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

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 (𝑋_𝑗)_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 𝔼𝑋_𝑖=0, and set 𝑆_𝑛=∑^𝑛_𝑗=1𝑋_𝑗. By the law of large numbers, ℙ{|𝑆_𝑛|>𝜀𝑛}→0 for 𝜀>0. Hsu and Robbins [1] introduced the notion of complete convergence and showed that ^∞_𝑛=1ℙ𝑆_𝑛>𝜀𝑛<∞∀𝜀>0 (1) if 𝔼𝑋²₁<∞; Erdös [2, 3] proved that the converse also holds. Spitzer [4] showed that ^∞_𝑛=1𝑛⁻¹ℙ𝑆_𝑛>𝜀𝑛<∞∀𝜀>0 (2) 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, ^∞_𝑛=1𝑛^𝑝𝛼−2ℙ𝑆_𝑛>𝜀𝑛^𝛼<∞∀𝜀>0 (3) 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 (𝑛^𝑝𝛼−2) and (𝑛^𝛼) 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 (𝑋_𝑗)_𝑗≥1 be a sequence of real-valued martingale differences defined on a probability space (Ω,ℱ,ℙ), adapted to a filtration (ℱ_𝑗), with ℱ₀={∅,Ω}. This means that for each (integer) 𝑗≥1, 𝑋_𝑗 is ℱ_𝑗-measurable and 𝔼[𝑋_𝑗∣ℱ_𝑗−1]=0 a.s. A natural question is whether the pre-mentioned theorem of Baum and Katz [6] is still valid for martingale differences (𝑋_𝑗). Lesigne and Volný [16] proved that, for 𝑝≥2, sup_𝑗≥1𝔼|𝑋_𝑗|^𝑝<∞ implies ℙ𝑆_𝑛>𝜀𝑛=𝑂𝑛^−𝑝/2 (4) (as usual, we write 𝑎_𝑛=𝑜(𝑏_𝑛) if lim_𝑛→∞𝑎_𝑛/𝑏_𝑛=0 and 𝑎_𝑛=𝑂(𝑏_𝑛) if the sequence (𝑎_𝑛/𝑏_𝑛) 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 (𝑋_𝑗)_𝑗≥1 if for some 𝛾∈(1/𝛼,2] and 𝑞∈[1,∞] with 𝑞>(𝑝𝛼−1)/(𝛾𝛼−1), sup_𝑛≥11𝑛^𝑛_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1𝑞<∞, (5) where ‖⋅‖_𝑞 denotes the 𝐿^𝑞 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 𝑆_𝑛=∑^𝑛_𝑗=1𝑗^𝑎𝑌_𝑗, where 𝑎>0 and 𝑌_𝑗 are i.i.d., as in this case the condition (5) (with 𝑋_𝑗=𝑗^𝑎𝑌_𝑗) 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 (𝑆_𝑛) into martingale differences (such as in the study of directed polymers in a random environment), the summands usually depend on 𝑛: 𝑆_𝑛=∑^𝑛_𝑗=1𝑋_𝑛𝑗, 𝑋_𝑛𝑗=𝔼[𝑆_𝑛∣ℱ_𝑗]−𝔼[𝑆_𝑛∣ℱ_𝑗−1], where ℱ₀={∅,Ω} and ℱ_𝑖=𝜎(𝑆₁,…,𝑆_𝑖) 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 ∑^𝑛_𝑗=1𝔼[‖𝑋_𝑛𝑗‖^𝛾∣ℱ_{𝑛,𝑗−1}] for some 𝛾∈(1,2], we will find sufficient conditions for ^∞_𝑛=1𝜙(𝑛)ℙ𝑆_𝑛,∞>𝜀<∞ (6) for a large class of sequences of Banach valued martingale differences {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1, 𝑛≥1, where 𝑆_𝑛,∞=∑^∞_𝑗=1𝑋_𝑛𝑗, 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)=𝑛^𝑏−1𝑙(𝑛)  (𝑏≥0), 𝑙(⋅)>0 being slowly varying at ∞; that is, 𝑙(⋅) is a positive measurable function defined on (0,∞) such that lim_𝑥→∞(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 𝑋_𝑗 with 𝔼𝑋_𝑗=0 and for each 𝑝≥1, ℙ(|𝑋₁|>𝑛)=𝑜(𝑛^−𝑝) if and only if ℙ(|𝑆_𝑛|>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 ∑^𝑛_𝑗=1𝔼[‖𝑋_𝑛𝑗‖^𝛾∣ℱ_{𝑛,𝑗−1}] for some 𝛾∈(1,2], we obtain sufficient conditions for 𝜙(𝑛)ℙ𝑆_𝑛,∞>𝜀=𝑜(1)resp.,𝑂(1), (7) where 𝜙 and 𝑆_𝑛,∞ 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 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 ∑^𝑛_𝑗=1𝑗^𝑎−1𝑋_𝑗. 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 ℕ^∗={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 𝕃^𝑝_𝔹 the space of 𝔹-valued random variables such that ‖𝑋‖_𝕃^𝑝_𝔹=(𝔼‖𝑋‖^𝑝)^1/𝑝 is finite. Let ℱ₀={∅,Ω}⊂ℱ₁⊂⋯ be an increasing sequence of sub-𝜎-fields of ℱ. Let {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1 be an adapted sequence of 𝔹-valued random variables defined on (Ω,ℱ,ℙ); that is, for every 𝑗≥1, 𝑋_𝑗 is ℱ_𝑗 measurable. We call it a sequence of 𝔹-valued martingale differences if additionally 𝔼[𝑋_𝑗∣ℱ_𝑗−1]=0 a.s. and 𝑋_𝑗 belongs to 𝕃¹_𝔹 for any 𝑗≥1 and a sequence of 𝔹-valued supermartingale differences if additionally 𝔼[𝑋_𝑗∣ℱ_𝑗−1]≤0 a.s. and 𝑋_𝑗 belongs to 𝕃¹_𝔹 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 sup_𝑡>01𝑡^𝛾sup𝑥+𝑡𝑦_𝔹+𝑥−𝑡𝑦_𝔹−2∶‖𝑥‖_𝔹=𝑦_𝔹=1<∞. (8) Set 𝑆₀=0,𝑆_𝑛=^𝑛_𝑗=1𝑋_𝑗,𝑆^∗_𝑛=max_{1≤𝑗≤𝑛}𝑆_𝑗∀𝑛∈ℕ^∗, (9) and set 𝑋^∗₀=0,𝑋^∗_𝑛=max_{1≤𝑗≤𝑛}𝑋_𝑗∀𝑛∈ℕ^∗. (10) For 𝛾>0, let 𝑚𝛾,𝑛=^𝑛_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (11) Accordingly, for an infinite 𝔹-valued adapted sequence {(𝑋_𝑗,ℱ_𝑗)}_𝑗≥1, we write 𝑋^∗_∞=sup_𝑗≥1𝑋_𝑗,𝑆^∗_∞=sup_𝑗≥1𝑆_𝑗,𝑆_∞=^∞_𝑗=1𝑋_𝑗 (12) if the series converges, and 𝑚𝛾,∞=^∞_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (13)

In the following, we consider relations among ∑^𝑛_𝑗=1ℙ{‖𝑋_𝑗‖>𝜀}, ℙ{𝑋^∗_𝑛>𝜀}, ℙ{𝑆^∗_𝑛>𝜀}, and ℙ{‖𝑆_𝑛‖>𝜀}.

Our first theorem describes relations between ℙ{𝑋^∗_𝑛>𝜀} and ∑^𝑛_𝑗=1ℙ{‖𝑋_𝑗‖>𝜀} for an adapted sequence of 𝔹-valued random variables {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1.

Theorem 1. Let {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1 be an adapted sequence of 𝔹-valued random variables. Then, for any 𝜀,𝛾>0, and 𝑞≥1, ℙ𝑋^∗_𝑛>𝜀≤^𝑛_𝑗=1ℙ𝑋_𝑗≤>𝜀1+𝜀^−𝛾ℙ𝑋^∗_𝑛>𝜀+𝜀^−𝛾𝔼𝑚^𝑞𝛾,𝑛. (14)

Our second theorem shows relations between ℙ{𝑆^∗_𝑛>𝜀} and ℙ{𝑋^∗_𝑛>𝜀} for a sequence of 𝔹-valued martingale differences {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1: that is, for each (integer) 1≤𝑗≤𝑛, 𝑋_𝑗 is ℱ_𝑗 measurable and belongs to 𝕃¹_𝔹, and 𝔼[𝑋_𝑗∣ℱ_𝑗−1]=0 a.s.

Theorem 2. Let {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1 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 ℙ𝑋^∗_𝑛>2𝜀≤ℙ𝑆^∗_𝑛𝑋>𝜀≤ℙ^∗_𝑛>𝜀4(𝐿+1)+𝜀^{−𝑞𝛾(1+𝐿)/(𝑞+𝐿)}𝐶×𝛾,𝑞,𝐿𝔼𝑚^𝑞𝛾,𝑛^{(1+𝐿)/(𝑞+𝐿)}, (15) where 𝐶(𝛾,𝑞,𝐿) is a constant only depending on 𝛾, 𝑞, and 𝐿.

Corollary 3. Let {(𝑋_𝑗,ℱ_𝑗)}_𝑗≥1 be a sequence of 𝔹-valued martingale differences. Suppose that, for some 𝛾∈(1,2], ^∞_𝑗=1𝔼𝑋_𝑗^𝛾<∞. (16) If 𝔹 is 𝛾-smooth, then 𝑆_∞ converges 𝑎.𝑠. and the inequalities (14) and (15) hold with 𝑛 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, ℙ𝑋^∗_𝑛>𝜀=^𝑛_𝑗=1ℙ𝑋^∗_𝑗−1≤𝜀,𝑋_𝑗=>𝜀_𝑛𝑗=1ℙ𝑋_𝑗>𝜀−^𝑛_𝑗=2ℙ𝑋^∗_𝑗−1>𝜀,𝑋_𝑗>𝜀. (17) Since {(𝑋_𝑗,ℱ_𝑗)}^𝑛_𝑗=1 is an adapted sequence of 𝔹-valued random variables, by Markov's inequality (conditional on ℱ_𝑗−1), ℙ𝑋^∗_𝑗−1>𝜀,𝑋_𝑗>𝜀=_{{𝑋∗𝑗−1>𝜀}}𝟏_{{‖𝑋𝑗‖>𝜀}}=dℙ_{{𝑋∗𝑗−1>𝜀}}ℙ𝑋_𝑗>𝜀∣ℱ_𝑗−1dℙ≤𝜀^−𝛾_{{𝑋∗𝑗−1>𝜀}}𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1dℙ≤𝜀^−𝛾_{{𝑋∗𝑛>𝜀}}𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1dℙ. (18) Hence, by summing, we obtain ^𝑛_𝑗=2ℙ𝑋^∗_𝑗−1>𝜀,𝑋_𝑗>𝜀≤𝜀^−𝛾_{{𝑋∗𝑛>𝜀}}𝑚𝛾,𝑛dℙ=𝜀^−𝛾_{{𝑚(𝛾,𝑛)≤1,𝑋∗𝑛>𝜀}}𝑚𝛾,𝑛dℙ+𝜀^−𝛾_{{𝑚(𝛾,𝑛)>1,𝑋∗𝑛>𝜀}}𝑚𝛾,𝑛dℙ≤𝜀^−𝛾ℙ𝑋^∗_𝑛>𝜀+𝜀^−𝛾_{{𝑚(𝛾,𝑛)>1}}𝑚^𝑞𝛾,𝑛dℙ≤𝜀^−𝛾ℙ𝑋^∗_𝑛>𝜀+𝜀^−𝛾𝔼𝑚^𝑞𝛾,𝑛. (19) Therefore, the upper bound in (19) gives a lower bound of ℙ{𝑋^∗_𝑛>𝜀} by (17), which implies the second inequality of (14).

