the result very adapted to study weighted sums of identically distributed -valued random variables of the form .

Remark 19. As explained in Remark 11, Corollary 18 also holds if is replaced by for some .

Proof of Theorem 17. It suffices to apply Theorem 10 for the array of -valued martingale differences defined by (80).

Proof of Corollary 18. Notice that

As , we can take .

Then,

Thus, the result holds by Theorem 17.

As a special case, we obtain the following extension of a result of Chow and Teicher [34, page 393] about the complete convergence on sums of independent random variables.

Corollary 20. Let   be sequences of identically distributed -valued martingale differences. Let . Suppose that (86) holds for some and with . If is -smooth, then if and only if

When are rowwise independent real-valued martingale differences, the sufficiency in Corollary 20 was proved in [34, page 393].

Proof of Corollary 20. It suffices to apply Corollary 18 with and : we just need to check that in the present case, (91) is equivalent to . In fact, we have

As , the last expectation is finite if and only if .

5. Convergence Rate for the Maxima of any Banach Valued Random Variables

In this section, we study the convergence rate for the maxima of a sequence of any Banach valued random variables to obtain further equivalent conditions about the convergence rate for a Banach valued martingale in Section 6.

Let a separable Banach space and be a sequence of any -valued random variables. For any , let be the integer part of . Set

Then, for any , . For any , set

Let be a function slowly varying at . Recall that a function slowly varying at has the representation form

for some , where is measurable and , . The function plays no role for our purpose. We can choose without loss of generality.

We are interested in the convergence rates of and . Notice that for any if and only if a.s. So, our results in this section describe the rate convergence for the almost surely convergence of .

The following result shows that and have similar asymptotic properties. More precise comparisons will be given in Theorems 22 and 24.

Lemma 21. Let . Then, for any and any ,

Let and . If there exists , such that for all ,

then there exists depending only on , and , such that for all ,

where .

Proof. The first inequality of (102) is obvious. If

then

Thus, the second inequality of (102) holds.

Assume that for some and all , (103) holds (with the notation introduced in the lemma). Then, there exists , such that for all ,

Set . Then, applying (103) for , we see that, for any ,

Set . Since is slowly varying at , by Potter's Theorem (cf. Theorem in [39, page 25]), for and , there exists such that for all , . Thus, there exists such that for all ,

Theorem 22. Let and . Then, the following assertions are equivalent:

Proof. We use Lemma 21. By the second inequality of (102), we see that (112) implies (111); by the first inequality of (102), we know that (111) implies (110). As (103) implies (104), we see that (110) implies (112). Thus (110), (111), and (112) are all equivalent.

Lemma 23. Let . Then for some and all ,

Proof. Without loss of generality, we suppose that has the form (101) with . Therefore, for , is increasing in for some large enough. Consequently, for some positive constants (which may depend on ) and all ,

Similarly, for ,

Since is slowly varying at , by Potter's Theorem, for and , there exists such that for all , . If , then for some positive constant ; if , then for some positive constant . So, for some constants , , and all ,

Theorem 24. Let and . Then, the following assertions are equivalent:

Proof. We proceed as in [34, page 394] where similar results were established for and real-valued random variables.

(a) We first prove that (117) is equivalent to

Let be large enough such that is decreasing in . Then, we have

for large enough. Here we have used the fact that . (To see this, notice that  for , so that ; in the same way, using the fact that is increasing in for large enough, we obtain for all .) Similarly,

for large enough. Let . As , the conclusion that (117) is equivalent to (121) follows from (122) and (123).

(b) We next remark that (119) is equivalent to

This can be seen by the same argument as in (a).

(c) We now prove that (121) implies (124). Set . We have

By Potter's Theorem, for , there exists such that

So,

Thus, (121) implies (124).

(d) We then conclude that (117), (118), and (119) are equivalent. By (a), (b), and (c), we see that (117) implies (119). By Lemma 21, we have the implications: (119)⇒(118)⇒(117).

