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

Authors: Shunli Hao
Journal: Abstract and Applied Analysis
 (2013)

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## 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).


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## Body

## 1. Introduction

The convergence rates in the law of large numbers have been considered by many authors. Let(Xj)j≥1 be a sequence of independent and identically distributed (i.i.d.) real valued random variables defined on a probability space (Ω,ℱ,ℙ) with 𝔼Xi=0, and set Sn=∑j=1n‍Xj. By the law of large numbers, ℙ{|Sn|>εn}→0 for ε>0. Hsu and Robbins [1] introduced the notion of complete convergence and showed that
(1)∑n=1∞‍ℙ{|Sn|>εn}<∞∀ε>0
if 𝔼X12<∞; Erdös [2, 3] proved that the converse also holds. Spitzer [4] showed that
(2)∑n=1∞‍n-1ℙ{|Sn|>εn}<∞∀ε>0
whenever 𝔼X1=0. Katz [5] and Baum and Katz [6] proved that, for p=1/α and α≥1/2, or p>1/α and α>1/2,
(3)∑n=1∞‍npα-2ℙ{|Sn|>εnα}<∞∀ε>0
if and only if 𝔼|X1|p<∞. Lai [7] studied the limiting case where p>2 and α=1/2. Gafurov and Slastnikov [8] considered the case where (npα-2) and (nα) 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(Xj)j≥1 be a sequence of real-valued martingale differences defined on a probability space (Ω,ℱ,ℙ), adapted to a filtration (ℱj), with ℱ0={∅,Ω}. This means that for each (integer) j≥1, Xj is ℱj-measurable and 𝔼[Xj∣ℱj-1]=0 a.s. A natural question is whether the pre-mentioned theorem of Baum and Katz [6] is still valid for martingale differences (Xj). Lesigne and Volný [16] proved that, for p≥2, supj≥1𝔼|Xj|p<∞ implies
(4)ℙ(|Sn|>εn)=O(n-p/2)
(as usual, we write an=o(bn) if limn→∞an/bn=0 and an=O(bn) if the sequence (an/bn) is bounded) and that the exponent p/2 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 p>2 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 p>2, we cannot choose α satisfying (6) of [17].) Alsmeyer [18] proved that the theorem of Baum and Katz for p>1/α and 1/2<α≤1 still holds for martingale differences (Xj)j≥1 if for some γ∈(1/α,2] and q∈[1,∞] with q>(pα-1)/(γα-1),
(5)supn≥1∥1n∑j=1n‍𝔼[|Xj|γ∣ℱj-1]∥q<∞,
where ∥·∥q denotes the Lq 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 Sn=∑j=1n‍jaYj, where a>0 and Yj are i.i.d., as in this case the condition (5) (with Xj=jaYj) 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 (Sn) into martingale differences (such as in the study of directed polymers in a random environment), the summands usually depend on n: Sn=∑j=1n‍Xnj, Xnj=𝔼[Sn∣ℱj]-𝔼[Sn∣ℱj-1], where ℱ0={∅,Ω} and ℱi=σ(S1,…,Si) for i≥1.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 ∑j=1n‍𝔼[∥Xnj∥γ∣ℱn,j-1] for some γ∈(1,2], we will find sufficient conditions for
(6)∑n=1∞‍ϕ(n)ℙ{∥Sn,∞∥>ε}<∞
for a large class of sequences of Banach valued martingale differences {(Xnj,ℱnj)}j≥1, n≥1, where Sn,∞=∑j=1∞‍Xnj, ϕ:ℕ→[0,∞) is a positive function, and ε>0. Of particular interest is the case where ϕ(n) is a regular function: ϕ(n)=nb-1l(n)  (b≥0), l(·)>0 being slowly varying at ∞; that is, l(·) is a positive measurable function defined on (0,∞) such that limx→∞(l(λx)/l(x))=1 for any λ>0. 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 Xj with 𝔼Xj=0 and for each p≥1, ℙ(|X1|>n)=o(n-p) if and only if ℙ(|Sn|>εn)=o(n-(p-1)) for all ε>0. In fact, we prove that a similar result holds for a large class of Banach valued martingale arrays: under a simple moment condition on ∑j=1n‍𝔼[∥Xnj∥γ∣ℱn,j-1] for some γ∈(1,2], we obtain sufficient conditions for
(7)ϕ(n)ℙ{∥Sn,∞∥>ε}=o(1)(resp.,O(1)),
where ϕ and Sn,∞ are defined as before, ε>0. 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 Section2, 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 ∑j=1n‍ja-1Xj. 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 writeℕ*={1,2,…}, ℕ={0}⋃‍ℕ* and ℝ=(-∞,∞).