Proof of Theorem 2. The first inequality is obvious, because if max_{1≤𝑗≤𝑛}‖𝑆_𝑗‖≤𝜀, then max_{1≤𝑗≤𝑛}𝑋_𝑗=max_{1≤𝑗≤𝑛}𝑆_𝑗−𝑆_𝑗−1≤2𝜀. (20) We will prove the second inequality. For any 𝜀>0,𝑛∈ℕ^∗, and 𝐿∈ℕ, ℙ𝑆^∗_𝑛𝑋>2𝜀≤ℙ^∗_𝑛>𝜀2()𝑆𝐿+1+ℙ^∗_𝑛>2𝜀,𝑋^∗_𝑛≤𝜀2().𝐿+1 (21) Define 𝑇(0)𝑇𝑗𝑇∶𝑆=0,=inf𝑖∈𝑗−1,𝑛_𝑖−𝑆_{𝑇(𝑗−1)}>𝜀𝐿+1for1≤𝑗≤𝐿+1, (22) 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⊂⋯⊂ℱ_𝑛⊂ℱ_𝑛+1⊂⋯, where we take ℱ_𝑘=ℱ_𝑛 for all 𝑘≥𝑛. As usual, we will write ℱ_𝑇(𝑗)={𝐴∈ℱ_𝑛∶𝐴∩{𝑇(𝑗)=𝑘}∈ℱ_𝑘,1≤𝑘≤𝑛}. Notice that for 1≤𝑗≤𝐿+1, if 𝑇(𝑗)<∞, then ‖𝑆_𝑇(𝑗)−𝑆_{𝑇(𝑗−1)}‖>𝜀/(𝐿+1); conversely, if there exists a positive integer 𝑖∈(𝑇(𝑗−1),𝑛] such that ‖𝑆_𝑖−𝑆_{𝑇(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 𝑆^∗_𝑛>2𝜀,𝑋^∗_𝑛≤𝜀2()⊂𝐿+1^𝐿+1_𝑗=1𝑇𝑗<∞, (23) which implies that ℙ𝑆^∗_𝑛>2𝜀,𝑋^∗_𝑛≤𝜀2()𝑉𝐿+1≤ℙ_𝑗>𝜀,𝐿+1,1≤𝑗≤𝐿+1 (24) where 𝑉_𝑗=𝑆_𝑇(𝑗)−𝑆_{𝑇(𝑗−1)}𝟏_{{𝑇(𝑗)<∞}},1≤𝑗≤𝐿+1. (25) Assume that the first event in (23) takes place. Since 𝑆^∗_𝑛>2𝜀, there exists 𝑚∈{1,2,…,𝑛} such that ‖𝑆_𝑚‖>2𝜀. As ‖𝑆_𝑚‖>2𝜀, and it is clear that 𝑇(1)≤𝑚<∞.
Let 𝑀 be the largest 𝑗∈[1,𝑚] such that 𝑇(𝑗)≤𝑚. Then, 𝑇(𝑀)≤𝑚.
We will prove that 𝑀≥𝐿+1. Suppose that 𝑀≤𝐿. Then, by the definition of 𝑀, 𝑇(𝑀+1)>𝑚 so that 𝑆_𝑚−𝑆_𝑇(𝑀)≤𝜀.𝐿+1 (26) As ‖𝑥+𝑦‖≤‖𝑥‖+‖𝑦‖, it follows that 𝑆_𝑇(𝑗)−𝑆_{𝑇(𝑗−1)}=𝑆_{𝑇(𝑗)−1}−𝑆_{𝑇(𝑗−1)}+𝑋_𝑇(𝑗)≤𝑆_{𝑇(𝑗)−1}−𝑆_{𝑇(𝑗−1)}+𝑋_𝑇(𝑗)≤𝜀+𝜀𝐿+12(),𝐿+1 (27) where the last step holds because ‖𝑆_{𝑇(𝑗)−1}−𝑆_{𝑇(𝑗−1)}‖≤𝜀/(𝐿+1) by the definition of 𝑇(𝑗) and ‖𝑋_𝑇(𝑗)‖≤𝑋^∗_𝑛≤𝜀/2(𝐿+1). As 𝑆_𝑚=∑^𝑀_𝑗=1(𝑆_𝑇(𝑗)−𝑆_{𝑇(𝑗−1)})+(𝑆_𝑚−𝑆_𝑇(𝑀)), by (26), (27), and the subadditivity of ‖⋅‖, we know that 𝑆_𝑚≤𝜀𝜀𝐿+1⋅𝑀+2()𝜀𝐿+1⋅𝑀+𝐿+1<2𝜀. (28) This is a contradiction with ‖𝑆_𝑚‖>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 of 𝔼[𝑉^𝛾_𝑗∣ℱ_{𝑇(𝑗−1)}]. For 0≤𝑟≤𝑛, 𝔼𝑉^𝛾_𝑗𝟏_{{𝑇(𝑗−1)=𝑟}}∣ℱ_{𝑇(𝑗−1)}=𝔼𝑉^𝛾_𝑗𝟏_{{𝑇(𝑗−1)=𝑟}}∣ℱ_𝑟≤𝔼𝑆_𝑇(𝑗)−𝑆_{𝑇(𝑗−1)}^𝛾𝟏_{{𝑇(𝑗−1)=𝑟,𝑇(𝑗)<∞}}∣ℱ_𝑟≤𝟏_{{𝑇(𝑗−1)=𝑟}}𝔼max_{𝑟+1≤𝑖≤𝑛}𝑆_𝑖−𝑆_𝑟^𝛾∣ℱ_𝑟. (29) Using successively Doob's inequality in a real separable Banach space [37, Theorem 3.1], the inequality in Assouad [38, Proposition 2] (conditional on ℱ_𝑟), and the subadditivity of the function 𝑥↦𝑥^𝛾/2(𝑥≥0), we obtain 𝔼max_{𝑟+1≤𝑖≤𝑛}𝑆_𝑖−𝑆_𝑟^𝛾∣ℱ_𝑟≤𝛾𝛾−1^𝛾𝔼^𝑛_𝑖=𝑟+1𝑋_𝑖^𝛾∣ℱ_𝑟≤𝛾𝛾−1^𝛾𝐶𝛾^𝑛_𝑖=𝑟+1𝔼𝑋_𝑖^𝛾∣ℱ_𝑟=𝛾𝛾−1^𝛾𝐶𝛾𝔼^𝑛_𝑖=𝑟+1𝔼𝑋_𝑖^𝛾∣ℱ_𝑖−1∣ℱ_𝑟≤𝛾𝛾−1^𝛾𝐶𝛾𝔼𝑚𝛾,𝑛∣ℱ_𝑟. (30) Therefore, by (29), 𝔼𝑉^𝛾_𝑗𝟏_{{𝑇(𝑗−1)=𝑟}}∣ℱ_{𝑇(𝑗−1)}≤𝛾𝛾−1^𝛾𝐶𝛾𝔼𝑚𝛾,𝑛𝟏_{{𝑇(𝑗−1)=𝑟}}∣ℱ_{𝑇(𝑗−1)}. (31) Summing over 𝑟, we obtain 𝔼𝑉^𝛾_𝑗∣ℱ_{𝑇(𝑗−1)}≤𝛾𝛾−1^𝛾𝐶𝛾𝔼𝑚𝛾,𝑛∣ℱ_{𝑇(𝑗−1)}. (32)
(c) We finally give un upper bound for the term of the right hand side of (24), using (32). Set 𝑄_𝑗=ℙ𝑉_𝑗>𝜀𝐿+1∣ℱ_{𝑇(𝑗−1)},1≤𝑗≤𝐿+1. (33) Applying (32) for 𝑗=1, we see that 𝑄₁=ℙ𝑉₁>𝜀𝐿+1≤𝜀^−𝛾(𝐿+1)^𝛾𝔼𝑉^𝛾₁≤𝜀^−𝛾(𝐿+1)^𝛾𝛾𝛾−1^𝛾𝐶𝛾𝔼𝑚𝛾,𝑛. (34) Now, for any 𝑥>0, ℙ𝑉_𝑗>𝜀≤𝐿+1,1≤𝑗≤𝐿+1^𝐿+1_𝑖=1𝑝_𝑖, (35) where 𝑝₁=ℙ𝑉_𝑗>𝜀𝐿+1,1≤𝑗≤𝐿+1;𝑄_𝑖≤𝑥⁻¹,𝑝,2≤𝑖≤𝐿+1_𝑖=ℙ𝑄_𝑖>𝑥⁻¹,2≤𝑖≤𝐿+1. (36) Notice that 𝑝₁=_{{𝑉𝑗𝑄>𝜀/(𝐿+1),1≤𝑗≤𝐿;𝑖≤𝑥−1,2≤𝑖≤𝐿+1}}𝔼𝟏_{{𝑉𝐿+1>𝜀/(𝐿+1)}}∣ℱ_𝑇(𝐿)=dℙ_{{𝑉𝑗>𝜀/(𝐿+1),1≤𝑗≤𝐿;𝑄𝑖≤𝑥−1,2≤𝑖≤𝐿+1}}𝑄_𝐿+1dℙ≤𝑥⁻¹ℙ𝑉_𝑗>𝜀𝐿+1,1≤𝑗≤𝐿;𝑄_𝑖≤𝑥⁻¹,2≤𝑖≤𝐿≤⋯≤𝑥^{−(𝐿−1)}ℙ𝑉_𝑗>𝜀𝐿+1,1≤𝑗≤2;𝑄₂≤𝑥⁻¹≤𝑥^−𝐿𝑄₁≤𝜀^−𝛾(𝐿+1)^𝛾𝛾𝛾−1^𝛾𝐶𝛾𝑥^−𝐿𝔼𝑚𝛾,𝑛, (37) 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 𝑝_𝑖=ℙℙ𝑉_𝑖>𝜀𝐿+1∣ℱ_{𝑇(𝑖−1)}>𝑥⁻¹≤ℙ𝔼𝑉^𝛾_𝑖∣ℱ_{𝑇(𝑖−1)}>𝜀^𝛾𝑥⁻¹(𝐿+1)^𝛾≤ℙ𝛾^𝛾𝛾−1^−𝛾𝐶𝛾𝔼𝑚𝛾,𝑛∣ℱ_{𝑇(𝑖−1)}>𝜀^𝛾𝑥⁻¹(𝐿+1)^𝛾≤𝜀^−𝑞𝛾(𝐿+1)^𝑞𝛾𝛾𝛾−1^𝑞𝛾𝐶^𝑞𝛾×𝑥^𝑞𝔼𝔼𝑚𝛾,𝑛∣ℱ_{𝑇(𝑖−1)}^𝑞≤𝜀^−𝑞𝛾(𝐿+1)^𝑞𝛾𝛾𝛾−1^𝑞𝛾𝐶^𝑞𝛾𝑥^𝑞𝔼𝑚^𝑞𝛾,𝑛. (38) Therefore, by (35), ℙ𝑉_𝑗>𝜀𝐿+1,1≤𝑗≤𝐿+1≤𝐶₁𝑥^−𝐿+𝐶₂𝑥^𝑞, (39) where 𝐶₁=𝜀^−𝛾(𝐿+1)^𝛾𝛾𝛾−1^𝛾𝐶𝛾,𝐶𝔼𝑚𝛾,𝑛₂=𝜀^−𝑞𝛾𝐿(𝐿+1)^𝑞𝛾𝛾𝛾−1^𝑞𝛾𝐶^𝑞𝛾𝔼𝑚^𝑞𝛾,𝑛. (40)
A simple calculation shows that inf_𝑥>0𝐶₁𝑥^−𝐿+𝐶₂𝑥^𝑞=𝜀^{−𝑞𝛾(1+𝐿)/(𝑞+𝐿)}𝐶₀𝛾,𝑞,𝐿𝔼𝑚𝛾,𝑛^{𝑞/(𝑞+𝐿)}×𝔼𝑚^𝑞𝛾,𝑛^{𝐿/(𝑞+𝐿)}, (41) where 𝐶₀(𝛾,𝑞,𝐿)=(𝐿+1)^{𝑞𝛾(1+𝐿)/(𝑞+𝐿)}[𝑞^{𝐿/(𝑞+𝐿)}+𝐿𝑞^{−𝑞/(𝑞+𝐿)}](𝛾^𝛾(𝛾−1)^−𝛾𝐶(𝛾))^{𝑞(1+𝐿)/(𝑞+𝐿)}. Therefore, ℙ𝑉_𝑗>𝜀1+𝐿,1≤𝑗≤𝐿+1≤𝜀^{−𝑞𝛾(1+𝐿)/(𝑞+𝐿)}𝐶₀𝛾,𝑞,𝐿𝔼𝑚𝛾,𝑛^{𝑞/(𝑞+𝐿)}×𝔼𝑚^𝑞𝛾,𝑛^{𝐿/(𝑞+𝐿)}≤𝜀^{−𝑞𝛾(1+𝐿)/(𝑞+𝐿)}𝐶₀𝛾,𝑞,𝐿𝔼𝑚^𝑞𝛾,𝑛^{(1+𝐿)/(𝑞+𝐿)}. (42) Together with (21) and (24), this proves (15).

Proof of Corollary 3. By Theorem 2 with 𝑞=1 and 𝐿=0, ℙsup_𝑗≥1𝑆_𝑛+𝑗−𝑆_𝑛>𝜀≤ℙsup_{𝑗≥𝑛+1}𝑋_𝑗>𝜀4+𝜀^−𝛾𝐶𝛾^∞_𝑗=𝑛+1𝔼𝑋_𝑗^𝛾. (43) By Theorem 1 and Markov's inequality, ℙsup_{𝑗≥𝑛+1}𝑋_𝑗>𝜀4≤^∞_𝑗=𝑛+1ℙ𝑋_𝑗>𝜀4≤4^𝛾𝜀^∞−𝛾_𝑗=𝑛+1𝔼𝑋_𝑗^𝛾. (44) Therefore, lim_𝑛→∞ℙsup_𝑗≥1𝑆_𝑛+𝑗−𝑆_𝑛>𝜀=0, (45) which is equivalent to lim_𝑛→∞sup_{𝑗,𝑘≥𝑛}𝑆_𝑘−𝑆_𝑗=0inprobability. (46) Since sup_{𝑗,𝑘≥𝑛}‖𝑆_𝑘−𝑆_𝑗‖ is decreasing in 𝑛, this implies that lim_𝑛→∞sup_{𝑗,𝑘≥𝑛}𝑆_𝑘−𝑆_𝑗=0a.s., (47) 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 ℱ_𝑛0={∅,Ω}⊂ℱ_𝑛1⊂⋯ be an increasing sequence of sub-𝜎-fields of ℱ. For every 𝑛≥1, let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1 be a sequence of 𝔹-valued martingale differences defined on (Ω,ℱ,ℙ), adapted to the filtration (ℱ_𝑛𝑗): that is, for every 𝑗≥1, 𝑋_𝑛𝑗 is ℱ_𝑛𝑗 measurable and belongs to 𝕃¹_𝔹, and 𝔼[𝑋_𝑛𝑗∣ℱ_{𝑛,𝑗−1}]=0 a.s. Set for 𝑛≥1, 𝑆_𝑛0=0,𝑆_𝑛𝑘=^𝑘_𝑗=1𝑋_𝑛𝑗𝑆for𝑘≥1,_𝑛,∞=^∞_𝑗=1𝑋_𝑛𝑗 (48) if the series converges. We will call the double sequence {(𝑋_𝑛𝑗,ℱ_𝑛𝑗),𝑗≥1,𝑛≥1} an array of 𝔹-valued martingale differences.

In the following, we give a sufficient condition for the convergence of 𝔹-valued martingale arrays. For 𝛾>0, let 𝑚_𝑛𝛾=^∞_𝑗=1𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}for𝑛≥1. (49)

Theorem 4. Assume that for some 𝛾∈(1,2], as 𝑛→∞, 𝔼𝑚_𝑛𝛾=^∞_𝑗=1𝔼𝑋_𝑛𝑗^𝛾⟶0. (50) If 𝔹 is 𝛾-smooth, then for all 𝜀>0, as 𝑛→∞, ^∞_𝑗=1ℙ𝑋_𝑛𝑗ℙ>𝜀⟶0;sup_𝑗≥1𝑋_𝑛𝑗ℙ>𝜀⟶0;sup_𝑗≥1𝑆_𝑛𝑗ℙ𝑆>𝜀⟶0;_𝑛,∞>𝜀⟶0. (51)(52)(53)(54)

Proof. Notice that, by Corollary 3, the condition (50) implies the a.s. convergence of 𝑆_𝑛,∞. Equation (51) comes from (50) as ^∞_𝑗=1ℙ𝑋_𝑛𝑗>𝜀≤𝜀^∞−𝛾_𝑗=1𝔼𝑋_𝑛𝑗^𝛾⟶0; (55) (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 ℙ{sup_𝑗≥1‖𝑆_𝑛𝑗‖>𝜀} and ℙ{‖𝑆_𝑛,∞‖>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=1ℙ{‖𝑋_𝑛𝑗‖>𝜀}, ℙ{sup_𝑗≥1‖𝑋_𝑛𝑗‖>𝜀}, ℙ{sup_𝑗≥1‖𝑆_𝑛𝑗‖>𝜀}, and ℙ{‖𝑆_𝑛,∞‖>𝜀}.