(e) We finally prove that (119) and (120) are equivalent. We have

By Lemma 23, there exist such that

By (128) and (129), we see that (119) implies

By Potter's Theorem, there exists such that when and ,

Therefore (130), is equivalent to (120). Hence, (119) implies (120). A similar argument (using again Lemma 23) shows that (120) implies (119). Thus, (119) and (120) are equivalent.

6. Convergence Rates for Banach Valued Martingales

In this section, we consider the convergence rate in the law of large numbers for a sequence of Banach valued martingales. We will obtain more equivalent conditions than in Section 4, using the results of Section 5.

Let a separable Banach space and a sequence of -valued martingale differences. We denote

Notice that by our notations. Set

Notice that a.s. if and only if for any . So, the following theorems describe the a.s. convergence of .

Theorem 25. Let , be a function slowly varying at and . Suppose that for some and ,

where . If is -smooth, then one has the following implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138):

Notice that, compared with Theorem 14, Theorem 25 contains the additional conditions (139) and (140). When and for i.i.d. real-valued random variables, the implications (135)⇒(139)⇒(138) with of Theorem 25 contain Theorem 4 of Baum and Katz [6].

Remark 26. As in Theorem 25, the conclusions of Theorem 25 remain valid if is replaced by

for some .

In fact, the case can be reduced to the case by considering the subsequences () of , which are still sequences of -valued martingale differences.

Corollary 27. Let and . Let be a function slowly varying at and . Suppose that for some , with ,

If is -smooth, then one has the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Theorem 25. Applying Theorem 14 to , and , we get the implications (135)⇔(136)⇔(137)⇒(138). Applying Theorem 22 to , we know that (137) and (140) are equivalent. Obviously, we have the implications (140)⇒(139)⇒(138). Therefore, we have proved the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Corollary 27. Since , we have

As , we can choose . For this ,

Thus, the condition (134) holds, so that the conclusion follows from Theorem 25.

Theorem 28. Let . Let be a function slowly varying at and . For any , set

Suppose that for some and ,

where . If is -smooth, then one has the following implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152):

(A)

(B)

Compared with Theorem 17, in Theorem 28 we have the additional conditions (149), (150), (151), (154), and (155).

Remark 29. As in Theorem 25, the conclusions of Theorem 28 remain valid if is replaced by

for some .

Corollary 30. Let . Let be a function slowly varying at and . Suppose that for some , with , (142) holds. If is -smooth, then one has the implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152).

If 's are identically distributed real-valued random variables, then (156) is equivalent to the moment condition . So, Corollary 30 contains Theorems 1, 2, and 3 of Baum and Katz [6] when and for i.i.d. real-valued random variables. When , and for real-valued martingale differences, Corollary 30 was proved by Alsmeyer [18, Theorems 1 and 2].

To see that Theorem 28 implies Corollary 30, it suffices to notice that (142) (with ) implies (147). In fact, when (142) holds for some with , then

and, for ,

Therefore, (147) holds for some and .

Proof of Theorem 28. Applying Theorem 10 to

we obtain the implications (153)⇔(156)⇔(148)⇒(152). By Theorem 24 applied to , we see that (148), (149), and (150) are equivalent. Since , (149) implies (151) and (151) implies (152). Thus, we have the implications (148)⇔(149)⇔(150)⇒(151)⇒(152). Again by Theorem 24 applied to , we know that (153), (154), and (155) are equivalent. Therefore, we have the implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152).

7. Convergence Rates for Weighted Sums of Banach Valued Martingale Differences of the Form

Let be a separable Banach space. In this section, we give a Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums

of -valued martingale differences , and we obtain a Baum-Katz type theorem for weighted sums of identically distributed -valued martingale differences which extends Theorems 2 and 3 of Lanzinger and Stadtmüller [21]. Our results will be obtained by means of our main Theorems 2 and 10.

We will need the following elementary result.