## 2. Maximal Inequalities for Banach Valued Martingales

In this section, we show new maximal inequalities for Banach valued martingales.Let(Ω,ℱ,ℙ) be a probability space and (𝔹,∥·∥) a real separable Banach space. For any real number p≥1, denote by 𝕃𝔹p the space of 𝔹-valued random variables such that ∥X∥𝕃𝔹p=(𝔼∥X∥p)1/p is finite. Let ℱ0={∅,Ω}⊂ℱ1⊂⋯ be an increasing sequence of sub-σ-fields of ℱ. Let {(Xj,ℱj)}j=1n be an adapted sequence of 𝔹-valued random variables defined on (Ω,ℱ,ℙ); that is, for every j≥1, Xj is ℱj measurable. We call it a sequence of 𝔹-valued martingale differences if additionally 𝔼[Xj∣ℱj-1]=0 a.s. and Xj belongs to 𝕃𝔹1 for any j≥1 and a sequence of 𝔹-valued supermartingale differences if additionally 𝔼[Xj∣ℱj-1]≤0 a.s. and Xj belongs to 𝕃𝔹1 for any j≥1. Following Pisier [36], we say that a Banach space (𝔹,∥·∥) is γ-smooth (1<γ≤2) if there exists an equivalent norm ∥·∥𝔹 such that
(8)supt>0{1tγsup{∥x+ty∥𝔹+∥x-ty∥𝔹-2:∥x∥𝔹=∥y∥𝔹=1}}<∞.
Set
(9)S0=0,Sn=∑j=1n‍Xj,Sn*=max1≤j≤n∥Sj∥∀n∈ℕ*,
and set
(10)X0*=0,Xn*=max1≤j≤n∥Xj∥∀n∈ℕ*.
For γ>0, let
(11)m(γ,n)=∑j=1n‍𝔼[∥Xj∥γ∣ℱj-1].
Accordingly, for an infinite 𝔹-valued adapted sequence {(Xj,ℱj)}j≥1, we write
(12)X∞*=supj≥1∥Xj∥,S∞*=supj≥1∥Sj∥,S∞=∑j=1∞‍Xj
if the series converges, and
(13)m(γ,∞)=∑j=1∞‍𝔼[∥Xj∥γ∣ℱj-1].In the following, we consider relations among∑j=1n‍ℙ{∥Xj∥>ε}, ℙ{Xn*>ε}, ℙ{Sn*>ε}, and ℙ{∥Sn∥>ε}.Our first theorem describes relations betweenℙ{Xn*>ε} and ∑j=1n‍ℙ{∥Xj∥>ε} for an adapted sequence of 𝔹-valued random variables {(Xj,ℱj)}j=1n.Theorem 1.
Let{(Xj,ℱj)}j=1n be an adapted sequence of 𝔹-valued random variables. Then, for any ε,γ>0, and q≥1,
(14)ℙ{Xn*>ε}≤∑j=1n‍ℙ{∥Xj∥>ε}≤(1+ε-γ)ℙ{Xn*>ε}+ε-γ𝔼mq(γ,n).Our second theorem shows relations betweenℙ{Sn*>ε} and ℙ{Xn*>ε} for a sequence of 𝔹-valued martingale differences {(Xj,ℱj)}j=1n: that is, for each (integer) 1≤j≤n, Xj is ℱj measurable and belongs to 𝕃𝔹1, and 𝔼[Xj∣ℱj-1]=0 a.s.Theorem 2.
Let{(Xj,ℱj)}j=1n be a finite sequence of 𝔹-valued martingale differences. For any ε>0,γ∈(1,2],q≥1, and L∈ℕ, if 𝔹 is γ-smooth, then
(15)ℙ{Xn*>2ε}≤ℙ{Sn*>ε}≤ℙ{Xn*>ε4(L+1)}+ε-qγ(1+L)/(q+L)C(γ,q,L)×(𝔼mq(γ,n))(1+L)/(q+L),
where C(γ,q,L) is a constant only depending on γ, q, and L.Corollary 3.
Let{(Xj,ℱj)}j≥1 be a sequence of 𝔹-valued martingale differences. Suppose that, for some γ∈(1,2],
(16)∑j=1∞‍𝔼∥Xj∥γ<∞.
If 𝔹 is γ-smooth, then S∞ converges a.s. and the inequalities (14) and (15) hold with n replaced by ∞.We get Theorems1 and 2 by a refinement of the method of Alsmeyer [18].Proof of Theorem1.
The first inequality is obvious. We only consider the second one. Clearly,(17)ℙ{Xn*>ε}=∑j=1n‍ℙ{Xj-1*≤ε,∥Xj∥>ε}=∑j=1n‍ℙ{∥Xj∥>ε}-∑j=2n‍ℙ{Xj-1*>ε,∥Xj∥>ε}.
Since {(Xj,ℱj)}j=1n is an adapted sequence of 𝔹-valued random variables, by Markov's inequality (conditional on ℱj-1),
(18)ℙ{Xj-1*>ε,∥Xj∥>ε}=∫{Xj-1*>ε}‍1{∥Xj∥>ε}dℙ=∫{Xj-1*>ε}‍ℙ{∥Xj∥>ε∣ℱj-1}dℙ≤ε-γ∫{Xj-1*>ε}‍𝔼[∥Xj∥γ∣ℱj-1]dℙ≤ε-γ∫{Xn*>ε}‍𝔼[∥Xj∥γ∣ℱj-1]dℙ.
Hence, by summing, we obtain
(19)∑j=2n‍ℙ{Xj-1*>ε,∥Xj∥>ε}≤ε-γ∫{Xn*>ε}‍m(γ,n)dℙ=ε-γ∫{m(γ,n)≤1,Xn*>ε}‍m(γ,n)dℙ+ε-γ∫{m(γ,n)>1,Xn*>ε}‍m(γ,n)dℙ≤ε-γℙ{Xn*>ε}+ε-γ∫{m(γ,n)>1}‍mq(γ,n)dℙ≤ε-γℙ{Xn*>ε}+ε-γ𝔼mq(γ,n).
Therefore, the upper bound in (19) gives a lower bound of ℙ{Xn*>ε} by (17), which implies the second inequality of (14).Proof of Theorem2.
The first inequality is obvious, because ifmax1≤j≤n∥Sj∥≤ε, then
(20)max1≤j≤n∥Xj∥=max1≤j≤n∥Sj-Sj-1∥≤2ε.
We will prove the second inequality. For any ε>0,n∈ℕ*, and L∈ℕ,
(21)ℙ{Sn*>2ε}≤ℙ{Xn*>ε2(L+1)}+ℙ{Sn*>2ε,Xn*≤ε2(L+1)}.
Define
(22)T(0)=0,T(j)=inf{i∈(T(j-1),n]:∥Si-ST(j-1)∥>εL+1}{i∈(T(j-1),n]:∥Si-ST(j-1)∥>}for1≤j≤L+1,
where by convention inf∅=+∞. It is easily seen that T(j) are stopping times (cf. e.g., [34] for the definition) with respect to the filtration ℱ0={∅,Ω}⊂ℱ1⊂⋯⊂ℱn⊂ℱn+1⊂⋯, where we take ℱk=ℱn for all k≥n. As usual, we will write ℱT(j)={A∈ℱn:A∩{T(j)=k}∈ℱk,1≤k≤n}. Notice that for 1≤j≤L+1, if T(j)<∞, then ∥ST(j)-ST(j-1)∥>ε/(L+1); conversely, if there exists a positive integer i∈(T(j-1),n] such that ∥Si-ST(j-1)∥>ε/(L+1), then T(j)<∞. We proceed by three steps to estimate the second term of the right hand side of (21).
(a) We first prove that(23){Sn*>2ε,Xn*≤ε2(L+1)}⊂∏j=1L+1‍{T(j)<∞},
which implies that
(24)ℙ{Sn*>2ε,Xn*≤ε2(L+1)}≤ℙ{Vj>εL+1,1≤j≤L+1},
where
(25)Vj=∥ST(j)-ST(j-1)∥1{T(j)<∞},1≤j≤L+1.
Assume that the first event in (23) takes place. Since Sn*>2ε, there exists m∈{1,2,…,n} such that ∥Sm∥>2ε. As ∥Sm∥>2ε, and it is clear that T(1)≤m<∞.
LetM be the largest j∈[1,m] such that T(j)≤m. Then, T(M)≤m.
We will prove thatM≥L+1. Suppose that M≤L. Then, by the definition of M, T(M+1)>m so that
(26)∥Sm-ST(M)∥≤εL+1.
As ∥x+y∥≤∥x∥+∥y∥, it follows that
(27)∥ST(j)-ST(j-1)∥=∥ST(j)-1-ST(j-1)+XT(j)∥≤∥ST(j)-1-ST(j-1)∥+∥XT(j)∥≤εL+1+ε2(L+1),
where the last step holds because ∥ST(j)-1-ST(j-1)∥≤ε/(L+1) by the definition of T(j) and ∥XT(j)∥≤Xn*≤ε/2(L+1). As Sm=∑j=1M‍(ST(j)-ST(j-1))+(Sm-ST(M)), by (26), (27), and the subadditivity of ∥·∥, we know that
(28)∥Sm∥≤εL+1·M+ε2(L+1)·M+εL+1<2ε.
This is a contradiction with ∥Sm∥>2ε, which proves that M≥L+1.
Therefore,T(j)<∞ for all j∈[1,L+1]. Thus, (23) holds.
(b) We next give an estimation of𝔼[Vjγ∣ℱT(j-1)]. For 0≤r≤n,
(29)𝔼[Vjγ1{T(j-1)=r}∣ℱT(j-1)]=𝔼[Vjγ1{T(j-1)=r}∣ℱr]≤𝔼[∥ST(j)-ST(j-1)∥γ1{T(j-1)=r,T(j)<∞}∣ℱr]≤1{T(j-1)=r}𝔼[maxr+1≤i≤n∥Si-Sr∥γ∣ℱr].
Using successively Doob's inequality in a real separable Banach space [37, Theorem 3.1], the inequality in Assouad [38, Proposition 2] (conditional on ℱr), and the subadditivity of the function x↦xγ/2(x≥0), we obtain
(30)𝔼[maxr+1≤i≤n∥Si-Sr∥γ∣ℱr]≤(γγ-1)γ𝔼[∥∑i=r+1n‍Xi∥γ∣ℱr]≤(γγ-1)γC(γ)∑i=r+1n‍𝔼[∥Xi∥γ∣ℱr]=(γγ-1)γC(γ)𝔼[∑i=r+1n‍𝔼[∥Xi∥γ∣ℱi-1]∣ℱr]≤(γγ-1)γC(γ)𝔼[m(γ,n)∣ℱr].
Therefore, by (29),
(31)𝔼[Vjγ1{T(j-1)=r}∣ℱT(j-1)]≤(γγ-1)γC(γ)𝔼[m(γ,n)1{T(j-1)=r}∣ℱT(j-1)].
Summing over r, we obtain
(32)𝔼[Vjγ∣ℱT(j-1)]≤(γγ-1)γC(γ)𝔼[m(γ,n)∣ℱT(j-1)].
(c) We finally give un upper bound for the term of the right hand side of (24), using (32). Set
(33)Qj=ℙ{Vj>εL+1∣ℱT(j-1)},1≤j≤L+1.
Applying (32) for j=1, we see that
(34)Q1=ℙ{V1>εL+1}≤ε-γ(L+1)γ𝔼V1γ≤ε-γ(L+1)γ(γγ-1)γC(γ)𝔼m(γ,n).
Now, for any x>0,
(35)ℙ{Vj>εL+1,1≤j≤L+1}≤∑i=1L+1‍pi,
where
(36)p1=ℙ{Vj>εL+1,1≤j≤L+1;Qi≤x-1,2≤i≤L+1},pi=ℙ{Qi>x-1},2≤i≤L+1.
Notice that
(37)p1=∫{Vj>ε/(L+1),1≤j≤L;Qi≤x-1,2≤i≤L+1}𝔼‍[1{VL+1>ε/(L+1)}∣ℱT(L)]dℙ=∫{Vj>ε/(L+1),1≤j≤L;Qi≤x-1,2≤i≤L+1}‍QL+1dℙ≤x-1ℙ{Vj>εL+1,1≤j≤L;Qi≤x-1,2≤i≤L}≤⋯≤x-(L-1)ℙ{Vj>εL+1,1≤j≤2;Q2≤x-1}≤x-LQ1≤ε-γ(L+1)γ(γγ-1)γC(γ)x-L𝔼m(γ,n),
where (34) has been used for the last inequality. For 2≤i≤L+1, by (32), together with Markov's inequality and Jensen's inequality, we have
(38)pi=ℙ{ℙ{Vi>εL+1∣ℱT(i-1)}>x-1}≤ℙ{𝔼[Viγ∣ℱT(i-1)]>εγx-1(L+1)γ}≤ℙ{γγ(γ-1)-γC(γ)𝔼[m(γ,n)∣ℱT(i-1)]>εγx-1(L+1)γ}≤ε-qγ(L+1)qγ(γγ-1)qγCq(γ)×xq𝔼(𝔼[m(γ,n)∣ℱT(i-1)])q≤ε-qγ(L+1)qγ(γγ-1)qγCq(γ)xq𝔼mq(γ,n).
Therefore, by (35),
(39)ℙ{Vj>εL+1,1≤j≤L+1}≤C1x-L+C2xq,
where
(40)C1=ε-γ(L+1)γ(γγ-1)γC(γ)𝔼m(γ,n),C2=ε-qγL(L+1)qγ(γγ-1)qγCq(γ)𝔼mq(γ,n).
A simple calculation shows that(41)infx>0{C1x-L+C2xq}=ε-qγ(1+L)/(q+L)C0(γ,q,L)(𝔼m(γ,n))q/(q+L)×(𝔼mq(γ,n))L/(q+L),
where C0(γ,q,L)=(L+1)qγ(1+L)/(q+L)[qL/(q+L)+Lq-q/(q+L)](γγ(γ-1)-γC(γ))q(1+L)/(q+L). Therefore,
(42)ℙ{Vj>ε1+L,1≤j≤L+1}≤ε-qγ(1+L)/(q+L)C0(γ,q,L)(𝔼m(γ,n))q/(q+L)×(𝔼mq(γ,n))L/(q+L)≤ε-qγ(1+L)/(q+L)C0(γ,q,L)(𝔼mq(γ,n))(1+L)/(q+L).
Together with (21) and (24), this proves (15).Proof of Corollary3.
By Theorem2 with q=1 and L=0,
(43)ℙ{supj≥1∥Sn+j-Sn∥>ε}≤ℙ{supj≥n+1∥Xj∥>ε4}+ε-γC(γ)∑j=n+1∞‍𝔼∥Xj∥γ.
By Theorem 1 and Markov's inequality,
(44)ℙ{supj≥n+1∥Xj∥>ε4}≤∑j=n+1∞‍ℙ{∥Xj∥>ε4}≤4γε-γ∑j=n+1∞‍𝔼∥Xj∥γ.
Therefore,
(45)limn→∞ℙ{supj≥1∥Sn+j-Sn∥>ε}=0,
which is equivalent to
(46)limn→∞supj,k≥n∥Sk-Sj∥=0inprobability.
Since supj,k≥n∥Sk-Sj∥ is decreasing in n, this implies that
(47)limn→∞supj,k≥n∥Sk-Sj∥=0a.s.,
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(Ω,ℱ,ℙ) a probability space and (𝔹,∥·∥) be a real separable Banach space. For every n≥1, let ℱn0={∅,Ω}⊂ℱn1⊂⋯ be an increasing sequence of sub-σ-fields of ℱ. For every n≥1, let {(Xnj,ℱnj)}j≥1 be a sequence of 𝔹-valued martingale differences defined on (Ω,ℱ,ℙ), adapted to the filtration (ℱnj): that is, for every j≥1, Xnj is ℱnj measurable and belongs to 𝕃𝔹1, and 𝔼[Xnj∣ℱn,j-1]=0 a.s. Set for n≥1,
(48)Sn0=0,Snk=∑j=1k‍Xnjfork≥1,Sn,∞=∑j=1∞‍Xnj
if the series converges. We will call the double sequence {(Xnj,ℱnj),j≥1,n≥1} an array of 𝔹-valued martingale differences.In the following, we give a sufficient condition for the convergence of𝔹-valued martingale arrays. For γ>0, let
(49)mn(γ)=∑j=1∞‍𝔼[∥Xnj∥γ∣ℱn,j-1]forn≥1.Theorem 4.
Assume that for someγ∈(1,2], as n→∞,
(50)𝔼mn(γ)=∑j=1∞‍𝔼∥Xnj∥γ⟶0.
If 𝔹 is γ-smooth, then for all ε>0, as n→∞,
(51)∑j=1∞‍ℙ{∥Xnj∥>ε}⟶0;(52)ℙ{supj≥1∥Xnj∥>ε}⟶0;(53)ℙ{supj≥1∥Snj∥>ε}⟶0;(54)ℙ{∥Sn,∞∥>ε}⟶0.Proof.
Notice that, by Corollary3, the condition (50) implies the a.s. convergence of Sn,∞. Equation (51) comes from (50) as
(55)∑j=1∞‍ℙ{∥Xnj∥>ε}≤ε-γ∑j=1∞‍𝔼∥Xnj∥γ⟶0;
(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 probabilitiesℙ{supj≥1∥Snj∥>ε} and ℙ{∥Sn,∞∥>ε}. We will describe their rates of convergence by comparing them with an auxilary function ϕ(n) and by considering the convergence of the related series.We begin with some relations among∑j=1∞‍ℙ{∥Xnj∥>ε}, ℙ{supj≥1∥Xnj∥>ε}, ℙ{supj≥1∥Snj∥>ε}, and ℙ{∥Sn,∞∥>ε}.Lemma 5.
Letϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2],q≥1 and some integer L≥0,
(56)ϕ(n)(𝔼mnq(γ))(1+L)/(q+L)=o(1)(resp.,O(1)).
If 𝔹 is γ-smooth, then the following conditions are equivalent:
(57)ϕ(n)ℙ{supj≥1∥Xnj∥>ε}=o(1)(1)(resp.,O(1))foranyε>0;ϕ(n)ℙ{supj≥1∥Snj∥>ε}=o(1)11(resp.,O(1))foranyε>0.Proof.
Equation (57) are equivalent by Theorem 2.Lemma 6.
Letϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2] and q∈[1,∞),
(58)ϕ(n)𝔼mnq(γ)=o(1)(resp.,O(1)).
Then, the following conditions are equivalent:
(59)ϕ(n)ℙ{supj≥1∥Xnj∥>ε}=o(1)111(resp.,O(1))foranyε>0;ϕ(n)∑j=1∞‍ℙ{∥Xnj∥>ε}=o(1)(1)(resp.,O(1))foranyε>0.Proof.
The conclusion comes directly from Theorem1.Lemma 7.
Letϕ:ℕ→[0,∞) be a positive function. Suppose that (58) holds for some γ∈(1,2] and q∈[1,∞). Then,
(60)ϕ(n)ℙ{∥Sn,∞∥>ε}=o(1)(resp.,O(1))foranyε>0
is implied by
(61)ϕ(n)ℙ{supj≥1∥Snj∥>ε}=o(1)11(resp.,O(1))foranyε>0.Proof.
The equivalence is an immediate consequence of Corollary3.Theorem 8.
Letϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(62)∥mn(γ)∥𝕃𝔹q⟶0,ϕ(n)∥mn(γ)∥𝕃𝔹qλ=O(1).
If 𝔹 is γ-smooth, then one has the following implications (63)⇔(64)⇔(65)⇒(66):
(63)ϕ(n)∑j=1∞‍ℙ{∥Xnj∥>ε}=o(1)11(resp.,O(1))foranyε>0;(64)ϕ(n)ℙ{supj≥1∥Xnj∥>ε}=o(1)11(resp.,O(1))foranyε>0;(65)ϕ(n)ℙ{supj≥1∥Snj∥>ε}=o(1)11(resp.,O(1))foranyε>0;(66)1ϕ(n)ℙ{∥Sn,∞∥>ε}=o(1)(resp.,O(1))foranyε>0.Remark 9.
The condition (62) holds if for some γ∈(1,2],r∈ℝ and ε1>0,
(67)ϕ(n)=O(nr),∥mn(γ)∥𝕃𝔹∞=O(n-ε1).Proof of Theorem8.
Notice that when (62) holds for q=∞ and some λ∈(0,∞), then for q∈(λ,∞),
(68)ϕ(n)∥mn(γ)∥𝕃𝔹qλ≤ϕ(n)∥mn(γ)∥𝕃𝔹∞λ=O(1).
Therefore, we can assume that (62) holds for some q∈[1,∞) and λ∈(0,q). Since as L→∞, q(1+L)/(q+L)→q>λ, we can choose an integer L≥0 large enough such that q(1+L)/(q+L)>λ. Therefore, the condition (56) holds with o(1). By Lemma 6, (63) and (64) are equivalent; by Lemma 5, (64) and (65) are equivalent; by Lemma 7, (65) implies (66).Theorem 10.
Letϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(69)∥mn(γ)∥𝕃𝔹q⟶0,∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ<∞.
If 𝔹 is γ-smooth, then one has the following implications (70)⇔(71)⇔(72)⇒(73):
(70)∑n=1∞‍ϕ(n)∑j=1∞‍ℙ{∥Xnj∥>ε}<∞∀ε>0;(71)∑n=1∞‍ϕ(n)ℙ{supj≥1∥Xnj∥>ε}<∞∀ε>0;(72)∑n=1∞‍ϕ(n)ℙ{supj≥1∥Snj∥>ε}<∞∀ε>0;(73)∑n=1∞‍ϕ(n)ℙ{∥Sn,∞∥>ε}<∞∀ε>0.Notice that, by (69), 𝔼mn(γ)<∞ for each n≥1 with ϕ(n)>0, so that Sn,∞ is well defined (cf. Corollary 3). When ϕ(n)=0, we use the convention that the associated term containing ϕ(n) as a factor is defined by 0.Whenq=∞, γ=2, and {(Xnj,ℱnj)}j≥1 is a sequence of real-valued martingale differences, the implication “(70)⇒(72)” reduces to Theorem 2 of Ghosal and Chandra [19]. (Although the condition ∥mn(2)∥𝕃𝔹∞→0 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.
Theorem10 also holds if mn(γ) is replaced by ∑j=1∞‍𝔼[∥Xnj∥γ∣ℱn,j-M] for  some M≥1. In fact, the case M≥2 can be reduced to the case M=1 by considering the subsequences {(Xn,lM+i,ℱn,lM+i)}l≥0 (1≤i≤M) of {(Xnk,ℱnk)}k≥1, which are still sequences of 𝔹-valued martingale differences.Corollary 12.
Suppose that (67) holds for some γ∈(1,2],r∈ℝ, and ε1>0. Then one has the implications (70)⇔(71)⇔(72)⇒(73).Proof of Theorem10.
As in the proof of Theorem8, we can assume that q<∞. Since as L→∞, q(1+L)/(q+L)→q>λ, we can choose an integer L≥0 large enough such that q(1+L)/(q+L)>λ. Let n0 be large enough such that ∥mn(γ)∥𝕃𝔹q≤1 for all n≥n0. Then,
(74)∑n=n0∞‍ϕ(n)(𝔼mnq(γ))(1+L)/(q+L)=∑n=n0∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qq(1+L)/(q+L)≤∑n=n0∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ<∞.
By Theorem 1, (70) and (71) are equivalent; by Theorem 2, (71) and (72) are equivalent; since (69) implies 𝔼mn(γ)<∞ for each n≥1 with ϕ(n)>0, by Corollary 3, (72) implies (73).Proof of Corollary12.
Chooseλ>(r+1)/ε1, then by (67), we have
(75)∥mn(γ)∥𝕃𝔹∞⟶0,∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹∞λ=∑n=1∞‍O(nr-ε1λ)<∞.
So, the condition (69) holds for q=∞, 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(Ω,ℱ,ℙ) be a probability space and (𝔹,∥·∥) a real separable Banach space. For every n≥1, let ℱn0={∅,Ω}⊂ℱn1⊂⋯⊂ℱnn be an increasing sequence of sub-σ-fields of ℱ. For each n≥1, let {(Xnj,ℱnj)}j=1n be a sequence of 𝔹-valued martingale differences defined on (Ω,ℱ,ℙ), adapted to the filtration (ℱnj): that is, for every 1≤j≤n and every n≥1, Xnj is ℱnj measurable and belongs to 𝕃𝔹1, and 𝔼[Xnj∣ℱn,j-1]=0 a.s. Set for n≥1,
(76)Sn0=0,Snk=∑j=1k‍Xnj,1≤k≤n.
We will call the double sequence {(Xnj,ℱnj),1≤j≤n,n≥1} a triangular array of 𝔹-valued martingale differences. In the following, we first give a sufficient condition for the convergence of triangular arrays of 𝔹-valued martingale. For α∈ℝ and γ>0, let
(77)mn(γ)=n-γα∑j=1n‍𝔼[∥Xnj∥γ∣ℱn,j-1]forn≥1.Theorem 13.
Letα∈ℝ. Assume that for some γ∈(1,2], as n→∞,
(78)𝔼mn(γ)=n-γα∑j=1n‍𝔼∥Xnj∥γ⟶0.
If 𝔹 is γ-smooth, then for all ε>0, as n→∞,
(79)∑j=1n‍ℙ{∥Xnj∥>εnα}⟶0;ℙ{max1≤j≤n∥Xnj∥>εnα}⟶0;ℙ{max1≤j≤n∥Snj∥>εnα}⟶0;ℙ{∥Snn∥>εnα}⟶0.Proof.
It suffices to apply Theorem4 for the array of 𝔹-valued martingale differences {(Ynj,𝒢nj),j≥1,n≥1} defined by
(80)Ynj={Xnjnαif1≤j≤n,0ifj>n,𝒢nj={ℱnjif1≤j≤n,ℱnnifj>n.We are interested in the convergence rates of the probabilitiesℙ{max1≤j≤n∥Snj∥>εnα} and ℙ{∥Snn∥>εnα}. We will describe their rates of convergence by comparing them with an auxilary function ϕ(n) and by considering the convergence of the related series.We begin with some relations among∑j=1n‍ℙ{∥Xnj∥>εnα}, ℙ{max1≤j≤n∥Xnj∥>εnα}, ℙ{max1≤j≤n∥Snj∥>εnα}, and ℙ{∥Snn∥>εnα}.Theorem 14.
Letα∈ℝ and ϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(81)∥mn(γ)∥𝕃𝔹q⟶0,ϕ(n)∥mn(γ)∥𝕃𝔹qλ=O(1).
If 𝔹 is γ-smooth, then one has the following implications (82)⇔(83)⇔(84)⇒(85):
(82)ϕ(n)∑j=1n‍ℙ{∥Xnj∥>εnα}=o(1)11(resp.,O(1))foranyε>0;(83)ϕ(n)ℙ{max1≤j≤n∥Xnj∥>εnα}=o(1)11111(resp.,O(1))foranyε>0;(84)ϕ(n)ℙ{max1≤j≤n∥Snj∥>εnα}=o(1)11111(resp.,O(1))foranyε>0;(85)ϕ(n)ℙ{∥Snn∥>εnα}=o(1)(resp.,O(1))foranyε>0.Corollary 15.
Let1/2<α≤1 and b≥0. Let l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nbl(n). Suppose that for some γ∈(1/α,2], q∈[1,∞] with q>b/(γα-1),
(86)supn≥1∥m_n(γ)∥𝕃𝔹q<∞,wherem_n(γ)=1n∑j=1n‍𝔼[∥Xnj∥γ∣ℱn,j-1].
If 𝔹 is γ-smooth, then one has the implications (82)⇔(83)⇔(84)⇒(85).Remark 16.
It is obvious that (86) holds with q=∞ if for some constant K>0, all n≥1 and j≥1,
(87)𝔼[∥Xnj∥γ∣ℱn,j-1]≤Ka.s.Proof of Theorem14.
It suffices to apply Theorem8 for the array of 𝔹-valued martingale differences {(Ynj,𝒢nj),j≥1,n≥1} defined by (80).Proof of Corollary15.
Sinceγ>1/α, we have
(88)∥mn(γ)∥𝕃𝔹q=n1-γα∥m_n(γ)∥𝕃𝔹q⟶0.
As q>b/(γα-1), we can choose λ∈(b/(γα-1),q). For this λ,
(89)ϕ(n)∥mn(γ)∥𝕃𝔹qλ=nbl(n)(n1-γα∥m_n(γ)∥𝕃𝔹q)λ=nb-λ(γα-1)l(n)∥m_n(γ)∥𝕃𝔹qλ=o(1).
Thus, the condition (81) holds, so that the conclusion follows from Theorem 14.Theorem 17.
Letα∈ℝ and ϕ:ℕ→[0,∞) be a positive function. Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(90)∥mn(γ)∥𝕃𝔹q⟶0,∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ<∞.
If 𝔹 is γ-smooth, then one has the following implications (91)⇔(92)⇔(93)⇒(94):
(91)∑n=1∞‍ϕ(n)∑j=1n‍ℙ{∥Xnj∥>εnα}<∞∀ε>0;(92)∑n=1∞‍ϕ(n)ℙ{max1≤j≤n∥Xnj∥>εnα}<∞∀ε>0;(93)∑n=1∞‍ϕ(n)ℙ{max1≤j≤n∥Snj∥>εnα}<∞∀ε>0;(94)∑n=1∞‍ϕ(n)ℙ{∥Snn∥>εnα}<∞∀ε>0.Corollary 18.
Letα>1/2 and b≥0. Let l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nb-1l(n). Suppose that (86) holds for some γ∈(max{1/α,1},2] and q∈[1,∞] with q>b/(γα-1). If 𝔹 is γ-smooth, then one has the implications (91)⇔(92)⇔(93)⇒(94).For a single real-valued martingale, whenl(n)=1 and α≤1, 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 𝔹-valued random variables of the form ∑j=1n‍anjXj.Remark 19.
As explained in Remark11, Corollary 18 also holds if m¯n(γ) is replaced by (1/n)∑j=1n‍𝔼[∥Xnj∥γ∣ℱn,j-M] for some M≥1.Proof of Theorem17.
It suffices to apply Theorem10 for the array of 𝔹-valued martingale differences {(Ynj,𝒢nj),j≥1,n≥1} defined by (80).Proof of Corollary18.
Notice that(95)∥mn(γ)∥𝕃𝔹q=n1-γα∥m¯n(γ)∥𝕃𝔹q⟶0.
As q>b/(γα-1), we can take λ∈(b/(γα-1),q).
Then,(96)∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ=∑n=1∞‍nb-1l(n)(n1-γα∥m¯n(γ)∥𝕃𝔹q)λ=∑n=1∞‍nb-1-λ(γα-1)l(n)∥m¯n(γ)∥𝕃𝔹qλ<∞.
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{(Xnj,ℱnj)}1≤j≤n (n≥1) be sequences of identically distributed 𝔹-valued martingale differences. Let p∈[1,4). Suppose that (86) holds for some γ∈(max{p/2,1},2] and q∈[1,∞] with q>p/(2γ-p). If 𝔹 is γ-smooth, then 𝔼∥X11∥p<∞ if and only if
(97)∑n=1∞‍ℙ{n-2/pmax1≤j≤n∥∑i=1j‍Xni∥>ε}<∞∀ε>0.When{Xnj} are rowwise independent real-valued martingale differences, the sufficiency in Corollary 20 was proved in [34, page 393].Proof of Corollary20.
It suffices to apply Corollary18 with α=2/p and l(n)=1: we just need to check that in the present case, (91) is equivalent to 𝔼∥X11∥p<∞. In fact, we have
(98)∑n=1∞‍∑j=1n‍ℙ{n-2/p∥Xnj∥>ε}=∑n=1∞‍nℙ{∥X11∥>εn2/p}=𝔼(∑1≤n<ε-p2∥X11∥p2‍n).
As ∑n=1m‍n=(1/2)m(m+1), the last expectation is finite if and only if 𝔼∥X11∥p<∞.