Lemma 5. Let 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2],𝑞≥1 and some integer 𝐿≥0, 𝜙(𝑛)𝔼𝑚^𝑞_𝑛𝛾^{(1+𝐿)/(𝑞+𝐿)}=𝑜(1)𝑟𝑒𝑠𝑝.,𝑂(1). (56) If 𝔹 is 𝛾-smooth, then the following conditions are equivalent: 𝜙(𝑛)ℙsup_𝑗≥1𝑋_𝑛𝑗(1)(1𝜙(𝑛)ℙ>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥1𝑆_𝑛𝑗(1)(1)>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (57)

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

Lemma 6. Let 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2] and 𝑞∈[1,∞), 𝜙(𝑛)𝔼𝑚^𝑞_𝑛𝛾=𝑜(1)𝑟𝑒𝑠𝑝.,𝑂(1). (58) Then, the following conditions are equivalent: 𝜙(𝑛)ℙsup_𝑗≥1𝑋_𝑛𝑗(1)(1𝜙(𝑛)>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_∞𝑗=1ℙ𝑋_𝑛𝑗(1)(1)>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (59)

Proof. The conclusion comes directly from Theorem 1.

Lemma 7. Let 𝜙∶ℕ→[0,∞) be a positive function. Suppose that (58) holds for some 𝛾∈(1,2] and 𝑞∈[1,∞). Then, 𝜙(𝑛)ℙ𝑆_𝑛,∞>𝜀=𝑜(1)𝑟𝑒𝑠𝑝.,𝑂(1)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0 (60) is implied by 𝜙(𝑛)ℙsup_𝑗≥1𝑆_𝑛𝑗(1)(1>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (61)

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

Theorem 8. Let 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑂(1). (62) If 𝔹 is 𝛾-smooth, then one has the following implications (63)⇔(64)⇔(65)⇒(66): 𝜙(𝑛)^∞_𝑗=1ℙ𝑋_𝑛𝑗(1)(1)𝜙(𝑛)ℙ>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥1𝑋_𝑛𝑗(1)(1)𝜙(𝑛)ℙ>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥1𝑆_𝑛𝑗(1)(1)𝜙(𝑛)ℙ𝑆>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_𝑛,∞(1)(1)>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (63)(64)(65)(66)

Remark 9. The condition (62) holds if for some 𝛾∈(1,2],𝑟∈ℝ and 𝜀₁>0, 𝜙(𝑛)=𝑂𝑛^𝑟,𝑚_𝑛(𝛾)_𝕃∞𝔹=𝑂𝑛^−𝜀1. (67)

Proof of Theorem 8. Notice that when (62) holds for 𝑞=∞ and some 𝜆∈(0,∞), then for 𝑞∈(𝜆,∞), 𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹≤𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃∞𝔹=𝑂(1). (68) 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. Let 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,^∞_𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞. (69) If 𝔹 is 𝛾-smooth, then one has the following implications (70)⇔(71)⇔(72)⇒(73): ^∞_𝑛=1𝜙(𝑛)^∞_𝑗=1ℙ𝑋_𝑛𝑗>𝜀<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥1𝑋_𝑛𝑗>𝜀<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥1𝑆_𝑛𝑗>𝜀<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙ𝑆_𝑛,∞>𝜀<∞∀𝜀>0. (70)(71)(72)(73)

Notice that, by (69), 𝔼𝑚_𝑛(𝛾)<∞ for each 𝑛≥1 with 𝜙(𝑛)>0, so that 𝑆_𝑛,∞ 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.

When 𝑞=∞, 𝛾=2, and {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_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 ‖𝑚_𝑛(2)‖_𝕃∞𝔹→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. Theorem 10 also holds if 𝑚_𝑛(𝛾) is replaced by ∑^∞_𝑗=1𝔼[‖𝑋_𝑛𝑗‖^𝛾∣ℱ_{𝑛,𝑗−𝑀}] for some 𝑀≥1. In fact, the case 𝑀≥2 can be reduced to the case 𝑀=1 by considering the subsequences {(𝑋_{𝑛,𝑙𝑀+𝑖},ℱ_{𝑛,𝑙𝑀+𝑖})}_𝑙≥0 (1≤𝑖≤𝑀) of {(𝑋_𝑛𝑘,ℱ_𝑛𝑘)}_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 Theorem 10. As in the proof of Theorem 8, 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 ‖𝑚_𝑛(𝛾)‖_𝕃^𝑞_𝔹≤1 for all 𝑛≥𝑛₀. Then, ^∞_𝑛=𝑛0𝜙(𝑛)𝔼𝑚^𝑞_𝑛𝛾^{(1+𝐿)/(𝑞+𝐿)}=^∞_𝑛=𝑛0𝜙(𝑛)𝑚_𝑛𝛾_𝕃^{𝑞(1+𝐿)/(𝑞+𝐿)}_𝑞𝔹≤^∞_𝑛=𝑛0𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞. (74) By Theorem 1, (70) and (71) are equivalent; by Theorem 2, (71) and (72) are equivalent; since (69) implies 𝔼𝑚_𝑛(𝛾)<∞ for each 𝑛≥1 with 𝜙(𝑛)>0, by Corollary 3, (72) implies (73).

Proof of Corollary 12. Choose 𝜆>(𝑟+1)/𝜀₁, then by (67), we have 𝑚_𝑛𝛾_𝕃∞𝔹⟶0,_∞𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃∞𝔹=^∞_𝑛=1𝑂𝑛^{𝑟−𝜀1𝜆}<∞. (75) 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 ℱ_𝑛0={∅,Ω}⊂ℱ_𝑛1⊂⋯⊂ℱ_𝑛𝑛 be an increasing sequence of sub-𝜎-fields of ℱ. For each 𝑛≥1, let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}^𝑛_𝑗=1 be a sequence of 𝔹-valued martingale differences defined on (Ω,ℱ,ℙ), adapted to the filtration (ℱ_𝑛𝑗): that is, for every 1≤𝑗≤𝑛 and every 𝑛≥1, 𝑋_𝑛𝑗 is ℱ_𝑛𝑗 measurable and belongs to 𝕃¹_𝔹, and 𝔼[𝑋_𝑛𝑗∣ℱ_{𝑛,𝑗−1}]=0 a.s. Set for 𝑛≥1, 𝑆_𝑛0=0,𝑆_𝑛𝑘=^𝑘_𝑗=1𝑋_𝑛𝑗,1≤𝑘≤𝑛. (76) We will call the double sequence {(𝑋_𝑛𝑗,ℱ_𝑛𝑗),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 𝑚_𝑛𝛾=𝑛^{𝑛−𝛾𝛼}_𝑗=1𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}for𝑛≥1. (77)

Theorem 13. Let 𝛼∈ℝ. Assume that for some 𝛾∈(1,2], as 𝑛→∞, 𝔼𝑚_𝑛𝛾=𝑛^{𝑛−𝛾𝛼}_𝑗=1𝔼𝑋_𝑛𝑗^𝛾⟶0. (78) If 𝔹 is 𝛾-smooth, then for all 𝜀>0, as 𝑛→∞, ^𝑛_𝑗=1ℙ𝑋_𝑛𝑗>𝜀𝑛^𝛼ℙ⟶0;max_{1≤𝑗≤𝑛}𝑋_𝑛𝑗>𝜀𝑛^𝛼ℙ⟶0;max_{1≤𝑗≤𝑛}𝑆_𝑛𝑗>𝜀𝑛^𝛼ℙ𝑆⟶0;_𝑛𝑛>𝜀𝑛^𝛼⟶0. (79)

Proof. It suffices to apply Theorem 4 for the array of 𝔹-valued martingale differences {(𝑌_𝑛𝑗,𝒢_𝑛𝑗),𝑗≥1,𝑛≥1} defined by 𝑌_𝑛𝑗=𝑋_𝑛𝑗𝑛^𝛼𝒢if1≤𝑗≤𝑛,0if𝑗>𝑛,_𝑛𝑗=ℱ_𝑛𝑗ℱif1≤𝑗≤𝑛,_𝑛𝑛if𝑗>𝑛. (80)

We are interested in the convergence rates of the probabilities ℙ{max_{1≤𝑗≤𝑛}‖𝑆_𝑛𝑗‖>𝜀𝑛^𝛼} and ℙ{‖𝑆_𝑛𝑛‖>𝜀𝑛^𝛼}. 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 ∑^𝑛_𝑗=1ℙ{‖𝑋_𝑛𝑗‖>𝜀𝑛^𝛼}, ℙ{max_{1≤𝑗≤𝑛}‖𝑋_𝑛𝑗‖>𝜀𝑛^𝛼}, ℙ{max_{1≤𝑗≤𝑛}‖𝑆_𝑛𝑗‖>𝜀𝑛^𝛼}, and ℙ{‖𝑆_𝑛𝑛‖>𝜀𝑛^𝛼}.

Theorem 14. Let 𝛼∈ℝ and 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑂(1). (81) If 𝔹 is 𝛾-smooth, then one has the following implications (82)⇔(83)⇔(84)⇒(85): 𝜙(𝑛)^𝑛_𝑗=1ℙ𝑋_𝑛𝑗>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)ℙ=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;max_{1≤𝑗≤𝑛}𝑋_𝑛𝑗>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)ℙ=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;max_{1≤𝑗≤𝑛}𝑆_𝑛𝑗>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)ℙ𝑆=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_𝑛𝑛>𝜀𝑛^𝛼(1)(1)=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (82)(83)(84)(85)

Corollary 15. Let 1/2<𝛼≤1 and 𝑏≥0. Let 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏𝑙(𝑛). Suppose that for some 𝛾∈(1/𝛼,2], 𝑞∈[1,∞] with 𝑞>𝑏/(𝛾𝛼−1), sup_𝑛≥1‖𝑚_𝑛𝛾‖_𝕃^𝑞_𝔹<∞,𝑤ℎ𝑒𝑟𝑒𝑚_𝑛𝛾=1𝑛^𝑛_𝑗=1𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}. (86) 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, 𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}≤𝐾a.s. (87)

Proof of Theorem 14. It suffices to apply Theorem 8 for the array of 𝔹-valued martingale differences {(𝑌_𝑛𝑗,𝒢_𝑛𝑗),𝑗≥1,𝑛≥1} defined by (80).

Proof of Corollary 15. Since 𝛾>1/𝛼, we have 𝑚_𝑛𝛾_𝕃^𝑞_𝔹=𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0. (88) As 𝑞>𝑏/(𝛾𝛼−1), we can choose 𝜆∈(𝑏/(𝛾𝛼−1),𝑞). For this 𝜆, 𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑛^𝑏𝑙(𝑛)𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹^𝜆=𝑛^{𝑏−𝜆(𝛾𝛼−1)}𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑜(1). (89) Thus, the condition (81) holds, so that the conclusion follows from Theorem 14.

Theorem 17. Let 𝛼∈ℝ and 𝜙∶ℕ→[0,∞) be a positive function. Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,^∞_𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞. (90) If 𝔹 is 𝛾-smooth, then one has the following implications (91)⇔(92)⇔(93)⇒(94): ^∞_𝑛=1𝜙(𝑛)^𝑛_𝑗=1ℙ𝑋_𝑛𝑗>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙmax_{1≤𝑗≤𝑛}𝑋_𝑛𝑗>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙmax_{1≤𝑗≤𝑛}𝑆_𝑛𝑗>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙ𝑆_𝑛𝑛>𝜀𝑛^𝛼<∞∀𝜀>0. (91)(92)(93)(94)

Corollary 18. Let 𝛼>1/2 and 𝑏≥0. Let 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏−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, when 𝑙(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 ∑^𝑛_𝑗=1𝑎_𝑛𝑗𝑋_𝑗.

Remark 19. As explained in Remark 11, Corollary 18 also holds if 𝑚_𝑛(𝛾) is replaced by (1/𝑛)∑^𝑛_𝑗=1𝔼[‖𝑋_𝑛𝑗‖^𝛾∣ℱ_{𝑛,𝑗−𝑀}] for some 𝑀≥1.

Proof of Theorem 17. It suffices to apply Theorem 10 for the array of 𝔹-valued martingale differences {(𝑌_𝑛𝑗,𝒢_𝑛𝑗),𝑗≥1,𝑛≥1} defined by (80).

Proof of Corollary 18. Notice that 𝑚_𝑛𝛾_𝕃^𝑞_𝔹=𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0. (95) As 𝑞>𝑏/(𝛾𝛼−1), we can take 𝜆∈(𝑏/(𝛾𝛼−1),𝑞).
Then, ^∞_𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹^𝜆=^∞_𝑛=1𝑛^{𝑏−1−𝜆(𝛾𝛼−1)}𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞. (96) 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 {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_{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 ^∞_𝑛=1ℙ𝑛^−2/𝑝max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀<∞∀𝜀>0. (97)

When {𝑋_𝑛𝑗} 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 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 ^∞𝑛_𝑛=1𝑗=1ℙ𝑛^−2/𝑝𝑋_𝑛𝑗>𝜀=^∞_𝑛=1𝑛ℙ𝑋₁₁>𝜀𝑛^2/𝑝=𝔼_{1≤𝑛<𝜀−}𝑝2_‖𝑋11‖𝑝2𝑛. (98) As ∑^𝑚_𝑛=1𝑛=(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 Section 6.

