A Note on Strong Convergence of Sums of Dependent Random Variables

Abstract

For a sequence of dependent square-integrable random variables and a sequence of positive constants {𝑏_𝑛,𝑛≥1}, conditions are provided under which the series ∑^𝑛_𝑖=1(𝑋_𝑖−𝐸𝑋_𝑖)/𝑏_𝑖 converges almost surely as 𝑛→∞. These conditions are weaker than those provided by Hu et al. (2008).

1. Introduction and Results

Let {𝑋_𝑛,𝑛≥1} be a sequence of square-integrable random variables defined on a probability space (Ω,ℱ,𝑃) and let {𝑏_𝑛,𝑛≥1} be a sequence of positive constants. The random variables {𝑋_𝑛,𝑛≥1} are not assumed to be independent. Past research has focussed on conditions that ensure the strong convergence of two distinct but related series:^𝑛_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖,𝑏_𝑛^𝑛−1_𝑖=1(𝑋_𝑖−𝐸𝑋_𝑖). (1.1) If the second sequence converges to 0 almost surely, then {𝑋_𝑛,𝑛≥1} is said to obey the strong law of large numbers (SLLN).

Assume that there exists a sequence of constants {𝜌_𝑘,𝑘≥1} such that sup_𝑛≥1|Cov(𝑋_𝑛,𝑋_𝑛+𝑘)|≤𝜌_𝑘,𝑘≥1. (1.2) Our interest is in conditions on the growth rates of {Var𝑋_𝑛,𝑛≥1}, {𝑏_𝑛,𝑛≥1}, and {𝜌_𝑘,𝑘≥1} which imply strong convergence of the above series.

There is an extensive literature on strong laws for independent random variables. Strong laws have been derived for various dependence structures such as negative association (e.g., Kuczmaszewska [1]), quasi-stationarity (e.g., Móricz [2], Chobanyan et al. [3]), and orthogonality (e.g., Stout [4]).

Hu et al. [5] focus on the strong convergence of the series without imposing strong conditions on the nature of the variances and covariances. Our aim is to weaken their condition on the covariances and establish the following theorem.

Theorem 1.1. Let {𝑋_𝑛,𝑛≥1} be a sequence of square-integrable random variables and suppose that there exists a sequence of constants {𝜌_𝑘,𝑘≥1} such that (1.2) holds. Let {𝑏_𝑛,𝑛≥1} be a sequence of positive constants. Assume that there exists a constant 𝐾 such that, for all 𝑛≥1, 𝑛𝑏_𝑛≤𝐾. (1.3) Suppose that ^∞_𝑛=1(Var𝑋_𝑛)(log𝑛)²𝑏²_𝑛<∞,_∞𝑘=1𝜌_𝑘𝑘(log𝑘)²<∞. (1.4)(1.5) Then ^𝑛_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖convergesa.s.as𝑛→∞. (1.6)

To motivate the general nature of our result consider the following example. Let {𝑋_𝑛} be a sequence of zero mean random variables where 𝑋_𝑛=𝜉_𝑛+𝜈_𝑛, (1.7) where {𝜉_𝑛} is a stationary time series with autocovariance function {𝛾_𝑘} and {𝜈_𝑛} is a sequence of independent, zero mean random variables distributed independently of {𝜉_𝑛}. Let Var(𝜈_𝑛)=𝜎²_𝑛. Thus what we observe is an underlying stationary series disturbed by a noise process with variance that can depend on 𝑛.

We have Var(𝑋_𝑛)=𝛾₀+𝜎²_𝑛 and Cov(𝑋_𝑛,𝑋_𝑛+𝑘)=𝛾_𝑘(=𝜌_𝑘), 𝑘≥1. Condition (3.1) in Theorem 1 of Hu et al. [5], which is the same as (1.4), is a constraint on the 𝜎²_𝑛 values whereas their condition (3.2) ^∞_𝑘=1𝜌_𝑘𝑘^𝑞<∞,forsome𝑞∈[0,1) (1.8) is a constraint on 𝛾_𝑘. In Chapter 2 of Stout [4] the condition on the variances is shown to be close to optimal for sequences of orthogonal random variables. Lyons [6] provides an SLLN for random variables with bounded variances under the condition ∑^∞_𝑘=1𝜌_𝑘/𝑘<∞. One might conjecture that the condition (1.8) could be relaxed to ∑𝜌_𝑘/𝑘<∞. The above theorem, whilst allowing for far more general models than (1.7), moves us closer to this constraint on the 𝜌_𝑘 values.

For long range dependent stationary processes we have 𝜌_𝑘=𝑂(𝑘^−𝑑𝐿(𝑘)), where 0<𝑑<1 and 𝐿(⋅) is a slowly varying function. Theorem 1.1 enables the strong convergence result to be extended to processes where the correlation decays at a slower rate than 𝑂(𝑘^−𝑑) for 𝑑>0.

Applying Kronecker's lemma the strong law of large numbers result is an immediate consequence of the above theorem.

Corollary 1.2. Under the conditions of Theorem 1.1, if 𝑏_𝑛 is monotone increasing, the strong law of large numbers holds, that is, lim_𝑛→∞∑^𝑛_𝑖=1(𝑋_𝑖−𝐸𝑋_𝑖)𝑏_𝑛=0a.s. (1.9)

There are strong law results under weaker conditions than (1.5) but with stronger conditions on the variance (see, e.g., Lyons [6], Chobanyan et al. [3]). Both papers show that if the summands have bounded variance, then (1.5) can be weakened to ∑^∞_𝑘=1𝜌_𝑘/𝑘<∞. Our approach focusses on the convergence of the series in (1.6) and relies on Kronecker's Lemma to obtain the strong law. If the aim is purely to obtain the SLLN, then alternative conditions might be possible as it is possible to construct sequences {𝑥_𝑛} and {𝑏_𝑛} such that 𝑏_𝑛⁻¹(𝑥₁+⋯𝑥_𝑛)→0 but ∑^𝑛_𝑖=1𝑏_𝑘⁻¹𝑥_𝑘 diverges. For example, take 𝑏_𝑛=𝑛 and 𝑥_𝑛=(log𝑛)⁻¹. Thus we can have the strong law holding but the series in (1.6) diverging.