## 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 Section6.Let(𝔹,∥·∥) a separable Banach space and (Yn)n≥1 be a sequence of any 𝔹-valued random variables. For any υ∈[1,∞), let [υ] be the integer part of υ. Set
(99)Y0=0,Yυ=Y[υ],Yυ*=max0≤k≤υ∥Yk∥111111111111111111111forυ∈[0,∞).
Then, for any n≥1, Yn*=max0≤k≤n∥Yk∥. For any α>0,ε>0, set
(100)M(ε)=M(ε,α)=supn≥1(∥Yn∥-εnα)+=supn≥1(Yn*-εnα)+=supn≥0(Yn*-εnα).
Let l(·)>0 be a function slowly varying at ∞. Recall that a function l(x)>0 slowly varying at ∞ has the representation form
(101)l(x)=c(x)exp{∫x0x‍ε(u)udu}(x≥x0)
for some x0>0, where c(·) is measurable and c(x)→c∈(0,∞), ε(x)→0asx→∞. The function c(·) plays no role for our purpose. We can choose c(x)≡1 without loss of generality.We are interested in the convergence rates ofℙ{Yn*>εnα} and ℙ{supj≥nj-αYj*>ε}. Notice that ℙ{supj≥nj-αYj*>ε}→0 for any ε>0 if and only if Yn*/nα→0 a.s. So, our results in this section describe the rate convergence for the almost surely convergence of Yn*/nα.The following result shows thatℙ{Yn*>εnα} and ℙ{M(ε)>εnα} have similar asymptotic properties. More precise comparisons will be given in Theorems 22 and 24.Lemma 21.
Letα>0. Then, for any υ≥1 and any ε>0,
(102)ℙ{Yυ*>2ευα}≤ℙ{supj≥υj-αYj*>2ε}≤ℙ{M(ε)>ευα}.
Let ε,α,b,δ>0 and ϕ(v)=vbl(v). If there exists v0>0, such that for all v≥v0,
(103)ϕ(v)ℙ{Yv*>12εvα}≤δ,
then there exists υ0′>0 depending only on v0,b, and α, such that for all υ≥υ0′,
(104)ϕ(υ)ℙ{M(ε)>ευα}≤δC(b,α),
where C(b,α)=2/(2b/(2α)-1).Proof.
The first inequality of (102) is obvious. If
(105)supj≥υj-αYj*>2ε,
then
(106)supn≥1(Yn*-εnα)+=supn≥1nα(n-αYn*-ε)+≥υαsupn≥υ(n-αYn*-ε)+>ευα.
Thus, the second inequality of (102) holds.
Assume that for somev0>0 and all v≥v0, (103) holds (with the notation introduced in the lemma). Then, there exists υ2=υ2(v0,b,α)>0, such that for all υ1≥υ2,
(107)ℙ{Y2k/αυ11/b*>2k-1ευ1α/b}≤δϕ(2k/αυ11/b).
Set β=b/α. Then, applying (103) for v=2k/αυ11/b, we see that, for any υ1≥υ2,
(108)ℙ{supn≥1(Yn*-εnα)+>ευ11/β}=ℙ{supn≥1(∥Yn∥-εnα)+>ευ11/β}≤∑k=1∞‍ℙ{max(2k-1-1)υ11/β<nα≤2kυ11/β(∥Yn∥-εnα)+>ευ11/β}≤∑k=1∞‍ℙ{Y(2kυ11/β)1/α*>2k-1ευ11/β}=∑k=1∞‍ℙ{Y2k/αυ11/b*>2k-1ευ1α/b}≤∑k=1∞‍δϕ(2k/αυ11/b).
Set υ=υ11/b. Since l is slowly varying at ∞, by Potter's Theorem (cf. Theorem 1.5.6 in [39, page 25]), for A=2 and δ1=b/2>0, there exists υ3=υ3(b,α) such that for all υ≥υ3, l(υ)/l(2k/αυ)≤2(2k/α)δ1=2·2bk/2α. Thus, there exists υ0′=υ0′(v0,b,α)=max{υ21/b,υ3}>0 such that for all υ≥υ0′,
(109)ϕ(υ)ℙ{supn≥1(Yn*-εnα)+>ευα}≤δ∑k=1∞‍ϕ(υ)ϕ(2k/αυ)=δ∑k=1∞‍2-bk/αl(υ)l(2k/αυ)≤2δ∑k=1∞‍2-bk/(2α)=δC(b,α).Theorem 22.
Letα,b>0 and ϕ(n)=nbl(n). Then, the following assertions are equivalent:
(110)ϕ(n)ℙ{Yn*>εnα}=o(1)(resp.,O(1))foranyε>0;(111)ϕ(n)ℙ{supj≥nj-αYj*>ε}=o(1)111(resp.,O(1))foranyε>0;(112)ϕ(n)ℙ{M(ε)>εnα}=o(1)1(resp.,O(1))foranyε>0.Proof.
We use Lemma21. 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.
Letα>-1. Then for some n0,c1,c2>0 and all N≥n0,
(113)c1Nα+1l(N)≤∑n=1N‍nαl(n)≤c2Nα+1l(N).Proof.
Without loss of generality, we suppose thatl(s) has the form (101) with c(s)≡1. Therefore, for δ∈(0,α+1), sδl(s) is increasing in [n1,∞) for some n1>0 large enough. Consequently, for some positive constants c0,c2,andc3 (which may depend on n1) and all N≥n1,
(114)∑n=1N‍nαl(n)=∑n=1N‍nα-δnδl(n)≤c0+∑n=n1N‍nα-δnδl(n)≤c0+Nα+1l(N)∑n=1N‍(nN)α-δ1N≤c0+Nα+1l(N)c3(aslimN→∞∑n=1N‍(nN)α-δ1N=∫01‍xα-δdx<∞)≤c2Nα+1l(N).
Similarly, for N≥2n1,
(115)∑n=1N‍nαl(n)≥∑N/2≤n≤N‍nα-δnδl(n)≥(N2)δl(N2)∑N/2<n≤N‍nα-δ.
Since l is slowly varying at ∞, by Potter's Theorem, for A=2 and δ1=1, there exists n2 such that for all N≥n2, l(N/2)≥2-2l(N). If δ≤α, then ∑N/2<n≤N‍nα-δ≥(N/2)α-δ(N-[N/2])≥c4Nα+1-δ for some positive constant c4; if δ>α, then ∑N/2<n≤N‍nα-δ≥Nα-δ(N-[N/2])≥c5Nα+1-δ for some positive constant c5. So, for some constants c1>0, n0=max{2n1,n2}, and all N≥n0,
(116)∑n=1N‍nαl(n)≥c1Nα+1l(N).Theorem 24.
Letα,b>0 and ϕ(n)=nb-1l(n). Then, the following assertions are equivalent:
(117)∑n=1∞‍ϕ(n)ℙ{Yn*>εnα}<∞∀ε>0;(118)∑n=1∞‍ϕ(n)ℙ{supj≥nj-αYj*>ε}<∞∀ε>0;(119)∑n=1∞‍ϕ(n)ℙ{M(ε)>εnα}<∞∀ε>0;(120)𝔼(M(ε)b/αl(M(ε)1/α))<∞∀ε>0.Proof.
We proceed as in [34, page 394] where similar results were established for l(n)=1 and real-valued random variables.
(a) We first prove that (117) is equivalent to
(121)∫1∞‍ϕ(u)ℙ{Yu*>12εuα}du<∞∀ε>0.
Let n0 be large enough such that u-1l(u) is decreasing in [n0,∞). Then, we have
(122)∫n-1n‍ϕ(u)ℙ{Yu*>12εuα}du=∫n-1n‍ub(u-1l(u))ℙ{Yu*>12εuα}du≤nb((n-1)-1l(n-1))ℙ{Yn*>12ε(n-1)α}≤2ϕ(n)ℙ{Yn*>14εnα}
for n≥n1≥n0+1 large enough. Here we have used the fact that limn→∞(l(n-1)/l(n))=1. (To see this, notice that l(n-1)/(n-1)≥l(n)/n for n≥n0+1, so that l(n-1)/l(n)≥(n-1)/n; in the same way, using the fact that ul(u) is increasing in [n0′,∞) for n0′ large enough, we obtain l(n-1)/l(n)≤n/(n-1) for all n≥n0′+1.) Similarly,
(123)∫n-1n‍ϕ(u)ℙ{Yu*>12εuα}du=∫n-1n‍ub(u-1l(u))ℙ{Yu*>12εuα}du≥(n-1)b(n-1l(n))ℙ{Yn-1*>12εnα}≥12ϕ(n-1)ℙ{Yn-1*>ε(n-1)α}
for n≥n2≥n0+1 large enough. Let N=max{n1,n2}. As ∫N-1∞‍=∑n=N∞‍∫n-1n‍, the conclusion that (117) is equivalent to (121) follows from (122) and (123).
(b) We next remark that (119) is equivalent to
(124)∫1∞‍ub-1l(u)ℙ{M(ε)>εuα}du<∞∀ε>0.
This can be seen by the same argument as in (a).
(c) We now prove that (121) implies (124). Set β=b/α. We have
(125)∫1∞‍ub-1l(u)ℙ{M(ε)>εuα}du=1b∫1∞‍l(υ1/(αβ))ℙ{supj≥1(∥Yj∥-εjα)+>ευ1/β}dυ≤1b∫1∞‍l(υ1/(αβ))×∑k=1∞‍ℙ{max(2k-1-1)υ1/β<jα≤2kυ1/β(∥Yj∥-εjα)+>ευ1/β}dυ≤1b∑k=1∞‍∫1∞‍l(υ1/(αβ))ℙ{Y(2kυ1/β)1/α*>2k-1ευ1/β}dυ=∑k=1∞‍2-kβ∫1∞‍ub-1l(2-k/αu)ℙ{Yu*>12εuα}du.
By Potter's Theorem, for 0<δ<b, there exists A=A(δ)>1 such that
(126)l(2-k/αu)≤A2kδ/αl(u)∀k≥1.
So,
(127)∫1∞‍ub-1l(u)ℙ{M(ε)>εuα}du≤A∑k=1∞‍2-k(β-δ/α)∫1∞‍ub-1l(u)ℙ{Yu*>12εuα}du=A2(b-δ)/α-1∫1∞‍ϕ(u)ℙ{Yu*>12εuα}du.
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
(128)∑n=1∞‍ϕ(n)ℙ{M(ε)>εnα}=𝔼(∑n=1∞‍ϕ(n)1{M(ε)>εnα})=𝔼(∑1≤n<(M(ε)/ε)1/α‍l(n)nb-1).
By Lemma 23, there exist n0,c1>0 such that
(129)𝔼((∑1≤n<(M(ε)/ε)1/α‍l(n)nb-1)1{(M(ε)/ε)1/α≥n0})≥c1𝔼((M(ε)ε)b/αl((M(ε)ε)1/α)1{(M(ε)/ε)1/α≥n0}).
By (128) and (129), we see that (119) implies
(130)𝔼((M(ε)ε)b/αl((M(ε)ε)1/α))<∞∀ε>0.
By Potter's Theorem, there exists x0>0 such that when (M(ε)/ε)1/α≥x0 and (M(ε))1/α≥x0,
(131)l(M(ε)1/α)2max{ε-1/α,ε1/α}≤l((M(ε)ε)1/α)≤2max{ε-1/α,ε1/α}l(M(ε)1/α).
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 Section4, using the results of Section 5.Let(𝔹,∥·∥) a separable Banach space and {(Xj,ℱj)}j≥1 a sequence of 𝔹-valued martingale differences. We denote
(132)S0=0,Sn=∑j=1n‍Xj,Sn*=max0≤j≤n∥Sj∥∀n∈ℕ.
Notice that ∥S0∥=0,S0*=0 by our notations. Set
(133)X0=0,Xn*=max0≤j≤n∥Xj∥∀n∈ℕ.
Notice that ∥Sn∥/nα→0 a.s. if and only if ℙ(supj≥nj-α∥Sj∥>ε)→0 for any ε>0. So, the following theorems describe the a.s. convergence of ∥Sn∥/nα.Theorem 25.
Letα,b>0, l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nbl(n). Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(134)ϕ(n)∥mn(γ)∥𝕃𝔹qλ=O(1),
where mn(γ)=n-γα∑j=1n‍𝔼[∥Xj∥γ∣ℱj-1]. If 𝔹 is γ-smooth, then one has the following implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138):
(135)ϕ(n)∑j=1n‍ℙ{∥Xj∥>εnα}=o(1)1111(resp.,O(1))foranyε>0;(136)ϕ(n)P{Xn*>εnα}=o(1)(resp.,O(1))foranyε>0;(137)ϕ(n)P{Sn*>εnα}=o(1)(resp.,O(1))foranyε>0;(138)ϕ(n)P{∥Sn∥>εnα}=o(1)(resp.,O(1))foranyε>0;(139)ϕ(n)P{supj≥nj-α∥Sj∥>ε}=o(1)1111(resp.,O(1))foranyε>0;(140)ϕ(n)P{supj≥nj-αSj*>ε}=o(1)11(resp.,O(1))foranyε>0.Notice that, compared with Theorem14, Theorem 25 contains the additional conditions (139) and (140). When ϕ(n)=nb and for i.i.d. real-valued random variables, the implications (135)⇒(139)⇒(138) with o(1) of Theorem 25 contain Theorem 4 of Baum and Katz [6].Remark 26.
As in Theorem25, the conclusions of Theorem 25 remain valid if mn(γ) is replaced by
(141)mn(γ,M)=n-γα∑j=1n‍𝔼[∥Xj∥γ∣ℱj-M]
for some M∈ℕ*.
In fact, the caseM≥2 can be reduced to the case M=1 by considering the subsequences {(XlM+i,ℱlM+i)}l≥0 (1≤i≤M) of {(Xj,ℱj)}j≥1, which are still sequences of 𝔹-valued martingale differences.Corollary 27.
Let1/2<α≤1 and b>0. Let l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nbl(n). Suppose that for some γ∈(1/α,2], q∈[1,∞] with q>b/(γα-1),
(142)supn≥1∥m_n(γ)∥𝕃𝔹q<∞,wherem_n(γ)=1n∑j=1n‍𝔼[∥Xj∥γ∣ℱj-1].
If 𝔹 is γ-smooth, then one has the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).Proof of Theorem25.
Applying Theorem14 to Xnj=Xj,ℱnj=ℱj(1≤j≤n), and ϕ(n)=nbl(n), we get the implications (135)⇔(136)⇔(137)⇒(138). Applying Theorem 22 to Yn=∥Sn∥, 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 Corollary27.
Sinceγ>1/α, we have
(143)∥mn(γ)∥𝕃𝔹q=n1-γα∥m_n(γ)∥𝕃𝔹q.
As q>b/(γα-1), we can choose λ∈(b/(γα-1),q). For this λ,
(144)ϕ(n)∥mn(γ)∥𝕃𝔹qλ=nbl(n)(n1-γα∥m_n(γ)∥𝕃𝔹q)λ=nb-λ(γα-1)l(n)∥m_n(γ)∥𝕃𝔹qλ=o(1).
Thus, the condition (134) holds, so that the conclusion follows from Theorem 25.Theorem 28.
Letα,b>0. Let l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nb-1l(n). For any ε>0, set
(145)M(S,ε)=M(S,ε,α)=supn≥0(∥Sn∥-εnα)=supn≥0(Sn*-εnα),(146)M(X,ε)=M(X,ε,α)=supn≥0(∥Xn∥-εnα)=supn≥0(Xn*-εnα).
Suppose that for some γ∈(1,2],q∈[1,∞] and λ∈(0,q),
(147)∥mn(γ)∥𝕃𝔹q⟶0,∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ<∞,
where mn(γ)=n-γα∑j=1n‍𝔼[∥Xj∥γ∣ℱj-1]. If 𝔹 is γ-smooth, then one has the following implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152):(A)(148)∑n=1∞‍ϕ(n)ℙ{Sn*>εnα}<∞∀ε>0;(149)∑n=1∞‍ϕ(n)ℙ{supj≥nj-αSj*>ε}<∞∀ε>0;(150)𝔼(M(S,ε)b/αl(M(S,ε)1/α))<∞∀ε>0;(151)∑n=1∞‍ϕ(n)ℙ{supj≥nj-α∥Sj∥>ε}<∞∀ε>0;(152)∑n=1∞‍ϕ(n)ℙ{∥Sn∥>εnα}<∞∀ε>0;(B)(153)∑n=1∞‍ϕ(n)ℙ{Xn*>εnα}<∞∀ε>0;(154)∑n=1∞‍ϕ(n)ℙ{supj≥nj-αXj*>ε}<∞∀ε>0;(155)𝔼(M(X,ε)b/αl(M(X,ε)1/α))<∞∀ε>0;(156)∑n=1∞‍ϕ(n)∑j=1n‍ℙ{∥Xj∥>εnα}<∞∀ε>0.Compared with Theorem17, in Theorem 28 we have the additional conditions (149), (150), (151), (154), and (155).Remark 29.
As in Theorem25, the conclusions of Theorem 28 remain valid if mn(γ) is replaced by
(157)mn(γ,M)=n-γα∑j=1n‍𝔼[∥Xj∥γ∣ℱj-M]
for some M∈ℕ*.Corollary 30.
Letα>1/2,b>0. Let l(·)>0 be a function slowly varying at ∞ and ϕ(n)=nb-1l(n). Suppose that for some γ∈(max{1,1/α},2], q∈[1,∞] with q>b/(γα-1), (142) holds. If 𝔹 is γ-smooth, then one has the implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152).IfXj's are identically distributed real-valued random variables, then (156) is equivalent to the moment condition 𝔼|X1|(b+1)/αl(|X1|1/α)<∞. So, Corollary 30 contains Theorems 1, 2, and 3 of Baum and Katz [6] when ϕ(n)=nb-1 and for i.i.d. real-valued random variables. When ϕ(n)=nb-1, α≤1 and for real-valued martingale differences, Corollary 30 was proved by Alsmeyer [18, Theorems 1 and 2].To see that Theorem28 implies Corollary 30, it suffices to notice that (142) (with q>b/(γα-1)) implies (147). In fact, when (142) holds for some q∈[1,∞] with q>b/(γα-1), then
(158)∥mn(γ)∥𝕃𝔹q=n1-γα∥m_n(γ)∥𝕃𝔹q⟶0,
and, for λ∈(b/(γα-1),q),
(159)∑n=1∞‍ϕ(n)∥mn(γ)∥𝕃𝔹qλ=∑n=1∞‍nb-1-λ(γα-1)l(n)∥m_n(γ)∥𝕃𝔹qλ<∞.
Therefore, (147) holds for some q∈[1,∞] and λ∈(0,q).Proof of Theorem28.
Applying Theorem10 to
(160)Xnj={n-αXjif1≤j≤n,0ifj>n,ℱnj={ℱjif1≤j≤n,ℱnifj>n,
we obtain the implications (153)⇔(156)⇔(148)⇒(152). By Theorem 24 applied to Yn=∥Sn∥, we see that (148), (149), and (150) are equivalent. Since n-α∥Sn∥≤supj≥nj-α∥Sj∥≤supj≥nj-αSj*, (149) implies (151) and (151) implies (152). Thus, we have the implications (148)⇔(149)⇔(150)⇒(151)⇒(152). Again by Theorem 24 applied to Yn=∥Xn∥, 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∑j=1n‍ja-1Xj