Let (𝔹,‖⋅‖) a separable Banach space and (𝑌_𝑛)_𝑛≥1 be a sequence of any 𝔹-valued random variables. For any 𝜐∈[1,∞), let [𝜐] be the integer part of 𝜐. Set 𝑌₀=0,𝑌_𝜐=𝑌_[𝜐],𝑌^∗_𝜐=max_{0≤𝑘≤𝜐}𝑌_𝑘for𝜐∈[0,∞). (99) Then, for any 𝑛≥1, 𝑌^∗_𝑛=max_{0≤𝑘≤𝑛}‖𝑌_𝑘‖. For any 𝛼>0,𝜀>0, set 𝑀(𝜀)=𝑀(𝜀,𝛼)=sup_𝑛≥1𝑌_𝑛−𝜀𝑛^𝛼₊=sup_𝑛≥1𝑌^∗_𝑛−𝜀𝑛^𝛼₊=sup_𝑛≥0𝑌^∗_𝑛−𝜀𝑛^𝛼. (100) Let 𝑙(⋅)>0 be a function slowly varying at ∞. Recall that a function 𝑙(𝑥)>0 slowly varying at ∞ has the representation form 𝑙(𝑥)=𝑐(𝑥)exp^𝑥_𝑥0𝜀(𝑢)𝑢d𝑢𝑥≥𝑥₀ (101) 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 of ℙ{𝑌^∗_𝑛>𝜀𝑛^𝛼} and ℙ{sup_𝑗≥𝑛𝑗^−𝛼𝑌^∗_𝑗>𝜀}. Notice that ℙ{sup_𝑗≥𝑛𝑗^−𝛼𝑌^∗_𝑗>𝜀}→0 for any 𝜀>0 if and only if 𝑌^∗_𝑛/𝑛^𝛼→0 a.s. So, our results in this section describe the rate convergence for the almost surely convergence of 𝑌^∗_𝑛/𝑛^𝛼.

The following result shows that ℙ{𝑌^∗_𝑛>𝜀𝑛^𝛼} and ℙ{𝑀(𝜀)>𝜀𝑛^𝛼} have similar asymptotic properties. More precise comparisons will be given in Theorems 22 and 24.

Lemma 21. Let 𝛼>0. Then, for any 𝜐≥1 and any 𝜀>0, ℙ𝑌^∗_𝜐>2𝜀𝜐^𝛼≤ℙsup_𝑗≥𝜐𝑗^−𝛼𝑌^∗_𝑗>2𝜀≤ℙ𝑀(𝜀)>𝜀𝜐^𝛼. (102) Let 𝜀,𝛼,𝑏,𝛿>0 and 𝜙(𝑣)=𝑣^𝑏𝑙(𝑣). If there exists 𝑣₀>0, such that for all 𝑣≥𝑣₀, 𝜙(𝑣)ℙ𝑌^∗_𝑣>12𝜀𝑣^𝛼≤𝛿, (103) then there exists 𝜐^′₀>0 depending only on 𝑣₀,𝑏, and 𝛼, such that for all 𝜐≥𝜐^′₀, 𝜙(𝜐)ℙ𝑀(𝜀)>𝜀𝜐^𝛼≤𝛿𝐶(𝑏,𝛼), (104) where 𝐶(𝑏,𝛼)=2/(2^𝑏/(2𝛼)−1).

Proof. The first inequality of (102) is obvious. If sup_𝑗≥𝜐𝑗^−𝛼𝑌^∗_𝑗>2𝜀, (105) then sup_𝑛≥1𝑌^∗_𝑛−𝜀𝑛^𝛼₊=sup_𝑛≥1𝑛^𝛼𝑛^−𝛼𝑌^∗_𝑛−𝜀₊≥𝜐^𝛼sup_𝑛≥𝜐𝑛^−𝛼𝑌^∗_𝑛−𝜀₊>𝜀𝜐^𝛼. (106) Thus, the second inequality of (102) holds.
Assume that for some 𝑣₀>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, ℙ𝑌^∗_{2𝑘/𝛼𝜐11/𝑏}>2^𝑘−1𝜀𝜐₁^𝛼/𝑏≤𝛿𝜙2^𝑘/𝛼𝜐₁^1/𝑏. (107) Set 𝛽=𝑏/𝛼. Then, applying (103) for 𝑣=2^𝑘/𝛼𝜐₁^1/𝑏, we see that, for any 𝜐₁≥𝜐₂, ℙsup_𝑛≥1𝑌^∗_𝑛−𝜀𝑛^𝛼₊>𝜀𝜐₁^1/𝛽=ℙsup_𝑛≥1𝑌_𝑛−𝜀𝑛^𝛼₊>𝜀𝜐₁^1/𝛽≤^∞_𝑘=1ℙmax_{2𝑘−1𝜐−111/𝛽<𝑛𝛼≤2𝑘𝜐11/𝛽}𝑌_𝑛−𝜀𝑛^𝛼₊>𝜀𝜐₁^1/𝛽≤^∞_𝑘=1ℙ𝑌^∗_{(2𝑘𝜐11/𝛽)1/𝛼}>2^𝑘−1𝜀𝜐₁^1/𝛽=^∞_𝑘=1ℙ𝑌^∗_{2𝑘/𝛼𝜐11/𝑏}>2^𝑘−1𝜀𝜐₁^𝛼/𝑏≤^∞_𝑘=1𝛿𝜙2^𝑘/𝛼𝜐₁^1/𝑏. (108) Set 𝜐=𝜐₁^1/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(2^𝑘/𝛼𝜐)≤2(2^𝑘/𝛼)^𝛿1=2⋅2^{𝑏𝑘/2𝛼}. Thus, there exists 𝜐^′₀=𝜐^′₀(𝑣₀,𝑏,𝛼)=max{𝜐₂^1/𝑏,𝜐₃}>0 such that for all 𝜐≥𝜐^′₀, 𝜙(𝜐)ℙsup_𝑛≥1𝑌^∗_𝑛−𝜀𝑛^𝛼₊>𝜀𝜐^𝛼≤𝛿^∞_𝑘=1𝜙(𝜐)𝜙2^𝑘/𝛼𝜐=𝛿^∞_𝑘=12^{−𝑏𝑘/𝛼}𝑙(𝜐)𝑙2^𝑘/𝛼𝜐≤2𝛿^∞_𝑘=12^{−𝑏𝑘/(2𝛼)}=𝛿𝐶(𝑏,𝛼). (109)

Theorem 22. Let 𝛼,𝑏>0 and 𝜙(𝑛)=𝑛^𝑏𝑙(𝑛). Then, the following assertions are equivalent: 𝜙(𝑛)ℙ𝑌^∗_𝑛>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)ℙ=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥𝑛𝑗^−𝛼𝑌^∗_𝑗(1)(1)𝜙(𝑛)ℙ𝑀(𝜀)>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;>𝜀𝑛^𝛼(1)(1)=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (110)(111)(112)

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

Lemma 23. Let 𝛼>−1. Then for some 𝑛₀,𝑐₁,𝑐₂>0 and all 𝑁≥𝑛₀, 𝑐₁𝑁^𝛼+1𝑙(𝑁)≤^𝑁_𝑛=1𝑛^𝛼𝑙(𝑛)≤𝑐₂𝑁^𝛼+1𝑙(𝑁). (113)

Proof. Without loss of generality, we suppose that 𝑙(𝑠) has the form (101) with 𝑐(𝑠)≡1. Therefore, for 𝛿∈(0,𝛼+1), 𝑠^𝛿𝑙(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, ^𝑁_𝑛=1𝑛^𝛼𝑙(𝑛)=^𝑁_𝑛=1𝑛^𝛼−𝛿𝑛^𝛿𝑙(𝑛)≤𝑐₀+^𝑁_𝑛=𝑛1𝑛^𝛼−𝛿𝑛^𝛿𝑙(𝑛)≤𝑐₀+𝑁^𝛼+1𝑙(𝑁)^𝑁_𝑛=1𝑛𝑁^𝛼−𝛿1𝑁≤𝑐₀+𝑁^𝛼+1𝑙(𝑁)𝑐₃aslim^𝑁_{𝑁→∞𝑛=1}𝑛𝑁^𝛼−𝛿1𝑁=¹₀𝑥^𝛼−𝛿d𝑥<∞≤𝑐₂𝑁^𝛼+1𝑙(𝑁). (114) Similarly, for 𝑁≥2𝑛₁, ^𝑁_𝑛=1𝑛^𝛼𝑙(𝑛)≥_{𝑁/2≤𝑛≤𝑁}𝑛^𝛼−𝛿𝑛^𝛿𝑙(𝑛)≥𝑁2^𝛿𝑙𝑁2_{𝑁/2<𝑛≤𝑁}𝑛^𝛼−𝛿. (115) 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≤𝑁}𝑛^𝛼−𝛿≥(𝑁/2)^𝛼−𝛿(𝑁−[𝑁/2])≥𝑐₄𝑁^{𝛼+1−𝛿} for some positive constant 𝑐₄; if 𝛿>𝛼, then ∑_{𝑁/2<𝑛≤𝑁}𝑛^𝛼−𝛿≥𝑁^𝛼−𝛿(𝑁−[𝑁/2])≥𝑐₅𝑁^{𝛼+1−𝛿} for some positive constant 𝑐₅. So, for some constants 𝑐₁>0, 𝑛₀=max{2𝑛₁,𝑛₂}, and all 𝑁≥𝑛₀, ^𝑁_𝑛=1𝑛^𝛼𝑙(𝑛)≥𝑐₁𝑁^𝛼+1𝑙(𝑁). (116)

Theorem 24. Let 𝛼,𝑏>0 and 𝜙(𝑛)=𝑛^𝑏−1𝑙(𝑛). Then, the following assertions are equivalent: ^∞_𝑛=1𝜙(𝑛)ℙ𝑌^∗_𝑛>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥𝑛𝑗^−𝛼𝑌^∗_𝑗>𝜀<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙ𝑀(𝜀)>𝜀𝑛^𝛼𝔼𝑀(𝜀)<∞∀𝜀>0;_𝑏/𝛼𝑙𝑀(𝜀)^1/𝛼<∞∀𝜀>0. (117)(118)(119)(120)

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 ^∞₁𝜙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼d𝑢<∞∀𝜀>0. (121) Let 𝑛₀ be large enough such that 𝑢⁻¹𝑙(𝑢) is decreasing in [𝑛₀,∞). Then, we have ^𝑛_𝑛−1𝜙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼=d𝑢_𝑛𝑛−1𝑢^𝑏𝑢⁻¹𝑙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼d𝑢≤𝑛^𝑏(𝑛−1)⁻¹𝑙(𝑛−1)ℙ𝑌^∗_𝑛>12𝜀(𝑛−1)^𝛼≤2𝜙(𝑛)ℙ𝑌^∗_𝑛>14𝜀𝑛^𝛼 (122) for 𝑛≥𝑛₁≥𝑛₀+1 large enough. Here we have used the fact that lim_𝑛→∞(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 [𝑛^′₀,∞) for 𝑛^′₀ large enough, we obtain 𝑙(𝑛−1)/𝑙(𝑛)≤𝑛/(𝑛−1) for all 𝑛≥𝑛^′₀+1.) Similarly, ^𝑛_𝑛−1𝜙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼=d𝑢_𝑛𝑛−1𝑢^𝑏𝑢⁻¹𝑙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼≥()d𝑢𝑛−1^𝑏𝑛⁻¹𝑙(𝑛)ℙ𝑌^∗_𝑛−1>12𝜀𝑛^𝛼≥12𝜙(𝑛−1)ℙ𝑌^∗_𝑛−1>𝜀(𝑛−1)^𝛼 (123) for 𝑛≥𝑛₂≥𝑛₀+1 large enough. Let 𝑁=max{𝑛₁,𝑛₂}. As ∫^∞_𝑁−1=∑^∞_𝑛=𝑁∫^𝑛_𝑛−1, the conclusion that (117) is equivalent to (121) follows from (122) and (123).
(b) We next remark that (119) is equivalent to ^∞₁𝑢^𝑏−1𝑙(𝑢)ℙ𝑀(𝜀)>𝜀𝑢^𝛼d𝑢<∞∀𝜀>0. (124) 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 ^∞₁𝑢^𝑏−1𝑙(𝑢)ℙ𝑀(𝜀)>𝜀𝑢^𝛼=1d𝑢𝑏^∞₁𝑙𝜐^1/(𝛼𝛽)ℙsup_𝑗≥1𝑌_𝑗−𝜀𝑗^𝛼₊>𝜀𝜐^1/𝛽≤1d𝜐𝑏^∞₁𝑙𝜐^1/(𝛼𝛽)×^∞_𝑘=1ℙmax_{2𝑘−1𝜐−11/𝛽<𝑗𝛼≤2𝑘𝜐1/𝛽}𝑌_𝑗−𝜀𝑗^𝛼₊>𝜀𝜐^1/𝛽≤1d𝜐𝑏^∞_𝑘=1^∞₁𝑙𝜐^1/(𝛼𝛽)ℙ𝑌^∗_{(2𝑘𝜐1/𝛽)1/𝛼}>2^𝑘−1𝜀𝜐^1/𝛽=d𝜐_∞𝑘=12^{−𝑘𝛽∞}₁𝑢^𝑏−1𝑙2^−𝑘/𝛼𝑢ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼d𝑢. (125) By Potter's Theorem, for 0<𝛿<𝑏, there exists 𝐴=𝐴(𝛿)>1 such that 𝑙2^−𝑘/𝛼𝑢≤𝐴2^{𝑘𝛿/𝛼}𝑙(𝑢)∀𝑘≥1. (126) So, ^∞₁𝑢^𝑏−1𝑙(𝑢)ℙ𝑀(𝜀)>𝜀𝑢^𝛼d𝑢≤𝐴^∞_𝑘=12^{−𝑘(𝛽−𝛿/𝛼)∞}₁𝑢^𝑏−1𝑙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼=𝐴d𝑢2^{(𝑏−𝛿)/𝛼}−1^∞₁𝜙(𝑢)ℙ𝑌^∗_𝑢>12𝜀𝑢^𝛼d𝑢. (127) 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 ^∞_𝑛=1𝜙(𝑛)ℙ𝑀(𝜀)>𝜀𝑛^𝛼=𝔼^∞_𝑛=1𝜙(𝑛)𝟏_{{𝑀(𝜀)>𝜀𝑛𝛼}}=𝔼_{1≤𝑛<(𝑀(𝜀)/𝜀)1/𝛼}𝑙(𝑛)𝑛^𝑏−1. (128) By Lemma 23, there exist 𝑛₀,𝑐₁>0 such that 𝔼_{1≤𝑛<(𝑀(𝜀)/𝜀)1/𝛼}𝑙(𝑛)𝑛^𝑏−1𝟏_{{(𝑀(𝜀)/𝜀)}^1/𝛼_≥𝑛0}≥𝑐₁𝔼𝑀(𝜀)𝜀^𝑏/𝛼𝑙𝑀(𝜀)𝜀^1/𝛼𝟏_{{(𝑀(𝜀)/𝜀)}^1/𝛼_≥𝑛0}. (129) By (128) and (129), we see that (119) implies 𝔼𝑀(𝜀)𝜀^𝑏/𝛼𝑙𝑀(𝜀)𝜀^1/𝛼<∞∀𝜀>0. (130) By Potter's Theorem, there exists 𝑥₀>0 such that when (𝑀(𝜀)/𝜀)^1/𝛼≥𝑥₀ and (𝑀(𝜀))^1/𝛼≥𝑥₀, 𝑙𝑀(𝜀)^1/𝛼2max𝜀^−1/𝛼,𝜀^1/𝛼𝑀(𝜀)≤𝑙𝜀^1/𝛼≤2max𝜀^−1/𝛼,𝜀^1/𝛼𝑙𝑀(𝜀)^1/𝛼. (131) Therefore (130), is equivalent to (120). Hence, (119) implies (120). A similar argument (using again Lemma 23) shows that (120) implies (119). Thus, (119) and (120) are equivalent.