Lemma 31. Let be a sequence of any -valued random variables. If there exist , such that for some and all ,

then

Proof. Let , , . Then,

As , it follows that

The following theorem is a Marcinkiewicz-Zygmund type strong law of large numbers for the weighted sums (161).

Theorem 32. Let be -valued martingale differences. Suppose that for some , there exist , such that for all ,

If is -smooth, then for and ,

Notice that when and for real-valued martingale differences, the result (167) is implied by the classical Marcinkiewicz-Zygmund strong laws of large numbers. Also, it is evident that (167) holds if and only if for any . So, (168) describes the convergence rates in the Marcinkiewicz-Zygmund strong laws of large numbers (167).

Proof of Theorem 32. Clearly,

By Theorem 22, we see that

if and only if

Write . By the proof of Lemma 31, we have

And by Theorem 2 with and , we know that

By Theorem 1, we see that

By (166), we have

From (173)–(175), we see that (171) holds. Thus, (170) holds, so that (168) holds.

To establish a general Baum-Katz type theorem for the weighted sums (161), we first introduce a definition and a technical lemma.

Definition 33. For a function regularly varying at of index , one define, as its inverse function.

Notice that when , is strictly increasing for large enough with , so that is well defined on for large enough. For simplicity, we always make the convention that if , so that is well defined on . We make a similar convention in the case where .

The following lemma shows that the inverse function of a regularly varying function of index remains regularly varying.

Lemma 34. If is regularly varying at of index , where is of the canonical form , then its inverse function is regularly varying at of index , where is slowly varying at .

Proof. Let . Define . We have

We will prove that is slowly varying at . We see that

After changing variable, we have

where . Thus, is slowly varying at and so is , which proves the desired result.

In the following Baum-Katz type theorem, and are functions slowly varying at . Without loss of generality, we suppose that and have the form (101) with . For and , define

and let and be, respectively, the inverse functions of and (cf. Definition 33), which are also regularly varying by Lemma 34. For , choose so large that

is locally bounded in , and set

Notice that may be finite or infinite.

Theorem 35. Let be identically distributed -valued martingale differences and . Suppose that for some and with , and ,

If is -smooth, then the following assertions hold.

(a) When ,

if and only if

(b) When , (183) is implied by

where is defined in (181); conversely, if and the function defined by (180) satisfies , then (183) implies (185).

Remark 36. Theorem 35 also holds if (182) is replaced by

for some and , where

Of particular interest are the cases where the slowly varying functions and are constants or powers of the logarithmic function, which will be studied in the following corollaries. We first consider the case where and are constants.

Corollary 37. Let be identically distributed -valued martingale differences and . Suppose that (182) holds for some and with , , and . If is -smooth, then

if and only if

Notice that the condition on implies in particular , giving . Therefore, the conclusion of the corollary is interesting only when the exponents in (189) are greater than .

When are i.i.d. real-valued random variables and , we get the sufficiencies of Theorems 2 and and of Lanzinger and Stadtmüller [21] by Corollary 37.

We then consider the case where and .

Corollary 38. Let be identically distributed -valued martingale differences and . Suppose that (182) holds for some and with , , and . If is -smooth, then for ,

if and only if

In the case where are i.i.d. real-valued random variables and the maximum is replaced by the , by Corollary 38, if , , and , we get the sufficiency of Theorem of Lanzinger and Stadtmüller [21]; if , , and , , we get the sufficiencies of Theorem of Lanzinger and Stadtmüller in [21].

Proof of Theorem 35. Notice that is slowly varying at by Proposition  1.5.9a in [39, page 26]. Set

and in Theorem 10, then are sequences of -valued martingale differences. For any , we have

By (193) and Lemma 31, we have

Thus, and

whenever . By Theorem 10, (183) is equivalent to

Notice that in view of the identically distributed assumption, (196) holds if and only if

By the monotonicity of the functions and for large enough and the fact that and as , it can be easily verified that the condition (197) is equivalent to

Therefore, Theorem 35 is a direct consequence of the following lemma.