Let(𝔹,∥·∥) be a separable Banach space. In this section, we give a Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums
(161)Tn=∑j=1n‍ja-1Xj(a∈ℝ)
of 𝔹-valued martingale differences {(Xj,ℱj)}j≥1, and we obtain a Baum-Katz type theorem for weighted sums of identically distributed 𝔹-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(Yj)j≥1 be a sequence of any 𝔹-valued random variables. If there exist n0,K>0, such that for some q∈[1,∞] and all n≥n0,
(162)∥1n∑j=1n‍Yj∥𝕃𝔹q≤K,
then
(163)∥∑j=1n‍jpYj∥𝕃𝔹q=O(np+1)foranyp>-1.Proof.
LetS0=0, Sn=∑j=1n‍Yj, n≥1. Then,
(164)∑j=1n‍jpYj=∑j=1n‍jp(Sj-Sj-1)=∑j=1n-1‍[jp-(j+1)p]Sj+npSn=∑j=1n-1‍jp[1-(1+1j)p]Sj+npSn.
As 1-(1+1/j)p~-p/j(j→∞), it follows that
(165)∥∑j=1n‍jpYj∥𝕃𝔹q≤∑j=1n-1‍jp|1-(1+1j)p|∥Sj∥𝕃𝔹q+np∥Sn∥𝕃𝔹q=O(np+1).The following theorem is a Marcinkiewicz-Zygmund type strong law of large numbers for the weighted sums (161).Theorem 32.
Let{(Xj,ℱj)}j≥1 be 𝔹-valued martingale differences. Suppose that for some γ∈(1,2], there exist n0,K>0, such that for all n≥n0,
(166)1n∑j=1n‍𝔼∥Xj∥γ≤K.
If 𝔹 is γ-smooth, then for α>-(1-1/γ) and a>1-1/γ,
(167)Tnna+α⟶0,a.s.,(168)ℙ{supm≥n∥Tm∥ma+α>ε}=O(n-(γ-1+γα))foranyε>0.Notice that whena=1 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 ℙ{supm≥n(∥Tm∥/ma+α)>ε}→0 for any ε>0. So, (168) describes the convergence rates in the Marcinkiewicz-Zygmund strong laws of large numbers (167).Proof of Theorem32.
Clearly,(169)ℙ{supm≥n∥Tm∥ma+α>ε}≤ℙ{supm≥n(m-(a+α)max1≤j≤m∥Tj∥)>ε}.
By Theorem 22, we see that
(170)ℙ{supm≥n(m-(a+α)max1≤j≤m∥Tj∥)>ε}=O(n-(γ-1+γα))∀ε>0
if and only if
(171)ℙ{max1≤j≤n∥Tj∥>εna+α}=O(n-(γ-1+γα))∀ε>0.
Write m(γ,n)=n-γ(a+α)∑j=1n‍𝔼[∥ja-1Xj∥γ∣ℱj-1]. By the proof of Lemma 31, we have
(172)𝔼m(γ,n)=n-γ(a+α)∑j=1n‍𝔼∥ja-1Xj∥γ=n-γ(a+α)∑j=1n‍jγ(a-1)𝔼∥Xj∥γ=O(n-(γ-1+γα)).
And by Theorem 2 with q=1 and L=0, we know that
(173)ℙ{n-(a+α)max1≤j≤n∥Tj∥>ε}≤ℙ{n-(a+α)max1≤j≤n∥ja-1Xj∥>14ε}+ε-γC2(γ)𝔼m(γ,n)=ℙ{max1≤j≤n∥ja-1Xj∥>14εna+α}+O(n-(γ-1+γα)).
By Theorem 1, we see that
(174)ℙ{n-(a+α)max1≤j≤n∥ja-1Xj∥>14ε}≤∑j=1n‍ℙ{n-(a+α)∥ja-1Xj∥>14ε}.
By (166), we have
(175)∑j=1n‍ℙ{∥ja-1Xj∥>14εna+α}≤4γε-γn-γ(a+α)∑j=1n‍jγ(a-1)𝔼∥Xj∥γ=O(n-(γ-1+γα)).
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 functionRρ regularly varying at ∞ of index ρ≠0, one define, Rρ← as its inverse function.Notice that whenρ>0, Rρ(x) is strictly increasing for x large enough with limx→∞Rρ(x)=+∞, so that Rρ← is well defined on [u0,∞) for u0>0 large enough. For simplicity, we always make the convention that Rρ←(u)=Rρ←(u0) if u∈[0,u0), so that Rρ← is well defined on [0,∞). We make a similar convention in the case where ρ<0.The following lemma shows that the inverse function of a regularly varying function of indexρ≠0 remains regularly varying.Lemma 34.
IfRρ(x)=xρl(x) is regularly varying at ∞ of index ρ≠0, where l(x) is of the canonical form l(x)=exp{∫x0x‍((ε(u))/u)du}(x≥x0)forsomex0>0, then its inverse function Rρ←(x)=x1/ρl*(x) is regularly varying at ∞ of index 1/ρ, where l*(x)=(l(Rρ←(x)))-1/ρ is slowly varying at ∞.Proof.
Lety=Rρ(x). Define l2(y):=l(Rρ←(y)). We have
(176)Rρ←(y)=y1/ρ(l2(y))-1/ρ=y1/ρl*(y).
We will prove that l2(y) is slowly varying at ∞. We see that
(177)l2(y)=exp{∫x0Rρ←(y)‍ε(u)udu}.
After changing variable, we have
(178)∫x0Rρ←(y)‍ε(u)udu=∫l(1)y‍ε(Rρ←(v))Rρ←(v)(Rρ←(v))′dv=∫l(1)y‍ε(Rρ←(v))Rρ←(v)Rρ←(v)v(ρ+ε(Rρ←(v)))dv=∫l(1)y‍ε(Rρ←(v))v(ρ+ε(Rρ←(v)))dv=∫l(1)y‍ε1(v)vdv,
where ε1(v):=ε(Rρ←(v))/(ρ+ε(Rρ←(v)))→0. Thus, l2(y)=exp{∫l(1)y‍(ε1(v)/v)dv} is slowly varying at ∞ and so is l*(y), which proves the desired result.In the following Baum-Katz type theorem,l(·) and l1(·) are functions slowly varying at ∞. Without loss of generality, we suppose that l and l1 have the form (101) with c(x)≡1. For α>-1 and a>-α, define
(179)R1+α(x)=x1+αl1(x),Ra+α(x)=xa+αl1(x),
and let R1+α← and Ra+α← be, respectively, the inverse functions of R1+α and Ra+α (cf. Definition 33), which are also regularly varying by Lemma 34. For a≠1, choose U so large that
(180)l0(x):=l(x)(l1(x))1/(a-1)
is locally bounded in [U,∞), and set
(181)l~(u)=0ifu∈[0,U),l~(u)=∫Uu‍x-1l0(x)dxifu∈[U,∞].
Notice that l~(∞) may be finite or infinite.Theorem 35.
Let{(Xj,ℱj)}j≥1 be identically distributed 𝔹-valued martingale differences and b≥0. Suppose that for some γ∈(1,2] and q∈[1,∞] with q>b/(γ(α+1)-1), γ(α+1)-1>0 and γ(a-1)+1>0,
(182)11111111111111supn≥1∥m_n(γ)∥𝕃𝔹q<∞,wherem_n(γ)=1n∑j=1n‍𝔼[∥Xj∥γ∣ℱj-1].
If 𝔹 is γ-smooth, then the following assertions hold.
(a) Whena≠(b-α)/(b+1),
(183)∑n=1∞‍nb-1l(n)ℙ{max1≤j≤n∥Tj∥>εna+αl1(n)}<∞1111111111111111111111111111111∀ε>0
if and only if
(184)𝔼(R1+α←(∥X1∥))b+1l(R1+α←(∥X1∥))<∞ifa>b-αb+1,𝔼(Ra+α←(∥X1∥))bl(Ra+α←(∥X1∥))<∞ifa<b-αb+1.
(b) Whena=(b-α)/(b+1), (183) is implied by
(185)𝔼(Ra+α←(∥X1∥))b(l1(Ra+α←(∥X1∥)))1/(1-a)l~(Ra+α←(∥X1∥))<∞,
where l~ is defined in (181); conversely, if l~(∞)=∞ and the function l0 defined by (180) satisfies limu→∞(l0(R1+α←(u))/l0(Ra+α←(u)))<(1+α)/(a+α), then (183) implies (185).Remark 36.
Theorem35 also holds if (182) is replaced by
(186)∥mn(γ)∥𝕃𝔹q⟶0,∑n=1∞‍nb-1l(n)∥mn(γ)∥𝕃𝔹qλ<∞
for some γ∈(1,2],q∈[1,∞] and λ∈(0,q), where
(187)mn(γ)=n-γ(a+α)(l1(n))-γ∑j=1n‍j-γ(1-a)𝔼[∥Xj∥γ∣ℱj-1].Of particular interest are the cases where the slowly varying functionsl and l1 are constants or powers of the logarithmic function, which will be studied in the following corollaries. We first consider the case where l and l1 are constants.Corollary 37.
Let{(Xj,ℱj)}j≥1 be identically distributed 𝔹-valued martingale differences and b≥0. Suppose that (182) holds for some γ∈(1,2] and q∈[1,∞] with q>b/(γ(α+1)-1), γ(α+1)-1>0, and γ(a-1)+1>0. If 𝔹 is γ-smooth, then
(188)∑n=1∞‍nb-1ℙ{max1≤j≤n∥Tj∥>εna+α}<∞∀ε>0
if and only if
(189)𝔼∥X1∥(b+1)/(1+α)<∞ifa>b-αb+1,𝔼∥X1∥1/(1-a)log+∥X1∥<∞ifa=b-αb+1,𝔼∥X1∥b/(a+α)<∞ifa<b-αb+1.Notice that the condition onm_n(γ) implies in particular 𝔼m_n(γ)<∞, giving 𝔼∥X1∥γ<∞. Therefore, the conclusion of the corollary is interesting only when the exponents in (189) are greater than γ.When(Xj)j≥1 are i.i.d. real-valued random variables and a∈(0,1), we get the sufficiencies of Theorems 2 and 3(a)(i) and (iii) of Lanzinger and Stadtmüller [21] by Corollary 37.We then consider the case wherel(x)=(log+x)p(p∈ℝ) and l1(x)=(log+x)β(β∈ℝ).Corollary 38.
Let{(Xj,ℱj)}j≥1 be identically distributed 𝔹-valued martingale differences and b≥0. Suppose that (182) holds for some γ∈(1,2] and q∈[1,∞] with q>b/(γ(α+1)-1), γ(α+1)-1>0, and γ(a-1)+1>0. If 𝔹 is γ-smooth, then for β,p∈ℝ,
(190)∑n=2∞‍nb-1(logn)pℙ{max1≤j≤n∥Tj∥>εna+α(logn)β}<∞11111111111111111111111111111111111111∀ε>0
if and only if
(191)𝔼∥X1∥(b+1)/(1+α)(log+∥X1∥)-β(b+1)/(1+α)+p<∞1111111111111111111111111111ifa>b-αb+1,𝔼∥X1∥1/(1-a)(log+∥X1∥)-β/(1-a)+p+1<∞111111111111111111111111ifa=b-αb+1,𝔼∥X1∥b/(a+α)(log+∥X1∥)-βb/(a+α)+p<∞11111111111111111111111ifa<b-αb+1.In the case where(Xj)j≥1 are i.i.d. real-valued random variables and the maximum max1≤j≤n∥Tj∥ is replaced by the ∥Tn∥, by Corollary 38, if a∈(0,1), β=0, and p=-1, we get the sufficiency of Theorem 3(a)(ii) of Lanzinger and Stadtmüller [21]; if a=0, β=1, and p<0, p≠-1, we get the sufficiencies of Theorem 3(b) of Lanzinger and Stadtmüller in [21].Proof of Theorem35.