6. Convergence Rates for Banach Valued Martingales

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

Let (𝔹,‖⋅‖) a separable Banach space and {(𝑋_𝑗,ℱ_𝑗)}_𝑗≥1 a sequence of 𝔹-valued martingale differences. We denote 𝑆₀=0,𝑆_𝑛=^𝑛_𝑗=1𝑋_𝑗,𝑆^∗_𝑛=max_{0≤𝑗≤𝑛}𝑆_𝑗∀𝑛∈ℕ. (132) Notice that ‖𝑆₀‖=0,𝑆^∗₀=0 by our notations. Set 𝑋₀=0,𝑋^∗_𝑛=max_{0≤𝑗≤𝑛}𝑋_𝑗∀𝑛∈ℕ. (133) Notice that ‖𝑆_𝑛‖/𝑛^𝛼→0 a.s. if and only if ℙ(sup_𝑗≥𝑛𝑗^−𝛼‖𝑆_𝑗‖>𝜀)→0 for any 𝜀>0. So, the following theorems describe the a.s. convergence of ‖𝑆_𝑛‖/𝑛^𝛼.

Theorem 25. Let 𝛼,𝑏>0, 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏𝑙(𝑛). Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑂(1), (134) where 𝑚_𝑛(𝛾)=𝑛^−𝛾𝛼∑^𝑛_𝑗=1𝔼[‖𝑋_𝑗‖^𝛾∣ℱ_𝑗−1]. If 𝔹 is 𝛾-smooth, then one has the following implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138): 𝜙(𝑛)^𝑛_𝑗=1ℙ𝑋_𝑗>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)𝑃𝑋=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_∗𝑛>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)𝑃𝑆=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_∗𝑛>𝜀𝑛^𝛼(1(1𝜙(𝑛)𝑃𝑆=𝑜)𝑟𝑒𝑠𝑝.,𝑂)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;_𝑛>𝜀𝑛^𝛼(1)(1)𝜙(𝑛)𝑃=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥𝑛𝑗^−𝛼𝑆_𝑗(1)(1)𝜙(𝑛)𝑃>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0;sup_𝑗≥𝑛𝑗^−𝛼𝑆^∗_𝑗(1)(1>𝜀=𝑜𝑟𝑒𝑠𝑝.,𝑂)𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (135)(136)(137)(138)(139)(140)

Notice that, compared with Theorem 14, Theorem 25 contains the additional conditions (139) and (140). When 𝜙(𝑛)=𝑛^𝑏 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 Theorem 25, the conclusions of Theorem 25 remain valid if 𝑚_𝑛(𝛾) is replaced by 𝑚_𝑛𝛾,𝑀=𝑛^{𝑛−𝛾𝛼}_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−𝑀 (141) for some 𝑀∈ℕ^∗.
In fact, the case 𝑀≥2 can be reduced to the case 𝑀=1 by considering the subsequences {(𝑋_{𝑙𝑀+𝑖},ℱ_{𝑙𝑀+𝑖})}_𝑙≥0 (1≤𝑖≤𝑀) of {(𝑋_𝑗,ℱ_𝑗)}_𝑗≥1, which are still sequences of 𝔹-valued martingale differences.

Corollary 27. Let 1/2<𝛼≤1 and 𝑏>0. Let 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏𝑙(𝑛). Suppose that for some 𝛾∈(1/𝛼,2], 𝑞∈[1,∞] with 𝑞>𝑏/(𝛾𝛼−1), sup_𝑛≥1𝑚_𝑛𝛾_𝕃^𝑞_𝔹<∞,𝑤ℎ𝑒𝑟𝑒𝑚_𝑛𝛾=1𝑛^𝑛_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (142) If 𝔹 is 𝛾-smooth, then one has the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Theorem 25. Applying Theorem 14 to 𝑋_𝑛𝑗=𝑋_𝑗,ℱ_𝑛𝑗=ℱ_𝑗(1≤𝑗≤𝑛), and 𝜙(𝑛)=𝑛^𝑏𝑙(𝑛), we get the implications (135)⇔(136)⇔(137)⇒(138). Applying Theorem 22 to 𝑌_𝑛=‖𝑆_𝑛‖, we know that (137) and (140) are equivalent. Obviously, we have the implications (140)⇒(139)⇒(138). Therefore, we have proved the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Corollary 27. Since 𝛾>1/𝛼, we have 𝑚_𝑛𝛾_𝕃^𝑞_𝔹=𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹. (143) As 𝑞>𝑏/(𝛾𝛼−1), we can choose 𝜆∈(𝑏/(𝛾𝛼−1),𝑞). For this 𝜆, 𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑛^𝑏𝑙(𝑛)𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹^𝜆=𝑛^{𝑏−𝜆(𝛾𝛼−1)}𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=𝑜(1). (144) Thus, the condition (134) holds, so that the conclusion follows from Theorem 25.

Theorem 28. Let 𝛼,𝑏>0. Let 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏−1𝑙(𝑛). For any 𝜀>0, set 𝑀(𝑆,𝜀)=𝑀(𝑆,𝜀,𝛼)=sup_𝑛≥0𝑆_𝑛−𝜀𝑛^𝛼=sup_𝑛≥0𝑆^∗_𝑛−𝜀𝑛^𝛼,𝑀(𝑋,𝜀)=𝑀(𝑋,𝜀,𝛼)=sup_𝑛≥0𝑋_𝑛−𝜀𝑛^𝛼=sup_𝑛≥0𝑋^∗_𝑛−𝜀𝑛^𝛼. (145)(146) Suppose that for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,^∞_𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞, (147) where 𝑚_𝑛(𝛾)=𝑛^−𝛾𝛼∑^𝑛_𝑗=1𝔼[‖𝑋_𝑗‖^𝛾∣ℱ_𝑗−1]. If 𝔹 is 𝛾-smooth, then one has the following implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152):

(A) ^∞_𝑛=1𝜙(𝑛)ℙ𝑆^∗_𝑛>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥𝑛𝑗^−𝛼𝑆^∗_𝑗𝔼𝑀()>𝜀<∞∀𝜀>0;𝑆,𝜀^𝑏/𝛼𝑙𝑀(𝑆,𝜀)^1/𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥𝑛𝑗^−𝛼𝑆_𝑗>𝜀<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙ𝑆_𝑛>𝜀𝑛^𝛼<∞∀𝜀>0; (148)(149)(150)(151)(152)

(B) ^∞_𝑛=1𝜙(𝑛)ℙ𝑋^∗_𝑛>𝜀𝑛^𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)ℙsup_𝑗≥𝑛𝑗^−𝛼𝑋^∗_𝑗𝔼𝑀()>𝜀<∞∀𝜀>0;𝑋,𝜀^𝑏/𝛼𝑙𝑀(𝑋,𝜀)^1/𝛼<∞∀𝜀>0;_∞𝑛=1𝜙(𝑛)^𝑛_𝑗=1ℙ𝑋_𝑗>𝜀𝑛^𝛼<∞∀𝜀>0. (153)(154)(155)(156)

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

Remark 29. As in Theorem 25, the conclusions of Theorem 28 remain valid if 𝑚_𝑛(𝛾) is replaced by 𝑚_𝑛𝛾,𝑀=𝑛^{𝑛−𝛾𝛼}_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−𝑀 (157) for some 𝑀∈ℕ^∗.

Corollary 30. Let 𝛼>1/2,𝑏>0. Let 𝑙(⋅)>0 be a function slowly varying at ∞ and 𝜙(𝑛)=𝑛^𝑏−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).

If 𝑋_𝑗's are identically distributed real-valued random variables, then (156) is equivalent to the moment condition 𝔼|𝑋₁|^{(𝑏+1)/𝛼}𝑙(|𝑋₁|^1/𝛼)<∞. So, Corollary 30 contains Theorems 1, 2, and 3 of Baum and Katz [6] when 𝜙(𝑛)=𝑛^𝑏−1 and for i.i.d. real-valued random variables. When 𝜙(𝑛)=𝑛^𝑏−1, 𝛼≤1 and for real-valued martingale differences, Corollary 30 was proved by Alsmeyer [18, Theorems 1 and 2].

To see that Theorem 28 implies Corollary 30, it suffices to notice that (142) (with 𝑞>𝑏/(𝛾𝛼−1)) implies (147). In fact, when (142) holds for some 𝑞∈[1,∞] with 𝑞>𝑏/(𝛾𝛼−1), then 𝑚_𝑛𝛾_𝕃^𝑞_𝔹=𝑛^1−𝛾𝛼𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0, (158) and, for 𝜆∈(𝑏/(𝛾𝛼−1),𝑞), ^∞_𝑛=1𝜙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=^∞_𝑛=1𝑛^{𝑏−1−𝜆(𝛾𝛼−1)}𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞. (159) Therefore, (147) holds for some 𝑞∈[1,∞] and 𝜆∈(0,𝑞).

Proof of Theorem 28. Applying Theorem 10 to 𝑋_𝑛𝑗=𝑛^−𝛼𝑋_𝑗ℱ𝑖𝑓1≤𝑗≤𝑛,0if𝑗>𝑛,_𝑛𝑗=ℱ_𝑗ℱ𝑖𝑓1≤𝑗≤𝑛,_𝑛𝑖𝑓𝑗>𝑛, (160) we obtain the implications (153)⇔(156)⇔(148)⇒(152). By Theorem 24 applied to 𝑌_𝑛=‖𝑆_𝑛‖, we see that (148), (149), and (150) are equivalent. Since 𝑛^−𝛼‖𝑆_𝑛‖≤sup_𝑗≥𝑛𝑗^−𝛼‖𝑆_𝑗‖≤sup_𝑗≥𝑛𝑗^−𝛼𝑆^∗_𝑗, (149) implies (151) and (151) implies (152). Thus, we have the implications (148)⇔(149)⇔(150)⇒(151)⇒(152). Again by Theorem 24 applied to 𝑌_𝑛=‖𝑋_𝑛‖, 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 ∑^𝑛_𝑗=1𝑗^𝑎−1𝑋_𝑗

Let (𝔹,‖⋅‖) be a separable Banach space. In this section, we give a Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums 𝑇_𝑛=^𝑛_𝑗=1𝑗^𝑎−1𝑋_𝑗(𝑎∈ℝ) (161) of 𝔹-valued martingale differences {(𝑋_𝑗,ℱ_𝑗)}_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 (𝑌_𝑗)_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, 1𝑛^𝑛_𝑗=1𝑌_{𝑗𝕃𝑞𝔹}≤𝐾, (162) then ^𝑛_𝑗=1𝑗^𝑝𝑌_{𝑗𝕃𝑞𝔹}=𝑂𝑛^𝑝+1𝑓𝑜𝑟𝑎𝑛𝑦𝑝>−1. (163)

Proof. Let 𝑆₀=0, 𝑆_𝑛=∑^𝑛_𝑗=1𝑌_𝑗, 𝑛≥1. Then, ^𝑛_𝑗=1𝑗^𝑝𝑌_𝑗=^𝑛_𝑗=1𝑗^𝑝𝑆_𝑗−𝑆_𝑗−1=^𝑛−1_𝑗=1𝑗^𝑝−𝑗+1^𝑝𝑆_𝑗+𝑛^𝑝𝑆_𝑛=^𝑛−1_𝑗=1𝑗^𝑝11−1+𝑗^𝑝𝑆_𝑗+𝑛^𝑝𝑆_𝑛. (164) As 1−(1+1/𝑗)^𝑝∼−𝑝/𝑗(𝑗→∞), it follows that ^𝑛_𝑗=1𝑗^𝑝𝑌_{𝑗𝕃𝑞𝔹}≤^𝑛−1_𝑗=1𝑗^𝑝11−1+𝑗^𝑝𝑆_{𝑗𝕃𝑞𝔹}+𝑛^𝑝𝑆_𝑛𝕃^𝑞_𝔹=𝑂𝑛^𝑝+1. (165)

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

Theorem 32. Let {(𝑋_𝑗,ℱ_𝑗)}_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, 1𝑛^𝑛_𝑗=1𝔼𝑋_𝑗^𝛾≤𝐾. (166) If 𝔹 is 𝛾-smooth, then for 𝛼>−(1−1/𝛾) and 𝑎>1−1/𝛾, 𝑇_𝑛𝑛^𝑎+𝛼ℙ⟶0,𝑎.𝑠.,sup_𝑚≥𝑛𝑇_𝑚𝑚^𝑎+𝛼>𝜀=𝑂𝑛^{−(𝛾−1+𝛾𝛼)}𝑓𝑜𝑟𝑎𝑛𝑦𝜀>0. (167)(168)