Lemma 39. Let , , , and and are slowly varying at . Let be any -valued random variable. Then, the following assertions hold.

(a)When , (198) holds if and only if (184) holds.(b)When , (198) is implied by (185), where is defined in (181).

Proof. We proceed as in the proof of Lemma 3.4 of Gut [20]. We distinguish three cases according to , or . By choosing a smooth version, we can suppose that is differentiable (cf. [39]).

Case  1 (). In (198), we use the change of variables

Notice that if and only if and . Therefore, (198) holds if and only if

By Proposition of [39, page 26], we have as ,

where and hereafter means that as . By Lemma 34, the right hand sides of (201) are regularly varying functions of index , and respectively, so that their ratio tends to as , since . Therefore, as ,

Thus, (200) is equivalent to

With the change of variable and the fact that (together with (179)), we see that (203) holds if and only if

which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).

Case  2 ). By the change of variables

we see that (198) holds if and only if

With the change of variable , we know that (206) holds if and only if

which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).

Case  3 (). By the change of variables

we see that (198) holds if and only if

We distinguish three cases according to , , or .

(i) Suppose that . By Proposition  1.5.8 of [39, page 26], we have as ,

so that (as in Case 1)

Thus, (209) is equivalent to

With the change of variable , we see that (212) holds if and only if

which is equivalent to the second condition of (184). Thus, (198) is equivalent to the second condition of (184).

(ii) Suppose that . By Proposition  1.5.10 of [39, page 27], we have as ,

so that as ,

Thus, (209) holds if and only if (203) holds. Notice that (203) is equivalent to the first condition of (184). Thus (198) is equivalent to the first condition of (184).

(iii) Suppose that . In this case, (209) reduces to

By Proposition  1.5.8 of [39, page 26], we have

So, (209) is implied by

With the change of variable , we see that (218) holds if and only if

which is equivalent to (185). Thus, (209) is implied by (185). Therefore, (198) is implied by (185). This ends the proof of Lemma 39.

Proof of Corollary 38. We are in the case where  . If , then

Thus,

Therefore,

Similarly,

Using (222), (223), and Theorem 35, we obtain the desired results.

8. Convergence of Weighted Sums of Banach Valued Martingale Differences of the Form

In this section, we consider more general weighted sums of Banach valued martingale differences than those considered in Section 7.

Let be a sequence of i.i.d. random variables with , and let be an array of real numbers. The study of the convergence of weighted sums as is a classical subject; see for example, Salem and Zygmund [40], Hill [41], Hanson and Koopman [42], Pruitt [43], Franck and Hanson [44], Chow [45], Chow and Lai [46], and Stout [47]. Pruitt [43] found a necessary and sufficient condition for in probability and a sufficient condition for a.s. Baxter et al. [35] also showed a sufficient condition for a.s. Li et al. [9] studied the complete convergence of weighted sums of independent random variables of the form . Yu [23] and Ghosal and Chandra [19] considered the same problem for martingale differences . We will extend or improve some of the aforementioned works.

8.1. Law of Large Numbers for Weighted Sums of Banach Valued Martingale Differences

Let be a separable Banach space. In this subsection, we find sufficient conditions for the convergence of weighted sums of -valued martingale differences .

Let us recall the famous theorem of Pruitt [43, Theorem 1] which states that for a sequence of i.i.d. random variables with and ,

if and only if , where is a Toeplitz summation matrix; that is, for every , , and .

In the following, we consider the same problem for -valued martingale differences .

Theorem 40. Let , , be sequences of -valued martingale differences. Let be an array of real numbers satisfying and . Suppose that for some , there exists , such that

If is -smooth, then

Proof. Since

we see that (226) follows from Theorem 4.