Notice thatl~(u) is slowly varying at ∞ by Proposition  1.5.9a in [39, page 26]. Set
(192)Xnj={n-a-α(l1(n))-1ja-1Xjif1≤j≤n,0ifj>n,ℱnj={ℱjif1≤j≤n,ℱnifj>n,
and ϕ(n)=nb-1l(n) in Theorem 10, then {(Xnj,ℱnj),j≥1,n≥1} are sequences of 𝔹-valued martingale differences. For any n≥1, we have
(193)mn(γ)=∑j=1n‍𝔼[∥n-a-α(l1(n))-1ja-1Xj∥γ∣ℱj-1]=n-γ(a+α)(l1(n))-γ∑j=1n‍jγ(a-1)𝔼[∥Xj∥γ∣ℱj-1].
By (193) and Lemma 31, we have
(194)∥mn(γ)∥𝕃𝔹q=n-γ(a+α)(l1(n))-γ∥∑j=1n‍jγ(a-1)𝔼[∥Xj∥γ∣ℱj-1]∥𝕃𝔹q=n-γ(a+α)(l1(n))-γO(nγ(a-1)+1)=(l1(n))-γO(n1-γ(α+1)).
Thus, ∥mn(γ)∥𝕃𝔹q→0 and
(195)∑n=1∞‍nb-1l(n)∥mn(γ)∥𝕃𝔹qλ=∑n=1∞‍l(n)(l1(n))-λγO(nb-1-λ(γ(α+1)-1))<∞
whenever λ∈(b/(γ(α+1)-1),q). By Theorem 10, (183) is equivalent to
(196)∑n=1∞‍nb-1l(n)∑j=1n‍ℙ{∥Xj∥>εj-a+1na+αl1(n)}<∞1111111111111111111111111111111111∀ε>0.
Notice that in view of the identically distributed assumption, (196) holds if and only if
(197)∑n=1∞‍nb-1l(n)∑j=1n‍ℙ{∥X1∥>εj-a+1na+αl1(n)}<∞1111111111111111111111111111111111∀ε>0.
By the monotonicity of the functions Rb-1(x):=xb-1l(x) and Ra+α(x)=xa+αl1(x) for x>0 large enough and the fact that Rb-1(n-1)~Rb-1(n) and Ra+α(n-1)~Ra+α(n) as n→∞, it can be easily verified that the condition (197) is equivalent to
(198)∫1∞‍xb-1l(x)(∫1x‍ℙ{∥X1∥>εy-a+1xa+αl1(x)}dy)dx<∞1111111111111111111111111111111111111111∀ε>0.
Therefore, Theorem 35 is a direct consequence of the following lemma.Lemma 39.
Letb≥0, α>-1, a>-α, and l(·) and l1(·) are slowly varying at ∞. Let X1 be any 𝔹-valued random variable. Then, the following assertions hold.(a)
Whena≠(b-α)/(b+1), (198) holds if and only if (184) holds.(b)
Whena=(b-α)/(b+1), (198) is implied by (185), where l~ is defined in (181).Proof.
We proceed as in the proof of Lemma 3.4 of Gut [20]. We distinguish three cases according to a>1, a=1 or a<1. By choosing a smooth version, we can suppose that l1 is differentiable (cf. [39]).
Case  1 (a>1). In (198), we use the change of variables
(199)u=y-a+1xa+αl1(x),x=x.
Notice that 1≤y≤x if and only if l1(1)≤u<∞ and Ra+α←(u)≤x≤R1+α←(u). Therefore, (198) holds if and only if
(200)∫l1(1)∞‍u-1/(a-1)-1ℙ{∥X1∥>εu}×(∫Ra+α←(u)R1+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx)du<∞∀ε>0.
By Proposition 1.5.8 of [39, page 26], we have as u→∞,
(201)∫UR1+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(R1+α←(u))b+((1+α)/(a-1))+1l(R1+α←(u))×(l1(R1+α←(u)))1/(a-1))×(b+1+αa-1+1)-1,∫URa+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(Ra+α←(u))b+(1+α)/(a-1)+1l(Ra+α←(u))×(l1(Ra+α←(u)))1/(a-1))×(b+1+αa-1+1)-1,
where and hereafter a(u)~b(u) means that a(u)/b(u)→1 as u→∞. By Lemma 34, the right hand sides of (201) are regularly varying functions of index 1/(1+α), and 1/(a+α) respectively, so that their ratio tends to ∞ as u→∞, since 1/(1+α)>1/(a+α). Therefore, as u→∞,
(202)∫Ra+α←(u)R1+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(R1+α←(u))b+(1+α)/(a-1)+1l(R1+α←(u))×(l1(R1+α←(u)))1/(a-1))×(b+1+αa-1+1)-1.
Thus, (200) is equivalent to
(203)∫l1(1)∞‍u-1/(a-1)-1[R1+α←(u)]b+(1+α)/(a-1)+1l(R1+α←(u))×[l1(R1+α←(u))]1/(a-1)ℙ{∥X1∥>εu}du<∞∀ε>0.
With the change of variable v=R1+α←(u) and the fact that R1+α′(v)~R1+α′(v)/(1+α)v (together with (179)), we see that (203) holds if and only if
(204)∫R1+α←(l1(1))∞‍vbl(v)ℙ{R1+α←(∥X1∥ε)>v}dv<∞11111111111111111111111111111111∀ε>0,
which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).
Case  2 (a=1). By the change of variables
(205)u=x1+αl1(x),y=y,
we see that (198) holds if and only if
(206)∫l1(1)∞‍(R1+α←(u)-1)(R1+α←(u))b-1l(R1+α←(u))×(R′(R1+α←(u)))-1ℙ{∥X1∥>εu}du<∞∀ε>0.
With the change of variable v=R1+α←(u), we know that (206) holds if and only if
(207)∫R1+α←(l1(1))∞‍(v-1)vb-1l(v)ℙ{R1+α←(∥X1∥ε)>v}dv<∞111111111111111111111111111111111111111∀ε>0,
which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).
Case  3 (-α<a<1). By the change of variables
(208)u=y-a+1xa+αl1(x),x=x,
we see that (198) holds if and only if
(209)∫l1(1)∞‍u-1/(a-1)-1ℙ{∥X1∥>εu}×(∫R1+α←(u)Ra+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx)du<∞∀ε>0.
We distinguish three cases according toa<(b-α)/(b+1), a>(b-α)/(b+1), or a=(b-α)/(b+1).
(i) Suppose thata<(b-α)/(b+1). By Proposition  1.5.8 of [39, page 26], we have as u→∞,
(210)∫URa+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(Ra+α←(u))b+(1+α)/(a-1)+1l(Ra+α←(u))×(l1(Ra+α←(u)))1/(a-1))×(b+1+αa-1+1)-1,∫UR1+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(R1+α←(u))b+(1+α)/(a-1)+1l(R1+α←(u))×(l1(R1+α←(u)))1/(a-1))×(b+1+αa-1+1)-1
so that (as in Case 1)
(211)∫R1+α←(u)Ra+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(Ra+α←(u))b+(1+α)/(a-1)+1l(Ra+α←(u))×(l1(Ra+α←(u)))1/(a-1))×(b+1+αa-1+1)-1.
Thus, (209) is equivalent to
(212)∫l1(1)∞‍u-1/(a-1)-1(Ra+α←(u))b+(a+α)/(a-1)+1l(Ra+α←(u))×(l1(Ra+α←(u)))1/(a-1)ℙ{∥X1∥>εu}du<∞∀ε>0.
With the change of variable v=Ra+α←(u), we see that (212) holds if and only if
(213)∫Ra+α←(l1(1))∞‍vb-1l(v)ℙ{Ra+α←(∥X1∥ε)>v}dv<∞∀ε>0,
which is equivalent to the second condition of (184). Thus, (198) is equivalent to the second condition of (184).
(ii) Suppose thata>(b-α)/(b+1). By Proposition  1.5.10 of [39, page 27], we have as u→∞,
(214)∫R1+α←(u)∞‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(R1+α←(u))b+(1+α)/(a-1)+1l(R1+α←(u))×(l1(R1+α←(u)))1/(a-1))×(-(b+1+αa-1+1))-1,∫Ra+α←(u)∞‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(Ra+α←(u))b+(a+α)/(a-1)+1l(Ra+α←(u))×(l1(Ra+α←(u)))1/(a-1))×(-(b+1+αa-1+1))-1
so that as u→∞,
(215)∫R1+α←(u)Ra+α←(u)‍xb+(1+α)/(a-1)l(x)(l1(x))1/(a-1)dx~((l1(R1+α←(u)))1/(a-1)(R1+α←(u))b+(1+α)/(a-1)+1l(R1+α←(u))×(l1(R1+α←(u)))1/(a-1))×(-(b+1+αa-1+1))-1.
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 thata=(b-α)/(b+1). In this case, (209) reduces to
(216)∫l1(1)∞‍u-1/(a-1)-1ℙ{∥X1∥>εu}×(∫R1+α←(u)Ra+α←(u)‍x-1l(x)(l1(x))1/(a-1)dx)du<∞∀ε>0.
By Proposition  1.5.8 of [39, page 26], we have
(217)∫R1+α←(u)Ra+α←(u)‍x-1l(x)(l1(x))1/(a-1)dx=l~(Ra+α←(u))-l~(R1+α←(u))<l~(Ra+α←(u)).
So, (209) is implied by
(218)∫l1(1)∞‍u-1/(a-1)-1l~(Ra+α←(u))ℙ{∥X1∥>εu}du<∞∀ε>0.
With the change of variable v=Ra+α←(u), we see that (218) holds if and only if
(219)∫Ra+α←(l1(1))∞‍vb-1(l1(v))-1/(a-1)l~(v)×ℙ{Ra+α←(∥X1∥ε)>v}dv<∞∀ε>0,
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 Corollary38.
We are in the case wherel1(x)=(logx)β (β≥0). If y=R1+α(x)=x1+α(logx)β, then
(220)logy~(1+α)logxasx⟶+∞.
Thus,
(221)x=y1/(1+α)(logx)-β/(1+α)~(1+α)β/(1+α)(y(logy)β)1/(1+α)1111111111111111111111111111111111111asy⟶+∞.
Therefore,
(222)R1+α←(y)~(1+α)β/(1+α)(y(logy)β)1/(1+α)asy⟶+∞.
Similarly,
(223)Ra+α←(y)~(a+α)β/(a+α)(y(logy)β)1/(a+α)asy⟶+∞.
Using (222), (223), and Theorem 35, we obtain the desired results.