Notice that when 𝑎=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{sup_𝑚≥𝑛(‖𝑇_𝑚‖/𝑚^𝑎+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 Theorem 32. Clearly, ℙsup_𝑚≥𝑛𝑇_𝑚𝑚^𝑎+𝛼>𝜀≤ℙsup_𝑚≥𝑛𝑚^{−(𝑎+𝛼)}max_{1≤𝑗≤𝑚}𝑇_𝑗>𝜀. (169) By Theorem 22, we see that ℙsup_𝑚≥𝑛𝑚^{−(𝑎+𝛼)}max_{1≤𝑗≤𝑚}𝑇_𝑗>𝜀=𝑂𝑛^{−(𝛾−1+𝛾𝛼)}∀𝜀>0 (170) if and only if ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝑎+𝛼=𝑂𝑛^{−(𝛾−1+𝛾𝛼)}∀𝜀>0. (171) Write 𝑚(𝛾,𝑛)=𝑛^{−𝛾(𝑎+𝛼)}∑^𝑛_𝑗=1𝔼[‖𝑗^𝑎−1𝑋_𝑗‖^𝛾∣ℱ_𝑗−1]. By the proof of Lemma 31, we have 𝔼𝑚𝛾,𝑛=𝑛^{𝑛−𝛾(𝑎+𝛼)}_𝑗=1𝔼𝑗^𝑎−1𝑋_𝑗^𝛾=𝑛^{𝑛−𝛾(𝑎+𝛼)}_𝑗=1𝑗^{𝛾(𝑎−1)}𝔼𝑋_𝑗^𝛾=𝑂𝑛^{−(𝛾−1+𝛾𝛼)}. (172) And by Theorem 2 with 𝑞=1 and 𝐿=0, we know that ℙ𝑛^{−(𝑎+𝛼)}max_{1≤𝑗≤𝑛}𝑇_𝑗𝑛>𝜀≤ℙ^{−(𝑎+𝛼)}max_{1≤𝑗≤𝑛}𝑗^𝑎−1𝑋_𝑗>14𝜀+𝜀^−𝛾𝐶₂𝛾𝔼𝑚𝛾,𝑛=ℙmax_{1≤𝑗≤𝑛}𝑗^𝑎−1𝑋_𝑗>14𝜀𝑛^𝑎+𝛼+𝑂𝑛^{−(𝛾−1+𝛾𝛼)}. (173) By Theorem 1, we see that ℙ𝑛^{−(𝑎+𝛼)}max_{1≤𝑗≤𝑛}𝑗^𝑎−1𝑋_𝑗>14𝜀≤^𝑛_𝑗=1ℙ𝑛^{−(𝑎+𝛼)}𝑗^𝑎−1𝑋_𝑗>14𝜀. (174) By (166), we have ^𝑛_𝑗=1ℙ𝑗^𝑎−1𝑋_𝑗>14𝜀𝑛^𝑎+𝛼≤4^𝛾𝜀^−𝛾𝑛^{𝑛−𝛾(𝑎+𝛼)}_𝑗=1𝑗^{𝛾(𝑎−1)}𝔼𝑋_𝑗^𝛾=𝑂𝑛^{−(𝛾−1+𝛾𝛼)}. (175) From (173)–(175), we see that (171) holds. Thus, (170) holds, so that (168) holds.

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

Definition 33. For a function 𝑅_𝜌 regularly varying at ∞ of index 𝜌≠0, one define, 𝑅^←_𝜌 as its inverse function.

Notice that when 𝜌>0, 𝑅_𝜌(𝑥) is strictly increasing for 𝑥 large enough with lim_𝑥→∞𝑅_𝜌(𝑥)=+∞, so that 𝑅^←_𝜌 is well defined on [𝑢₀,∞) for 𝑢₀>0 large enough. For simplicity, we always make the convention that 𝑅^←_𝜌(𝑢)=𝑅^←_𝜌(𝑢₀) if 𝑢∈[0,𝑢₀), so that 𝑅^←_𝜌 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 index 𝜌≠0 remains regularly varying.

Lemma 34. If 𝑅_𝜌(𝑥)=𝑥^𝜌𝑙(𝑥) is regularly varying at ∞ of index 𝜌≠0, where 𝑙(𝑥) is of the canonical form 𝑙(𝑥)=exp{∫^𝑥_𝑥0((𝜀(𝑢))/𝑢)𝑑𝑢}(𝑥≥𝑥₀)𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑥₀>0, then its inverse function 𝑅^←_𝜌(𝑥)=𝑥^1/𝜌𝑙^∗(𝑥) is regularly varying at ∞ of index 1/𝜌, where 𝑙^∗(𝑥)=(𝑙(𝑅^←_𝜌(𝑥)))^−1/𝜌 is slowly varying at ∞.

Proof. Let 𝑦=𝑅_𝜌(𝑥). Define 𝑙₂(𝑦)∶=𝑙(𝑅^←_𝜌(𝑦)). We have 𝑅^←_𝜌𝑦=𝑦^1/𝜌𝑙₂𝑦^−1/𝜌=𝑦^1/𝜌𝑙^∗𝑦. (176) We will prove that 𝑙₂(𝑦) is slowly varying at ∞. We see that 𝑙₂𝑦=exp^𝑅←𝜌_𝑥^(𝑦)₀𝜀(𝑢)𝑢d𝑢. (177) After changing variable, we have ^𝑅←𝜌_𝑥^(𝑦)₀𝜀(𝑢)𝑢d𝑢=^𝑦_𝑙(1)𝜀𝑅^←_𝜌(𝑣)𝑅^←_𝜌(𝑣)𝑅^←_𝜌(𝑣)^′=d𝑣_𝑦𝑙(1)𝜀𝑅^←_𝜌(𝑣)𝑅^←_𝜌(𝑣)𝑅^←_𝜌(𝑣)𝑣𝜌+𝜀𝑅^←_𝜌(𝑣)d𝑣=^𝑦_𝑙(1)𝜀𝑅^←_𝜌(𝑣)𝑣𝜌+𝜀𝑅^←_𝜌(𝑣)=d𝑣_𝑦𝑙(1)𝜀₁(𝑣)𝑣d𝑣, (178) where 𝜀₁(𝑣)∶=𝜀(𝑅^←_𝜌(𝑣))/(𝜌+𝜀(𝑅^←_𝜌(𝑣)))→0. Thus, 𝑙₂(𝑦)=exp{∫^𝑦_𝑙(1)(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 𝑅_1+𝛼(𝑥)=𝑥^1+𝛼𝑙₁(𝑥),𝑅_𝑎+𝛼(𝑥)=𝑥^𝑎+𝛼𝑙₁(𝑥), (179) and let 𝑅^←_1+𝛼 and 𝑅^←_𝑎+𝛼 be, respectively, the inverse functions of 𝑅_1+𝛼 and 𝑅_𝑎+𝛼 (cf. Definition 33), which are also regularly varying by Lemma 34. For 𝑎≠1, choose 𝑈 so large that 𝑙₀(𝑥)∶=𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1) (180) is locally bounded in [𝑈,∞), and set 𝑙̃(𝑢)[),𝑙̃(𝑢)==0if𝑢∈0,𝑈_𝑢𝑈𝑥⁻¹𝑙₀(𝑥)d𝑥if𝑢∈[𝑈,∞]. (181) Notice that 𝑙(̃∞) may be finite or infinite.

Theorem 35. Let {(𝑋_𝑗,ℱ_𝑗)}_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, sup_𝑛≥1𝑚_𝑛𝛾_𝕃^𝑞_𝔹<∞,𝑤ℎ𝑒𝑟𝑒𝑚_𝑛𝛾=1𝑛^𝑛_𝑗=1𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (182) If 𝔹 is 𝛾-smooth, then the following assertions hold.
(a) When 𝑎≠(𝑏−𝛼)/(𝑏+1), ^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝑎+𝛼𝑙₁(𝑛)<∞∀𝜀>0 (183) if and only if 𝔼𝑅^←_1+𝛼𝑋₁^𝑏+1𝑙𝑅^←_1+𝛼𝑋₁<∞𝑖𝑓𝑎>𝑏−𝛼,𝔼𝑅𝑏+1_←𝑎+𝛼𝑋₁^𝑏𝑙𝑅^←_𝑎+𝛼𝑋₁<∞𝑖𝑓𝑎<𝑏−𝛼.𝑏+1 (184)
(b) When 𝑎=(𝑏−𝛼)/(𝑏+1), (183) is implied by 𝔼𝑅^←_𝑎+𝛼𝑋₁^𝑏𝑙₁𝑅^←_𝑎+𝛼𝑋₁^1/(1−𝑎)𝑙̃𝑅^←_𝑎+𝛼𝑋₁<∞, (185) where 𝑙̃ is defined in (181); conversely, if 𝑙(̃∞)=∞ and the function 𝑙₀ defined by (180) satisfies lim_𝑢→∞(𝑙₀(𝑅^←_1+𝛼(𝑢))/𝑙₀(𝑅^←_𝑎+𝛼(𝑢)))<(1+𝛼)/(𝑎+𝛼), then (183) implies (185).

Remark 36. Theorem 35 also holds if (182) is replaced by 𝑚_𝑛𝛾_𝕃^𝑞_𝔹⟶0,^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹<∞ (186) for some 𝛾∈(1,2],𝑞∈[1,∞] and 𝜆∈(0,𝑞), where 𝑚_𝑛𝛾=𝑛^{−𝛾(𝑎+𝛼)}𝑙₁(𝑛)^𝑛−𝛾_𝑗=1𝑗^{−𝛾(1−𝑎)}𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (187)

Of particular interest are the cases where the slowly varying functions 𝑙 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 {(𝑋_𝑗,ℱ_𝑗)}_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 ^∞_𝑛=1𝑛^𝑏−1ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝑎+𝛼<∞∀𝜀>0 (188) if and only if 𝔼𝑋₁^{(𝑏+1)/(1+𝛼)}<∞𝑖𝑓𝑎>𝑏−𝛼,𝔼𝑋𝑏+1_{11/(1−𝑎)}log₊𝑋₁<∞𝑖𝑓𝑎=𝑏−𝛼,𝔼𝑋𝑏+1_{1𝑏/(𝑎+𝛼)}<∞𝑖𝑓𝑎<𝑏−𝛼.𝑏+1 (189)

Notice that the condition on 𝑚_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 (𝑋_𝑗)_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 where 𝑙(𝑥)=(log₊𝑥)^𝑝(𝑝∈ℝ) and 𝑙₁(𝑥)=(log₊𝑥)^𝛽(𝛽∈ℝ).

Corollary 38. Let {(𝑋_𝑗,ℱ_𝑗)}_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, ^∞_𝑛=2𝑛^𝑏−1log𝑛^𝑝ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝑎+𝛼log𝑛^𝛽<∞∀𝜀>0 (190) if and only if 𝔼𝑋₁^{(𝑏+1)/(1+𝛼)}log₊𝑋₁^{−𝛽(𝑏+1)/(1+𝛼)+𝑝}<∞if𝑎>𝑏−𝛼,𝔼𝑋𝑏+1_{11/(1−𝑎)}log₊𝑋₁^{−𝛽/(1−𝑎)+𝑝+1}<∞if𝑎=𝑏−𝛼,𝔼𝑋𝑏+1_{1𝑏/(𝑎+𝛼)}log₊𝑋₁^{−𝛽𝑏/(𝑎+𝛼)+𝑝}<∞if𝑎<𝑏−𝛼.𝑏+1 (191)

In the case where (𝑋_𝑗)_𝑗≥1 are i.i.d. real-valued random variables and the maximum max_{1≤𝑗≤𝑛}‖𝑇_𝑗‖ is replaced by the ‖𝑇_𝑛‖, 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 Theorem 35. Notice that 𝑙(̃𝑢) is slowly varying at ∞ by Proposition  1.5.9a in [39, page 26]. Set 𝑋_𝑛𝑗=𝑛^{−𝑎−𝛼}𝑙₁(𝑛)⁻¹𝑗^𝑎−1𝑋_𝑗ℱ𝑖𝑓1≤𝑗≤𝑛,0𝑖𝑓𝑗>𝑛,_𝑛𝑗=ℱ_𝑗ℱ𝑖𝑓1≤𝑗≤𝑛,_𝑛𝑖𝑓𝑗>𝑛, (192) and 𝜙(𝑛)=𝑛^𝑏−1𝑙(𝑛) in Theorem 10, then {(𝑋_𝑛𝑗,ℱ_𝑛𝑗),𝑗≥1,𝑛≥1} are sequences of 𝔹-valued martingale differences. For any 𝑛≥1, we have 𝑚_𝑛𝛾=^𝑛_𝑗=1𝔼𝑛^{−𝑎−𝛼}𝑙₁(𝑛)⁻¹𝑗^𝑎−1𝑋_𝑗^𝛾∣ℱ_𝑗−1=𝑛^{−𝛾(𝑎+𝛼)}𝑙₁(𝑛)^𝑛−𝛾_𝑗=1𝑗^{𝛾(𝑎−1)}𝔼𝑋_𝑗^𝛾∣ℱ_𝑗−1. (193) By (193) and Lemma 31, we have 𝑚_𝑛𝛾_𝕃^𝑞_𝔹=𝑛^{−𝛾(𝑎+𝛼)}𝑙₁(𝑛)^−𝛾𝑛_𝑗=1𝑗^{𝛾(𝑎−1)}𝔼𝑋_𝑗^𝛾∣ℱ_{𝑗−1𝕃𝑞𝔹}=𝑛^{−𝛾(𝑎+𝛼)}𝑙₁(𝑛)^−𝛾𝑂𝑛^{𝛾(𝑎−1)+1}=𝑙₁(𝑛)^−𝛾𝑂𝑛^{1−𝛾(𝛼+1)}. (194) Thus, ‖𝑚_𝑛(𝛾)‖_𝕃^𝑞_𝔹→0 and ^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)𝑚_𝑛𝛾^𝜆_𝕃^𝑞_𝔹=^∞_𝑛=1𝑙(𝑛)𝑙₁(𝑛)^−𝜆𝛾𝑂𝑛^{𝑏−1−𝜆(𝛾(𝛼+1)−1)}<∞ (195) whenever 𝜆∈(𝑏/(𝛾(𝛼+1)−1),𝑞). By Theorem 10, (183) is equivalent to ^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)^𝑛_𝑗=1ℙ𝑋_𝑗>𝜀𝑗^−𝑎+1𝑛^𝑎+𝛼𝑙₁(𝑛)<∞∀𝜀>0. (196) Notice that in view of the identically distributed assumption, (196) holds if and only if ^∞_𝑛=1𝑛^𝑏−1𝑙(𝑛)^𝑛_𝑗=1ℙ𝑋₁>𝜀𝑗^−𝑎+1𝑛^𝑎+𝛼𝑙₁(𝑛)<∞∀𝜀>0. (197) By the monotonicity of the functions 𝑅_𝑏−1(𝑥)∶=𝑥^𝑏−1𝑙(𝑥) and 𝑅_𝑎+𝛼(𝑥)=𝑥^𝑎+𝛼𝑙₁(𝑥) for 𝑥>0 large enough and the fact that 𝑅_𝑏−1(𝑛−1)∼𝑅_𝑏−1(𝑛) and 𝑅_𝑎+𝛼(𝑛−1)∼𝑅_𝑎+𝛼(𝑛) as 𝑛→∞, it can be easily verified that the condition (197) is equivalent to ^∞₁𝑥^𝑏−1𝑙(𝑥)^𝑥₁ℙ𝑋₁>𝜀𝑦^−𝑎+1𝑥^𝑎+𝛼𝑙₁(𝑥)d𝑦d𝑥<∞∀𝜀>0. (198) Therefore, Theorem 35 is a direct consequence of the following lemma.