Let be a Toeplitz summation matrix; Theorem 2 of Pruitt [43] states that for a sequence of i.i.d. random variables with and , if , , then implies that

In the following, we also consider the similar problem for arrays of -valued martingale differences .

Theorem 41. Let , , be sequences of -valued martingale differences. Let be an array of real numbers satisfying and , . Suppose that, for some , there exists a constant such that for all . If is -smooth, then

Consequently, if is -smooth and for some , then

Proof. Set , then is an array of -valued martingale differences and for any ,

By Corollary 3, we see that

By Theorem 2 with and , we know that

By Theorem 1, we see that

Since (231), we know that

Since (231)–(235), we have

Thus, (229) holds.

If additionally , then (229) implies that

Therefore, (230) holds.

The following theorem extends Theorem 3.3 of Baxter et al. [35].

Theorem 42. Let be -valued martingale differences. Suppose that for some , there exists , such that for all ,

Let satisfy

and set . If is -smooth, then

When are i.i.d. real-valued random variables, , and , (240) reduces to Theorem 3.3 of Baxter et al. [35].

Proof of Theorem 42. Clearly,

By Theorem 22, we see that

if and only if

By (238) and (239), we have

And by Theorem 2 with and , we know that

By Theorem 1, we see that

By (238) and (239), we know that

From (246)–(248), we see that (244) holds. Thus, (241) holds.

8.2. Complete Convergence of Weighted Sums of Banach Valued Martingale Differences

Let be a separable Banach space. In this subsection, we consider complete convergence of weighted sums of -valued martingale differences of the form . We extend and improve Corollary 1 of Ghosal and Chandra [19] and Theorems 2.2–2.4 of Li et al. [9]. We also generalize Theorem 2 of Yu [23].

Theorem 43. Let , , be sequences of -valued martingale differences. Let and be an array of real numbers. If there exists such that

and is -smooth, then

In the square-integrable real-valued martingale differences case, the result was proved by Ghosal and Chandra in Corollary 1 [19] if, additionally, for some .

We generalize Theorem 2 of Yu [23] from two directions by Theorem 43: first, we extend sequences of martingale differences to sequences of -valued martingale differences; secondly, we do not need the condition for some .

Proof of Theorem 43. Set . From (249), we have

By Corollary 3 and Theorem 2 with and , we see that

By Corollary 3 and Theorem 1, we know that

By Markov's inequality, we see that

Since (249) and (251)–(254), we see that (250) holds.

Theorem 44. Let , , be sequences of -valued martingale differences. Let be an array of real numbers. Suppose that for some constants , , and ,

If is -smooth, then (250) holds.

When , , is a sequence of zero mean independent real-valued random variables and , Theorem 44 reduces to Theorem  2.3 of Li et al. [9].

Proof of Theorem 44. Set , then is an array of -valued martingale differences. Since (255) and (257), for any , we have

By Corollary 3 and Theorem 2 with and , we see that

By Corollary 3 and Theorem 1, we know that

By Markov's inequality, we see that

By (256), (257), and (258)–(261), we know that (250) holds.

Theorem 45. Let be a triangular array of identically distributed -valued martingale differences. Let , and let be a triangular array of positive numbers satisfying

with for every , and some constants and . Suppose that there exists some constant , such that, for some ,

Suppose that

then for all ,

When are the same sequence of i.i.d. real-valued random variables and for all , Theorem 45 reduces to the sufficiency of Theorem 2.4 of Li et al. [9].

Proof of Theorem 45. Set

and , then are sequences of -valued martingale differences. For any , since (262) and (263), we have

where . By Corollary 12 of Theorem 10, (265) is implied by

Equation (268) holds if and only if (264) holds. (cf. [9, pages 62-63])

Acknowledgment

The author is most grateful to editor Dumitru Motreanu and an anonymous referee for their careful reading and insightful comments. This work has been partially supported by the Research Fund of Beijing International Studies University (no. 13Bb023) and Doctoral Research Start-up Funds Projects of Beijing International Studies University.

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