## 8. Convergence of Weighted Sums of Banach Valued Martingale Differences of the Form∑j=1∞‍anjXnj

In this section, we consider more general weighted sums of Banach valued martingale differences than those considered in Section7.Let(Xj)j≥1 be a sequence of i.i.d. random variables with 𝔼Xj=0, and let {anj,j≥1,n≥1} be an array of real numbers. The study of the convergence of weighted sums ∑j=1∞‍anjXj as n→∞ 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 ∑j=1∞‍anjXj→0 in probability and a sufficient condition for ∑j=1∞‍anjXj→0 a.s. Baxter et al. [35] also showed a sufficient condition for ∑j=1∞‍anjXj→0 a.s. Li et al. [9] studied the complete convergence of weighted sums of independent random variables of the form ∑j=1∞‍anjXnj. Yu [23] and Ghosal and Chandra [19] considered the same problem for martingale differences (Xnj). 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(𝔹,∥·∥) be a separable Banach space. In this subsection, we find sufficient conditions for the convergence of weighted sums of 𝔹-valued martingale differences (Xnj).Let us recall the famous theorem of Pruitt [43, Theorem 1] which states that for a sequence of i.i.d. random variables Xj with 𝔼|X1|<∞ and 𝔼X1=0,
(224)∑j=1∞‍anjXj⟶0inprobability
if and only if maxj≥1|anj|→0asn→∞, where {anj,j≥1,n≥1} is a Toeplitz summation matrix; that is, limn→∞anj=0 for every j≥1, limn→∞∑j=1∞‍anj=1, and supn≥1∑j=1∞‍|anj|<∞.In the following, we consider the same problem for𝔹-valued martingale differences (Xnj).Theorem 40.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers satisfying supn≥1∑j=1∞‍|anj|<∞ and An=supj≥1|anj|→0asn→∞. Suppose that for some γ∈(1,2], there exists K>0, such that
(225)𝔼∥Xnj∥γ≤K.
If 𝔹 is γ-smooth, then
(226)∑j=1∞‍anjXnj⟶0inprobability.Proof.
Since(227)𝔼mn(γ)=∑j=1∞‍𝔼∥anjXnj∥γ≤∑j=1∞‍|anj|Anγ-1𝔼∥Xnj∥γ≤Anγ-1K∑j=1∞‍|anj|⟶0,
we see that (226) follows from Theorem 4.Let{anj,j≥1,n≥1} be a Toeplitz summation matrix; Theorem 2 of Pruitt [43] states that for a sequence (Xj)j≥1 of i.i.d. random variables with 𝔼|X1|<∞ and 𝔼X1=0, if maxj≥1|anj|=O(n-β), β>0, then 𝔼|X1|1+1/β<∞ implies that
(228)∑j=1∞‍anjXj⟶0a.s.In the following, we also consider the similar problem for arrays of𝔹-valued martingale differences (Xnj).Theorem 41.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers satisfying supn≥1∑j=1∞‍|anj|<∞ and maxj≥1|anj|=O(n-β), β>0. Suppose that, for some γ∈(1,2], there exists a constant K>0 such that 𝔼∥Xnj∥γ≤K for all j≥1,n≥1. If 𝔹 is γ-smooth, then
(229)ℙ{∥∑j=1∞‍anjXnj∥>ε}=O(n-β(γ-1))∀ε>0.
Consequently, if 𝔹 is γ-smooth and supn,j≥1𝔼∥Xnj∥γ<∞ for some γ>1+1/β, then
(230)∑j=1∞‍anjXnj⟶0a.s.Proof.
SetYnj=anjXnj,j≥1,n≥1, then {(Ynj,ℱnj),j≥1,n≥1} is an array of 𝔹-valued martingale differences and for any n≥1,
(231)𝔼mn(γ)=𝔼[∑j=1∞‍𝔼[∥Ynj∥γ∣ℱn,j-1]]=∑j=1∞‍𝔼∥Ynj∥γ=∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤(supn≥1∑j=1∞‍|anj|)(Cn-β)γ-1K≤C1(γ)n-β(γ-1).
By Corollary 3, we see that
(232)ℙ{∥∑j=1∞‍anjXnj∥>ε}≤ℙ{supj≥1∥∑i=1j‍anjXnj∥>ε}.
By Theorem 2 with q=1 and L=0, we know that
(233)ℙ{supj≥1∥∑i=1j‍anjXnj∥>ε}≤ℙ{supj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Theorem 1, we see that
(234)ℙ{supj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{∥anjXnj∥>ε4}.
Since (231), we know that
(235)∑j=1∞‍ℙ{∥anjXnj∥>ε4}≤4γε-γ∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤4γε-γC1(γ)n-β(γ-1).
Since (231)–(235), we have
(236)ℙ{∥∑j=1∞‍anjXnj∥>ε}≤4γε-γC1(γ)n-β(γ-1)+ε-γC2(γ)C1(γ)n-β(γ-1).
Thus, (229) holds.
If additionallyγ>1+1/β, then (229) implies that
(237)∑n=1∞‍ℙ{∥∑j=1∞‍anjXnj∥>ε}<∞∀ε>0.
Therefore, (230) holds.The following theorem extends Theorem 3.3 of Baxter et al. [35].Theorem 42.
Let{(Xj,ℱj)}j≥1 be 𝔹-valued martingale differences. Suppose that for some γ∈(1,2], there exists K>0, such that for all j≥1,
(238)𝔼∥Xj∥γ≤K.
Let {aj,j≥1} satisfy
(239)1n∑j=1n‍|aj|γ=O(nδ)forsomeδ∈ℝ,
and set Tn=∑j=1n‍ajXj. If 𝔹 is γ-smooth, then
(240)Tnnα⟶0a.s.forα>δγ,(241)ℙ{supm≥n∥Tm∥mα>ε}=O(n-(γα-δ)).When(Xj,j≥1) are i.i.d. real-valued random variables, α≤1, and δ=1, (240) reduces to Theorem 3.3 of Baxter et al. [35].Proof of Theorem42.
Clearly,(242)ℙ{supm≥n∥Tm∥mα>ε}≤ℙ{supm≥n(m-αmax1≤j≤m∥Tj∥)>ε}.
By Theorem 22, we see that
(243)ℙ{supm≥n(m-αmax1≤j≤m∥Tj∥)>ε}=O(n-(γα-δ))∀ε>0
if and only if
(244)ℙ{max1≤j≤n∥Tj∥>εnα}=O(n-(γα-δ))∀ε>0.
By (238) and (239), we have
(245)𝔼m(γ,n)=∑j=1n‍𝔼∥ajXj∥γ=∑j=1n‍|aj|γ𝔼∥Xj∥γ≤CKnδ.
And by Theorem 2 with q=1 and L=0, we know that
(246)ℙ{max1≤j≤n∥Tj∥>εnα}≤ℙ{max1≤j≤n∥ajXj∥>14εnα}+(εnα)-γC1(γ)𝔼m(γ,n)≤ℙ{max1≤j≤n∥ajXj∥>14εnα}+CKC1(γ)ε-γn-(γα-δ).
By Theorem 1, we see that
(247)ℙ{max1≤j≤n∥ajXj∥>14εnα}≤∑j=1n‍ℙ{∥ajXj∥>14εnα}.
By (238) and (239), we know that
(248)∑j=1n‍ℙ{∥ajXj∥>14εnα}≤4γε-γn-γα∑j=1n‍|aj|γ𝔼∥Xj∥γ≤4γCKε-γn-(γα-δ).
From (246)–(248), we see that (244) holds. Thus, (241) holds.


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

Let(𝔹,∥·∥) be a separable Banach space. In this subsection, we consider complete convergence of weighted sums of 𝔹-valued martingale differences (Xnj) of the form ∑j=1∞‍anjXnj. 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{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let α>0 and {anj,j≥1,n≥1} be an array of real numbers. If there exists γ∈(1,2] such that
(249)∑n=1∞‍∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ<∞
and 𝔹 is γ-smooth, then
(250)∑n=1∞‍ℙ{n-αsupj≥1∥∑i=1j‍aniXni∥>ε}<∞∀ε>0.In the square-integrable real-valued martingale differences case, the result was proved by Ghosal and Chandra in Corollary 1 [19] if, additionally, ∑j=1∞‍anj2=O(nδ) for some δ<2α.We generalize Theorem 2 of Yu [23] from two directions by Theorem 43: first, we extend sequences of Lp(p≥2) martingale differences to Lγ(γ∈(1,2]) sequences of 𝔹-valued martingale differences; secondly, we do not need the condition ∑j=1∞‍|anj|2=O(nδ) for some δ<α.Proof of Theorem43.
SetYnj=n-αanjXnj. From (249), we have
(251)∑n=1∞‍𝔼mn(γ)=∑n=1∞‍∑j=1∞‍𝔼∥Ynj∥γ=∑n=1∞‍∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ<∞.
By Corollary 3 and Theorem 2 with q=1 and L=0, we see that
(252)ℙ{n-αsupj≥1∥∑i=1j‍aniXni∥>ε}≤ℙ{n-αsupj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Corollary 3 and Theorem 1, we know that
(253)ℙ{n-αsupj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}.
By Markov's inequality, we see that
(254)∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}≤4γε-γ∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ.
Since (249) and (251)–(254), we see that (250) holds.Theorem 44.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers. Suppose that for some constants γ∈(1,2], K>0, α∈(0,1/2] and δ<γα,
(255)∑j=1∞‍|anj|γ≤Knδ,n≥1,(256)∑n=1∞‍n-1∑j=1∞‍|anj|1/α<∞,(257)supn,j≥1𝔼∥Xnj∥1/α<∞.
If 𝔹 is γ-smooth, then (250) holds.When(Xnj)j≥1, n≥1, is a sequence of zero mean independent real-valued random variables and γ=2, Theorem 44 reduces to Theorem  2.3 of Li et al. [9].Proof of Theorem44.
SetYnj=n-αanjXnj,j≥1,n≥1, then {(Ynj,ℱnj),j≥1,n≥1} is an array of 𝔹-valued martingale differences. Since (255) and (257), for any n≥1, we have
(258)∑n=1∞‍𝔼mn(γ)=∑n=1∞‍∑j=1∞‍𝔼∥n-αanjXnj∥γ=∑n=1∞‍n-γα∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤CK∑n=1∞‍n-(γα-δ)<∞.
By Corollary 3 and Theorem 2 with q=1 and L=0, we see that
(259)ℙ{n-αsupj≥1∥∑i=1j‍anjXnj∥>ε}≤ℙ{n-αsupj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Corollary 3 and Theorem 1, we know that
(260)ℙ{n-αsupj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}.
By Markov's inequality, we see that
(261)∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}≤ε-1/αn-1∑j=1∞‍|anj|1/α𝔼∥Xnj∥1/α<∞.
By (256), (257), and (258)–(261), we know that (250) holds.Theorem 45.
Let{(Xnj,ℱnj)}j=1n be a triangular array of identically distributed 𝔹-valued martingale differences. Let β>-1, and let {anj,1≤j≤n} be a triangular array of positive numbers satisfying
(262)anj=cnjn-β-1jβ
with 0<c≤cnj≤C<∞ for every 1≤j≤n, n≥1 and some constants c and C. Suppose that there exists some constant K, such that, for some γ∈(1,2],
(263)supn,j≥1𝔼[∥Xnj∥γ∣ℱn,j-1]≤K.
Suppose that
(264)𝔼∥X11∥1/(1+β)<∞if-1<β<-12,𝔼∥X11∥2log(1+∥X11∥)<∞ifβ=-12,𝔼∥X11∥2<∞ifβ>-12,
then for all ε>0,
(265)∑n=1∞‍ℙ{∥∑j=1n‍anjXnj∥>ε}<∞.When{Xnj}={Xj} are the same sequence of i.i.d. real-valued random variables and ∑j=1n‍anj=1 for all n≥1, Theorem 45 reduces to the sufficiency of Theorem 2.4 of Li et al. [9].Proof of Theorem45.
Set(266)Ynj={anjXnjif1≤j≤n,0ifj>n,
and cn=1, then {(Ynj,ℱnj)}j=1n are sequences of 𝔹-valued martingale differences. For any n≥1, since (262) and (263), we have
(267)mn(γ)=∑j=1n‍𝔼[∥anjXnj∥γ∣ℱn,j-1]=∑j=1n‍anjγ𝔼[∥Xnj∥γ∣ℱn,j-1]≤KCγn-(γ-1)(n-1∑j=1n‍(jn)γβ)~KCγn-(γ-1)∫1/n1‍xγβdx=O(n-b),
where b=min{γ(β+1),γ-1}. By Corollary 12 of Theorem 10, (265) is implied by
(268)∑n=1∞‍∑j=1n‍ℙ{∥anjXnj∥>ε}<∞∀ε>0.
Equation (268) holds if and only if (264) holds. (cf. [9, pages 62-63])