Lemma 39. Let 𝑏≥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)When 𝑎≠(𝑏−𝛼)/(𝑏+1), (198) holds if and only if (184) holds.(b)When 𝑎=(𝑏−𝛼)/(𝑏+1), (198) is implied by (185), where 𝑙̃ 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 𝑢=𝑦^−𝑎+1𝑥^𝑎+𝛼𝑙₁(𝑥),𝑥=𝑥. (199) Notice that 1≤𝑦≤𝑥 if and only if 𝑙₁(1)≤𝑢<∞ and 𝑅^←_𝑎+𝛼(𝑢)≤𝑥≤𝑅^←_1+𝛼(𝑢). Therefore, (198) holds if and only if ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}ℙ𝑋₁×>𝜀𝑢_{𝑅←1+𝛼𝑅(𝑢)←𝑎+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)d𝑥d𝑢<∞∀𝜀>0. (200) By Proposition 1.5.8 of [39, page 26], we have as 𝑢→∞, ^{𝑅←1+𝛼}_𝑈^(𝑢)𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←1+𝛼(𝑢)^{𝑏+((1+𝛼)/(𝑎−1))+1}𝑙𝑅^←_1+𝛼(𝑢)×𝑙₁𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹,_{𝑅←𝑎+𝛼𝑈(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←𝑎+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_𝑎+𝛼(𝑢)×𝑙₁𝑅^←_𝑎+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹, (201) 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→∞, ^{𝑅←1+𝛼}_𝑅^(𝑢)_{←𝑎+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←1+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_1+𝛼(𝑢×𝑙₁)𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹. (202) Thus, (200) is equivalent to ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}𝑅^←_1+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_1+𝛼(𝑢)×𝑙₁𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)ℙ𝑋₁>𝜀𝑢d𝑢<∞∀𝜀>0. (203) With the change of variable 𝑣=𝑅^←_1+𝛼(𝑢) and the fact that 𝑅^′_1+𝛼(𝑣)∼𝑅^′_1+𝛼(𝑣)/(1+𝛼)𝑣 (together with (179)), we see that (203) holds if and only if ^∞_{𝑅←1+𝛼𝑙1(1)}𝑣^𝑏𝑙(𝑣)ℙ𝑅^←_1+𝛼𝑋₁𝜀>𝑣d𝑣<∞∀𝜀>0, (204) 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 𝑢=𝑥^1+𝛼𝑙₁(𝑥),𝑦=𝑦, (205) we see that (198) holds if and only if ^∞_𝑙1(1)𝑅^←_1+𝛼(𝑢)−1𝑅^←_1+𝛼(𝑢)^𝑏−1𝑙𝑅^←_1+𝛼(𝑢)×𝑅^′𝑅^←_1+𝛼(𝑢)⁻¹ℙ𝑋₁>𝜀𝑢d𝑢<∞∀𝜀>0. (206) With the change of variable 𝑣=𝑅^←_1+𝛼(𝑢), we know that (206) holds if and only if ^∞_{𝑅←1+𝛼𝑙1(1)}(𝑣−1)𝑣^𝑏−1𝑙(𝑣)ℙ𝑅^←_1+𝛼𝑋₁𝜀>𝑣d𝑣<∞∀𝜀>0, (207) 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 𝑢=𝑦^−𝑎+1𝑥^𝑎+𝛼𝑙₁(𝑥),𝑥=𝑥, (208) we see that (198) holds if and only if ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}ℙ𝑋₁×>𝜀𝑢_{𝑅←𝑎+𝛼𝑅(𝑢)←1+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)d𝑥d𝑢<∞∀𝜀>0. (209)
We distinguish three cases according to 𝑎<(𝑏−𝛼)/(𝑏+1), 𝑎>(𝑏−𝛼)/(𝑏+1), or 𝑎=(𝑏−𝛼)/(𝑏+1).
(i) Suppose that 𝑎<(𝑏−𝛼)/(𝑏+1). By Proposition  1.5.8 of [39, page 26], we have as 𝑢→∞, ^{𝑅←𝑎+𝛼}_𝑈^(𝑢)𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←𝑎+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_𝑎+𝛼(𝑢)×𝑙₁𝑅^←_𝑎+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹,_{𝑅←1+𝛼𝑈(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←1+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_1+𝛼(𝑢)×𝑙₁𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹ (210) so that (as in Case 1) ^{𝑅←𝑎+𝛼}_𝑅^(𝑢)_{←1+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←𝑎+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_𝑎+𝛼(𝑢)×𝑙₁𝑅^←_𝑎+𝛼(𝑢)^1/(𝑎−1)×𝑏+1+𝛼𝑎−1+1⁻¹. (211) Thus, (209) is equivalent to ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}𝑅^←_𝑎+𝛼(𝑢)^{𝑏+(𝑎+𝛼)/(𝑎−1)+1}𝑙𝑅^←_𝑎+𝛼(𝑢)×𝑙₁𝑅^←_𝑎+𝛼(𝑢)^1/(𝑎−1)ℙ𝑋₁>𝜀𝑢d𝑢<∞∀𝜀>0. (212) With the change of variable 𝑣=𝑅^←_𝑎+𝛼(𝑢), we see that (212) holds if and only if ^∞_{𝑅←𝑎+𝛼𝑙1(1)}𝑣^𝑏−1𝑙(𝑣)ℙ𝑅^←_𝑎+𝛼𝑋₁𝜀>𝑣d𝑣<∞∀𝜀>0, (213) which is equivalent to the second condition of (184). Thus, (198) is equivalent to the second condition of (184).
(ii) Suppose that 𝑎>(𝑏−𝛼)/(𝑏+1). By Proposition  1.5.10 of [39, page 27], we have as 𝑢→∞, ^∞_{𝑅←1+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←1+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_1+𝛼(𝑢)×𝑙₁𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)×−𝑏+1+𝛼𝑎−1+1⁻¹,_{∞𝑅←𝑎+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←𝑎+𝛼(𝑢)^{𝑏+(𝑎+𝛼)/(𝑎−1)+1}𝑙𝑅^←_𝑎+𝛼(𝑢)×𝑙₁𝑅^←_𝑎+𝛼(𝑢)^1/(𝑎−1)×−𝑏+1+𝛼𝑎−1+1⁻¹ (214) so that as 𝑢→∞, ^{𝑅←𝑎+𝛼}_𝑅^(𝑢)_{←1+𝛼(𝑢)}𝑥^{𝑏+(1+𝛼)/(𝑎−1)}𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)∼𝑅d𝑥_←1+𝛼(𝑢)^{𝑏+(1+𝛼)/(𝑎−1)+1}𝑙𝑅^←_1+𝛼(𝑢)×𝑙₁𝑅^←_1+𝛼(𝑢)^1/(𝑎−1)×−𝑏+1+𝛼𝑎−1+1⁻¹. (215) Thus, (209) holds if and only if (203) holds. Notice that (203) is equivalent to the first condition of (184). Thus (198) is equivalent to the first condition of (184).
(iii) Suppose that 𝑎=(𝑏−𝛼)/(𝑏+1). In this case, (209) reduces to ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}ℙ𝑋₁×>𝜀𝑢_{𝑅←𝑎+𝛼𝑅(𝑢)←1+𝛼(𝑢)}𝑥⁻¹𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)d𝑥d𝑢<∞∀𝜀>0. (216) By Proposition  1.5.8 of [39, page 26], we have ^{𝑅←𝑎+𝛼}_𝑅^(𝑢)_{←1+𝛼(𝑢)}𝑥⁻¹𝑙(𝑥)𝑙₁(𝑥)^1/(𝑎−1)=𝑙̃𝑅d𝑥_←𝑎+𝛼(𝑢)−𝑙̃𝑅^←_1+𝛼(𝑢)<𝑙̃𝑅^←_𝑎+𝛼(𝑢). (217) So, (209) is implied by ^∞_𝑙1(1)𝑢^{−1/(𝑎−1)−1}𝑙̃𝑅^←_𝑎+𝛼(𝑢)ℙ𝑋₁>𝜀𝑢d𝑢<∞∀𝜀>0. (218) With the change of variable 𝑣=𝑅^←_𝑎+𝛼(𝑢), we see that (218) holds if and only if ^∞_{𝑅←𝑎+𝛼𝑙1(1)}𝑣^𝑏−1𝑙₁(𝑣)^{−1/(𝑎−1)}𝑙̃(𝑣)×ℙ𝑅^←_𝑎+𝛼𝑋₁𝜀>𝑣d𝑣<∞∀𝜀>0, (219) which is equivalent to (185). Thus, (209) is implied by (185). Therefore, (198) is implied by (185). This ends the proof of Lemma 39.

Proof of Corollary 38. We are in the case where 𝑙₁(𝑥)=(log𝑥)^𝛽 (𝛽≥0). If 𝑦=𝑅_1+𝛼(𝑥)=𝑥^1+𝛼(log𝑥)^𝛽, then log𝑦∼(1+𝛼)log𝑥as𝑥⟶+∞. (220) Thus, 𝑥=𝑦^1/(1+𝛼)log𝑥^{−𝛽/(1+𝛼)}∼(1+𝛼)^{𝛽/(1+𝛼)}𝑦log𝑦^{𝛽1/(1+𝛼)}as𝑦⟶+∞. (221) Therefore, 𝑅^←_1+𝛼𝑦∼(1+𝛼)^{𝛽/(1+𝛼)}𝑦log𝑦^{𝛽1/(1+𝛼)}as𝑦⟶+∞. (222) Similarly, 𝑅^←_𝑎+𝛼𝑦∼(𝑎+𝛼)^{𝛽/(𝑎+𝛼)}𝑦log𝑦^{𝛽1/(𝑎+𝛼)}as𝑦⟶+∞. (223) 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=1𝑎_𝑛𝑗𝑋_𝑛𝑗

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

Let (𝑋_𝑗)_𝑗≥1 be a sequence of i.i.d. random variables with 𝔼𝑋_𝑗=0, and let {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} be an array of real numbers. The study of the convergence of weighted sums ∑^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑗 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=1𝑎_𝑛𝑗𝑋_𝑗→0 in probability and a sufficient condition for ∑^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑗→0 a.s. Baxter et al. [35] also showed a sufficient condition for ∑^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑗→0 a.s. Li et al. [9] studied the complete convergence of weighted sums of independent random variables of the form ∑^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗. Yu [23] and Ghosal and Chandra [19] considered the same problem for martingale differences (𝑋_𝑛𝑗). 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 (𝑋_𝑛𝑗).

Let us recall the famous theorem of Pruitt [43, Theorem 1] which states that for a sequence of i.i.d. random variables 𝑋_𝑗 with 𝔼|𝑋₁|<∞ and 𝔼𝑋₁=0, ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑗⟶0𝑖𝑛𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (224) if and only if max_𝑗≥1|𝑎_𝑛𝑗|→0as𝑛→∞, where {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} is a Toeplitz summation matrix; that is, lim_𝑛→∞𝑎_𝑛𝑗=0 for every 𝑗≥1, lim_𝑛→∞∑^∞_𝑗=1𝑎_𝑛𝑗=1, and sup_𝑛≥1∑^∞_𝑗=1|𝑎_𝑛𝑗|<∞.

In the following, we consider the same problem for 𝔹-valued martingale differences (𝑋_𝑛𝑗).

Theorem 40. Let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1, 𝑛≥1, be sequences of 𝔹-valued martingale differences. Let {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} be an array of real numbers satisfying sup_𝑛≥1∑^∞_𝑗=1|𝑎_𝑛𝑗|<∞ and 𝐴_𝑛=sup_𝑗≥1|𝑎_𝑛𝑗|→0𝑎𝑠𝑛→∞. Suppose that for some 𝛾∈(1,2], there exists 𝐾>0, such that 𝔼𝑋_𝑛𝑗^𝛾≤𝐾. (225) If 𝔹 is 𝛾-smooth, then ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗⟶0𝑖𝑛𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦. (226)

Proof. Since 𝔼𝑚_𝑛𝛾=^∞_𝑗=1𝔼𝑎_𝑛𝑗𝑋_𝑛𝑗^𝛾≤^∞_𝑗=1𝑎_𝑛𝑗𝐴_𝑛^𝛾−1𝔼𝑋_𝑛𝑗^𝛾≤𝐴_𝑛^𝛾−1𝐾^∞_𝑗=1𝑎_𝑛𝑗⟶0, (227) we see that (226) follows from Theorem 4.

