#### 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