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

Let(𝔹,∥·∥) be a separable Banach space. In this subsection, we find sufficient conditions for the convergence of weighted sums of 𝔹-valued martingale differences (Xnj).Let us recall the famous theorem of Pruitt [43, Theorem 1] which states that for a sequence of i.i.d. random variables Xj with 𝔼|X1|<∞ and 𝔼X1=0,
(224)∑j=1∞‍anjXj⟶0inprobability
if and only if maxj≥1|anj|→0asn→∞, where {anj,j≥1,n≥1} is a Toeplitz summation matrix; that is, limn→∞anj=0 for every j≥1, limn→∞∑j=1∞‍anj=1, and supn≥1∑j=1∞‍|anj|<∞.In the following, we consider the same problem for𝔹-valued martingale differences (Xnj).Theorem 40.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers satisfying supn≥1∑j=1∞‍|anj|<∞ and An=supj≥1|anj|→0asn→∞. Suppose that for some γ∈(1,2], there exists K>0, such that
(225)𝔼∥Xnj∥γ≤K.
If 𝔹 is γ-smooth, then
(226)∑j=1∞‍anjXnj⟶0inprobability.Proof.
Since(227)𝔼mn(γ)=∑j=1∞‍𝔼∥anjXnj∥γ≤∑j=1∞‍|anj|Anγ-1𝔼∥Xnj∥γ≤Anγ-1K∑j=1∞‍|anj|⟶0,
we see that (226) follows from Theorem 4.Let{anj,j≥1,n≥1} be a Toeplitz summation matrix; Theorem 2 of Pruitt [43] states that for a sequence (Xj)j≥1 of i.i.d. random variables with 𝔼|X1|<∞ and 𝔼X1=0, if maxj≥1|anj|=O(n-β), β>0, then 𝔼|X1|1+1/β<∞ implies that
(228)∑j=1∞‍anjXj⟶0a.s.In the following, we also consider the similar problem for arrays of𝔹-valued martingale differences (Xnj).Theorem 41.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers satisfying supn≥1∑j=1∞‍|anj|<∞ and maxj≥1|anj|=O(n-β), β>0. Suppose that, for some γ∈(1,2], there exists a constant K>0 such that 𝔼∥Xnj∥γ≤K for all j≥1,n≥1. If 𝔹 is γ-smooth, then
(229)ℙ{∥∑j=1∞‍anjXnj∥>ε}=O(n-β(γ-1))∀ε>0.
Consequently, if 𝔹 is γ-smooth and supn,j≥1𝔼∥Xnj∥γ<∞ for some γ>1+1/β, then
(230)∑j=1∞‍anjXnj⟶0a.s.Proof.
SetYnj=anjXnj,j≥1,n≥1, then {(Ynj,ℱnj),j≥1,n≥1} is an array of 𝔹-valued martingale differences and for any n≥1,
(231)𝔼mn(γ)=𝔼[∑j=1∞‍𝔼[∥Ynj∥γ∣ℱn,j-1]]=∑j=1∞‍𝔼∥Ynj∥γ=∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤(supn≥1∑j=1∞‍|anj|)(Cn-β)γ-1K≤C1(γ)n-β(γ-1).
By Corollary 3, we see that
(232)ℙ{∥∑j=1∞‍anjXnj∥>ε}≤ℙ{supj≥1∥∑i=1j‍anjXnj∥>ε}.
By Theorem 2 with q=1 and L=0, we know that
(233)ℙ{supj≥1∥∑i=1j‍anjXnj∥>ε}≤ℙ{supj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Theorem 1, we see that
(234)ℙ{supj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{∥anjXnj∥>ε4}.
Since (231), we know that
(235)∑j=1∞‍ℙ{∥anjXnj∥>ε4}≤4γε-γ∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤4γε-γC1(γ)n-β(γ-1).
Since (231)–(235), we have
(236)ℙ{∥∑j=1∞‍anjXnj∥>ε}≤4γε-γC1(γ)n-β(γ-1)+ε-γC2(γ)C1(γ)n-β(γ-1).
Thus, (229) holds.
If additionallyγ>1+1/β, then (229) implies that
(237)∑n=1∞‍ℙ{∥∑j=1∞‍anjXnj∥>ε}<∞∀ε>0.
Therefore, (230) holds.The following theorem extends Theorem 3.3 of Baxter et al. [35].Theorem 42.
Let{(Xj,ℱj)}j≥1 be 𝔹-valued martingale differences. Suppose that for some γ∈(1,2], there exists K>0, such that for all j≥1,
(238)𝔼∥Xj∥γ≤K.
Let {aj,j≥1} satisfy
(239)1n∑j=1n‍|aj|γ=O(nδ)forsomeδ∈ℝ,
and set Tn=∑j=1n‍ajXj. If 𝔹 is γ-smooth, then
(240)Tnnα⟶0a.s.forα>δγ,(241)ℙ{supm≥n∥Tm∥mα>ε}=O(n-(γα-δ)).When(Xj,j≥1) are i.i.d. real-valued random variables, α≤1, and δ=1, (240) reduces to Theorem 3.3 of Baxter et al. [35].Proof of Theorem42.
Clearly,(242)ℙ{supm≥n∥Tm∥mα>ε}≤ℙ{supm≥n(m-αmax1≤j≤m∥Tj∥)>ε}.
By Theorem 22, we see that
(243)ℙ{supm≥n(m-αmax1≤j≤m∥Tj∥)>ε}=O(n-(γα-δ))∀ε>0
if and only if
(244)ℙ{max1≤j≤n∥Tj∥>εnα}=O(n-(γα-δ))∀ε>0.
By (238) and (239), we have
(245)𝔼m(γ,n)=∑j=1n‍𝔼∥ajXj∥γ=∑j=1n‍|aj|γ𝔼∥Xj∥γ≤CKnδ.
And by Theorem 2 with q=1 and L=0, we know that
(246)ℙ{max1≤j≤n∥Tj∥>εnα}≤ℙ{max1≤j≤n∥ajXj∥>14εnα}+(εnα)-γC1(γ)𝔼m(γ,n)≤ℙ{max1≤j≤n∥ajXj∥>14εnα}+CKC1(γ)ε-γn-(γα-δ).
By Theorem 1, we see that
(247)ℙ{max1≤j≤n∥ajXj∥>14εnα}≤∑j=1n‍ℙ{∥ajXj∥>14εnα}.
By (238) and (239), we know that
(248)∑j=1n‍ℙ{∥ajXj∥>14εnα}≤4γε-γn-γα∑j=1n‍|aj|γ𝔼∥Xj∥γ≤4γCKε-γn-(γα-δ).
From (246)–(248), we see that (244) holds. Thus, (241) holds.


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

Let(𝔹,∥·∥) be a separable Banach space. In this subsection, we consider complete convergence of weighted sums of 𝔹-valued martingale differences (Xnj) of the form ∑j=1∞‍anjXnj. 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{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let α>0 and {anj,j≥1,n≥1} be an array of real numbers. If there exists γ∈(1,2] such that
(249)∑n=1∞‍∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ<∞
and 𝔹 is γ-smooth, then
(250)∑n=1∞‍ℙ{n-αsupj≥1∥∑i=1j‍aniXni∥>ε}<∞∀ε>0.In the square-integrable real-valued martingale differences case, the result was proved by Ghosal and Chandra in Corollary 1 [19] if, additionally, ∑j=1∞‍anj2=O(nδ) for some δ<2α.We generalize Theorem 2 of Yu [23] from two directions by Theorem 43: first, we extend sequences of Lp(p≥2) martingale differences to Lγ(γ∈(1,2]) sequences of 𝔹-valued martingale differences; secondly, we do not need the condition ∑j=1∞‍|anj|2=O(nδ) for some δ<α.Proof of Theorem43.
SetYnj=n-αanjXnj. From (249), we have
(251)∑n=1∞‍𝔼mn(γ)=∑n=1∞‍∑j=1∞‍𝔼∥Ynj∥γ=∑n=1∞‍∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ<∞.
By Corollary 3 and Theorem 2 with q=1 and L=0, we see that
(252)ℙ{n-αsupj≥1∥∑i=1j‍aniXni∥>ε}≤ℙ{n-αsupj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Corollary 3 and Theorem 1, we know that
(253)ℙ{n-αsupj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}.
By Markov's inequality, we see that
(254)∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}≤4γε-γ∑j=1∞‍n-γα|anj|γ𝔼∥Xnj∥γ.
Since (249) and (251)–(254), we see that (250) holds.Theorem 44.
Let{(Xnj,ℱnj)}j≥1, n≥1, be sequences of 𝔹-valued martingale differences. Let {anj,j≥1,n≥1} be an array of real numbers. Suppose that for some constants γ∈(1,2], K>0, α∈(0,1/2] and δ<γα,
(255)∑j=1∞‍|anj|γ≤Knδ,n≥1,(256)∑n=1∞‍n-1∑j=1∞‍|anj|1/α<∞,(257)supn,j≥1𝔼∥Xnj∥1/α<∞.
If 𝔹 is γ-smooth, then (250) holds.When(Xnj)j≥1, n≥1, is a sequence of zero mean independent real-valued random variables and γ=2, Theorem 44 reduces to Theorem  2.3 of Li et al. [9].Proof of Theorem44.
SetYnj=n-αanjXnj,j≥1,n≥1, then {(Ynj,ℱnj),j≥1,n≥1} is an array of 𝔹-valued martingale differences. Since (255) and (257), for any n≥1, we have
(258)∑n=1∞‍𝔼mn(γ)=∑n=1∞‍∑j=1∞‍𝔼∥n-αanjXnj∥γ=∑n=1∞‍n-γα∑j=1∞‍|anj|γ𝔼∥Xnj∥γ≤CK∑n=1∞‍n-(γα-δ)<∞.
By Corollary 3 and Theorem 2 with q=1 and L=0, we see that
(259)ℙ{n-αsupj≥1∥∑i=1j‍anjXnj∥>ε}≤ℙ{n-αsupj≥1∥anjXnj∥>ε4}+ε-γC2(γ)𝔼mn(γ).
By Corollary 3 and Theorem 1, we know that
(260)ℙ{n-αsupj≥1∥anjXnj∥>ε4}≤∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}.
By Markov's inequality, we see that
(261)∑j=1∞‍ℙ{n-α∥anjXnj∥>ε4}≤ε-1/αn-1∑j=1∞‍|anj|1/α𝔼∥Xnj∥1/α<∞.
By (256), (257), and (258)–(261), we know that (250) holds.Theorem 45.
Let{(Xnj,ℱnj)}j=1n be a triangular array of identically distributed 𝔹-valued martingale differences. Let β>-1, and let {anj,1≤j≤n} be a triangular array of positive numbers satisfying
(262)anj=cnjn-β-1jβ
with 0<c≤cnj≤C<∞ for every 1≤j≤n, n≥1 and some constants c and C. Suppose that there exists some constant K, such that, for some γ∈(1,2],
(263)supn,j≥1𝔼[∥Xnj∥γ∣ℱn,j-1]≤K.
Suppose that
(264)𝔼∥X11∥1/(1+β)<∞if-1<β<-12,𝔼∥X11∥2log(1+∥X11∥)<∞ifβ=-12,𝔼∥X11∥2<∞ifβ>-12,
then for all ε>0,
(265)∑n=1∞‍ℙ{∥∑j=1n‍anjXnj∥>ε}<∞.When{Xnj}={Xj} are the same sequence of i.i.d. real-valued random variables and ∑j=1n‍anj=1 for all n≥1, Theorem 45 reduces to the sufficiency of Theorem 2.4 of Li et al. [9].Proof of Theorem45.
Set(266)Ynj={anjXnjif1≤j≤n,0ifj>n,
and cn=1, then {(Ynj,ℱnj)}j=1n are sequences of 𝔹-valued martingale differences. For any n≥1, since (262) and (263), we have
(267)mn(γ)=∑j=1n‍𝔼[∥anjXnj∥γ∣ℱn,j-1]=∑j=1n‍anjγ𝔼[∥Xnj∥γ∣ℱn,j-1]≤KCγn-(γ-1)(n-1∑j=1n‍(jn)γβ)~KCγn-(γ-1)∫1/n1‍xγβdx=O(n-b),
where b=min{γ(β+1),γ-1}. By Corollary 12 of Theorem 10, (265) is implied by
(268)∑n=1∞‍∑j=1n‍ℙ{∥anjXnj∥>ε}<∞∀ε>0.
Equation (268) holds if and only if (264) holds. (cf. [9, pages 62-63])



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