Let {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} be a Toeplitz summation matrix; Theorem 2 of Pruitt [43] states that for a sequence (𝑋_𝑗)_𝑗≥1 of i.i.d. random variables with 𝔼|𝑋₁|<∞ and 𝔼𝑋₁=0, if max_𝑗≥1|𝑎_𝑛𝑗|=𝑂(𝑛^−𝛽), 𝛽>0, then 𝔼|𝑋₁|^1+1/𝛽<∞ implies that ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑗⟶0𝑎.𝑠. (228)

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

Theorem 41. Let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1, 𝑛≥1, be sequences of 𝔹-valued martingale differences. Let {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} be an array of real numbers satisfying sup_𝑛≥1∑^∞_𝑗=1|𝑎_𝑛𝑗|<∞ and max_𝑗≥1|𝑎_𝑛𝑗|=𝑂(𝑛^−𝛽), 𝛽>0. Suppose that, for some 𝛾∈(1,2], there exists a constant 𝐾>0 such that 𝔼‖𝑋_𝑛𝑗‖^𝛾≤𝐾 for all 𝑗≥1,𝑛≥1. If 𝔹 is 𝛾-smooth, then ℙ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀=𝑂𝑛^{−𝛽(𝛾−1)}∀𝜀>0. (229) Consequently, if 𝔹 is 𝛾-smooth and sup_{𝑛,𝑗≥1}𝔼‖𝑋_𝑛𝑗‖^𝛾<∞ for some 𝛾>1+1/𝛽, then ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗⟶0𝑎.𝑠. (230)

Proof. Set 𝑌_𝑛𝑗=𝑎_𝑛𝑗𝑋_𝑛𝑗,𝑗≥1,𝑛≥1, then {(𝑌_𝑛𝑗,ℱ_𝑛𝑗),𝑗≥1,𝑛≥1} is an array of 𝔹-valued martingale differences and for any 𝑛≥1, 𝔼𝑚_𝑛𝛾=𝔼^∞_𝑗=1𝔼𝑌_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}=^∞_𝑗=1𝔼𝑌_𝑛𝑗^𝛾=^∞_𝑗=1𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾≤sup^∞_{𝑛≥1𝑗=1}𝑎_𝑛𝑗𝐶𝑛^{−𝛽𝛾−1}𝐾≤𝐶₁𝛾𝑛^{−𝛽(𝛾−1)}. (231) By Corollary 3, we see that ℙ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀≤ℙsup_𝑗≥1^𝑗_𝑖=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀. (232) By Theorem 2 with 𝑞=1 and 𝐿=0, we know that ℙsup_𝑗≥1^𝑗_𝑖=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀≤ℙsup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4+𝜀^−𝛾𝐶₂𝛾𝔼𝑚_𝑛𝛾. (233) By Theorem 1, we see that ℙsup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤^∞_𝑗=1ℙ𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4. (234) Since (231), we know that ^∞_𝑗=1ℙ𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤4^𝛾𝜀^∞−𝛾_𝑗=1𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾≤4^𝛾𝜀^−𝛾𝐶₁𝛾𝑛^{−𝛽(𝛾−1)}. (235) Since (231)–(235), we have ℙ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀≤4^𝛾𝜀^−𝛾𝐶₁𝛾𝑛^{−𝛽(𝛾−1)}+𝜀^−𝛾𝐶₂𝛾𝐶₁𝛾𝑛^{−𝛽(𝛾−1)}. (236) Thus, (229) holds.
If additionally 𝛾>1+1/𝛽, then (229) implies that ^∞_𝑛=1ℙ^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀<∞∀𝜀>0. (237) Therefore, (230) holds.

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

Theorem 42. Let {(𝑋_𝑗,ℱ_𝑗)}_𝑗≥1 be 𝔹-valued martingale differences. Suppose that for some 𝛾∈(1,2], there exists 𝐾>0, such that for all 𝑗≥1, 𝔼𝑋_𝑗^𝛾≤𝐾. (238) Let {𝑎_𝑗,𝑗≥1} satisfy 1𝑛^𝑛_𝑗=1𝑎_𝑗^𝛾=𝑂𝑛^𝛿𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛿∈ℝ, (239) and set 𝑇_𝑛=∑^𝑛_𝑗=1𝑎_𝑗𝑋_𝑗. If 𝔹 is 𝛾-smooth, then 𝑇_𝑛𝑛^𝛼𝛿⟶0𝑎.𝑠.for𝛼>𝛾,ℙsup_𝑚≥𝑛𝑇_𝑚𝑚^𝛼>𝜀=𝑂𝑛^{−(𝛾𝛼−𝛿)}. (240)(241)

When (𝑋_𝑗,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 Theorem 42. Clearly, ℙsup_𝑚≥𝑛𝑇_𝑚𝑚^𝛼>𝜀≤ℙsup_𝑚≥𝑛𝑚^−𝛼max_{1≤𝑗≤𝑚}𝑇_𝑗>𝜀. (242) By Theorem 22, we see that ℙsup_𝑚≥𝑛𝑚^−𝛼max_{1≤𝑗≤𝑚}𝑇_𝑗>𝜀=𝑂𝑛^{−(𝛾𝛼−𝛿)}∀𝜀>0 (243) if and only if ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝛼=𝑂𝑛^{−(𝛾𝛼−𝛿)}∀𝜀>0. (244) By (238) and (239), we have 𝔼𝑚𝛾,𝑛=^𝑛_𝑗=1𝔼𝑎_𝑗𝑋_𝑗^𝛾=^𝑛_𝑗=1𝑎_𝑗^𝛾𝔼𝑋_𝑗^𝛾≤𝐶𝐾𝑛^𝛿. (245) And by Theorem 2 with 𝑞=1 and 𝐿=0, we know that ℙmax_{1≤𝑗≤𝑛}𝑇_𝑗>𝜀𝑛^𝛼≤ℙmax_{1≤𝑗≤𝑛}𝑎_𝑗𝑋_𝑗>14𝜀𝑛^𝛼+𝜀𝑛^𝛼−𝛾𝐶₁𝛾𝔼𝑚𝛾,𝑛≤ℙmax_{1≤𝑗≤𝑛}𝑎_𝑗𝑋_𝑗>14𝜀𝑛^𝛼+𝐶𝐾𝐶₁𝛾𝜀^−𝛾𝑛^{−(𝛾𝛼−𝛿)}. (246) By Theorem 1, we see that ℙmax_{1≤𝑗≤𝑛}𝑎_𝑗𝑋_𝑗>14𝜀𝑛^𝛼≤^𝑛_𝑗=1ℙ𝑎_𝑗𝑋_𝑗>14𝜀𝑛^𝛼. (247) By (238) and (239), we know that ^𝑛_𝑗=1ℙ𝑎_𝑗𝑋_𝑗>14𝜀𝑛^𝛼≤4^𝛾𝜀^−𝛾𝑛^{𝑛−𝛾𝛼}_𝑗=1𝑎_𝑗^𝛾𝔼𝑋_𝑗^𝛾≤4^𝛾𝐶𝐾𝜀^−𝛾𝑛^{−(𝛾𝛼−𝛿)}. (248) 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 (𝑋_𝑛𝑗) of the form ∑^∞_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗. 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 {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1, 𝑛≥1, be sequences of 𝔹-valued martingale differences. Let 𝛼>0 and {𝑎_𝑛𝑗,𝑗≥1,𝑛≥1} be an array of real numbers. If there exists 𝛾∈(1,2] such that ^∞∞_𝑛=1𝑗=1𝑛^−𝛾𝛼𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾<∞ (249) and 𝔹 is 𝛾-smooth, then ^∞_𝑛=1ℙ𝑛^−𝛼sup_𝑗≥1^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀<∞∀𝜀>0. (250)

In the square-integrable real-valued martingale differences case, the result was proved by Ghosal and Chandra in Corollary 1 [19] if, additionally, ∑^∞_𝑗=1𝑎²_𝑛𝑗=𝑂(𝑛^𝛿) for some 𝛿<2𝛼.

We generalize Theorem 2 of Yu [23] from two directions by Theorem 43: first, we extend sequences of 𝐿^𝑝(𝑝≥2) martingale differences to 𝐿^𝛾(𝛾∈(1,2]) sequences of 𝔹-valued martingale differences; secondly, we do not need the condition ∑^∞_𝑗=1|𝑎_𝑛𝑗|²=𝑂(𝑛^𝛿) for some 𝛿<𝛼.

Proof of Theorem 43. Set 𝑌_𝑛𝑗=𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗. From (249), we have ^∞_𝑛=1𝔼𝑚_𝑛𝛾=^∞∞_𝑛=1𝑗=1𝔼𝑌_𝑛𝑗^𝛾=^∞∞_𝑛=1𝑗=1𝑛^−𝛾𝛼𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾<∞. (251) By Corollary 3 and Theorem 2 with 𝑞=1 and 𝐿=0, we see that ℙ𝑛^−𝛼sup_𝑗≥1^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖𝑛>𝜀≤ℙ^−𝛼sup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4+𝜀^−𝛾𝐶₂𝛾𝔼𝑚_𝑛𝛾. (252) By Corollary 3 and Theorem 1, we know that ℙ𝑛^−𝛼sup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤^∞_𝑗=1ℙ𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4. (253) By Markov's inequality, we see that ^∞_𝑗=1ℙ𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤4^𝛾𝜀^∞−𝛾_𝑗=1𝑛^−𝛾𝛼𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾. (254) Since (249) and (251)–(254), we see that (250) holds.

Theorem 44. Let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}_𝑗≥1, 𝑛≥1, be sequences of 𝔹-valued martingale differences. Let {𝑎_𝑛𝑗,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, ^∞_𝑗=1𝑎_𝑛𝑗^𝛾≤𝐾𝑛^𝛿,𝑛≥1,_∞𝑛=1𝑛^∞−1_𝑗=1𝑎_𝑛𝑗^1/𝛼<∞,sup_{𝑛,𝑗≥1}𝔼𝑋_𝑛𝑗^1/𝛼<∞. (255)(256)(257) If 𝔹 is 𝛾-smooth, then (250) holds.

When (𝑋_𝑛𝑗)_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 Theorem 44. Set 𝑌_𝑛𝑗=𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗,𝑗≥1,𝑛≥1, then {(𝑌_𝑛𝑗,ℱ_𝑛𝑗),𝑗≥1,𝑛≥1} is an array of 𝔹-valued martingale differences. Since (255) and (257), for any 𝑛≥1, we have ^∞_𝑛=1𝔼𝑚_𝑛𝛾=^∞∞_𝑛=1𝑗=1𝔼𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗^𝛾=^∞_𝑛=1𝑛^{∞−𝛾𝛼}_𝑗=1𝑎_𝑛𝑗^𝛾𝔼𝑋_𝑛𝑗^𝛾≤𝐶𝐾^∞_𝑛=1𝑛^{−(𝛾𝛼−𝛿)}<∞. (258) By Corollary 3 and Theorem 2 with 𝑞=1 and 𝐿=0, we see that ℙ𝑛^−𝛼sup_𝑗≥1^𝑗_𝑖=1𝑎_𝑛𝑗𝑋_𝑛𝑗𝑛>𝜀≤ℙ^−𝛼sup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4+𝜀^−𝛾𝐶₂𝛾𝔼𝑚_𝑛𝛾. (259) By Corollary 3 and Theorem 1, we know that ℙ𝑛^−𝛼sup_𝑗≥1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤^∞_𝑗=1ℙ𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4. (260) By Markov's inequality, we see that ^∞_𝑗=1ℙ𝑛^−𝛼𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀4≤𝜀^−1/𝛼𝑛^∞−1_𝑗=1𝑎_𝑛𝑗^1/𝛼𝔼𝑋_𝑛𝑗^1/𝛼<∞. (261) By (256), (257), and (258)–(261), we know that (250) holds.

Theorem 45. Let {(𝑋_𝑛𝑗,ℱ_𝑛𝑗)}^𝑛_𝑗=1 be a triangular array of identically distributed 𝔹-valued martingale differences. Let 𝛽>−1, and let {𝑎_𝑛𝑗,1≤𝑗≤𝑛} be a triangular array of positive numbers satisfying 𝑎_𝑛𝑗=𝑐_𝑛𝑗𝑛^−𝛽−1𝑗^𝛽 (262) with 0<𝑐≤𝑐_𝑛𝑗≤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], sup_{𝑛,𝑗≥1}𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}≤𝐾. (263) Suppose that 𝔼𝑋₁₁^1/(1+𝛽)1<∞𝑖𝑓−1<𝛽<−2,𝔼𝑋₁₁²log1+𝑋₁₁1<∞𝑖𝑓𝛽=−2,𝔼𝑋₁₁²1<∞𝑖𝑓𝛽>−2, (264) then for all 𝜀>0, ^∞_𝑛=1ℙ^𝑛_𝑗=1𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀<∞. (265)

When {𝑋_𝑛𝑗}={𝑋_𝑗} are the same sequence of i.i.d. real-valued random variables and ∑^𝑛_𝑗=1𝑎_𝑛𝑗=1 for all 𝑛≥1, Theorem 45 reduces to the sufficiency of Theorem 2.4 of Li et al. [9].

Proof of Theorem 45. Set 𝑌_𝑛𝑗=𝑎_𝑛𝑗𝑋_𝑛𝑗if1≤𝑗≤𝑛,0if𝑗>𝑛, (266) and 𝑐_𝑛=1, then {(𝑌_𝑛𝑗,ℱ_𝑛𝑗)}^𝑛_𝑗=1 are sequences of 𝔹-valued martingale differences. For any 𝑛≥1, since (262) and (263), we have 𝑚_𝑛𝛾=^𝑛_𝑗=1𝔼𝑎_𝑛𝑗𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}=^𝑛_𝑗=1𝑎^𝛾_𝑛𝑗𝔼𝑋_𝑛𝑗^𝛾∣ℱ_{𝑛,𝑗−1}≤𝐾𝐶^𝛾𝑛^{−(𝛾−1)}𝑛^𝑛−1_𝑗=1𝑗𝑛^𝛾𝛽∼𝐾𝐶^𝛾𝑛^{−(𝛾−1)1}_1/𝑛𝑥^𝛾𝛽𝑛d𝑥=𝑂^−𝑏, (267) where 𝑏=min{𝛾(𝛽+1),𝛾−1}. By Corollary 12 of Theorem 10, (265) is implied by ^∞𝑛_𝑛=1𝑗=1ℙ𝑎_𝑛𝑗𝑋_𝑛𝑗>𝜀<∞∀𝜀>0. (268) Equation (268) holds if and only if (264) holds. (cf. [9, pages 62-63])

Acknowledgment

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