2. Proofs

Throughout this paper, the symbol 𝐶 denotes a generic constant (0<𝐶<∞) which is not necessarily the same at each appearance. We first prove a number of lemmas that enable us to obtain tighter bounds for key expressions in the proof of Theorem 1 of Hu et al. [5].

Lemma 2.1. Let {𝑋_𝑛,𝑛≥1} be a sequence of square-integrable random variables and suppose that there exists a sequence of constants {𝜌_𝑘,𝑘≥1} such that (1.2) holds and a sequence {𝑏_𝑛} satisfying (1.3). Then for all 𝑛≥0, 𝑚≥𝑛+2, 𝐸(^𝑚_𝑖=𝑛+1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖)²≤^𝑚_𝑖=𝑛+1Var𝑋_𝑖𝑏²_𝑖+^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘𝑘log(1+𝑛). (2.1)

Proof. For all 𝑛≥0, 𝑚≥𝑛+2, 𝐸(^𝑚_𝑖=𝑛+1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖)²=^𝑚_𝑖=𝑛+1Var(𝑋_𝑖)𝑏²_𝑖+2^𝑚−1𝑚_{𝑖=𝑛+1𝑗=𝑖+1}Cov(𝑋_𝑖,𝑋_𝑗)𝑏_𝑖𝑏_𝑗≤^𝑚_𝑖=𝑛+1Var𝑋_𝑖𝑏²_𝑖+𝐶^𝑚−1𝑚_{𝑖=𝑛+1𝑗=𝑖+1}𝜌_𝑗−𝑖=𝑖𝑗_{𝑚𝑖=𝑛+1}Var𝑋_𝑖𝑏²_𝑖+𝐶^𝑚−1_𝑖=𝑛+1^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘(1𝑖−1)=𝑖+𝑘_{𝑚𝑖=𝑛+1}Var𝑋_𝑖𝑏²_𝑖+𝐶^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘(^𝑚−1_𝑖=𝑛+11𝑖−^{𝑚−1+𝑘}_{𝑖=𝑛+𝑘+1}1𝑖)≤^𝑚_𝑖=𝑛+1Var𝑋_𝑖𝑏²_𝑖+𝐶^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘≤(log(𝑛+𝑘)−log(𝑛))_{𝑚𝑖=𝑛+1}Var𝑋_𝑖𝑏²_𝑖+𝐶^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘𝑘log(1+𝑛). (2.2)

Lemma 2.2. For 0<𝑒²≤𝑘≤𝑛, 𝑘log(1+𝑛)≤(log𝑘)log𝑛². (2.3)

Proof. Note that 𝑥/(log𝑥)² is an increasing function for 𝑥≥𝑒²>0. Thus, for 𝑥≥𝑘>𝑒², 𝑥(log𝑥)²≥𝑘(log𝑘)². (2.4) Hence for 𝑛≥𝑘>𝑒², 𝑘log(1+𝑛𝑘)≤𝑛≤(log𝑘)log𝑛². (2.5)

Lemma 2.3. For 𝑎>0, define 𝑆_𝑖(𝑎)=^∞_𝑛=𝑎𝑛^𝑖2^𝑛,𝑖=0,1,…. (2.6) Then 𝑆₀(𝑎)=2^{−(𝑎−1)}, 𝑆₁(𝑎)=(𝑎+1)2^{−(𝑎−1)} and, in general, 𝑆_𝑗𝑎(𝑎)=^𝑗2^𝑎−1+^𝑗−1_𝑖=0𝑗𝑖𝑆_𝑖(𝑎). (2.7)

Proof. The result for 𝑆₀(𝑎) is the sum of a standard geometric progression. The general result follows by noting 2𝑆_𝑗(𝑎)=^∞_𝑛=𝑎𝑛^𝑗2^𝑛−1=𝑎^𝑗2^𝑎−1+^∞_𝑛=𝑎(𝑛+1)^𝑗2^𝑛=𝑎^𝑗2^𝑎−1+^∞_𝑛=𝑎𝑛^𝑗2^𝑛+^𝑗−1∞_{𝑖=0𝑛=𝑎}𝑗𝑖𝑛^𝑖2^𝑛. (2.8) Thus 𝑆_𝑗𝑎(𝑎)=^𝑗2^𝑎−1+^𝑗−1_𝑖=0𝑗𝑖𝑆_𝑖(𝑎). (2.9)

Proof of Theorem 1.1. We will follow the method of proof in Theorem 1 in Hu et al. [5]. To prove (1.6) we first show that {∑^𝑛_𝑖=1((𝑋_𝑖−𝐸𝑋_𝑖)/𝑏_𝑖),𝑛≥1} is a Cauchy sequence for convergence in 𝐿₂ which will imply convergence in probability. Using Lemmas 2.1 and 2.2, sup_𝑚>𝑛𝐸(^𝑚_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖−^𝑛_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖)²=sup_𝑚>𝑛𝐸(^𝑚_𝑖=𝑛+1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖)²≤sup_𝑚>𝑛(^𝑚_𝑖=𝑛+1Var𝑋_𝑖𝑏²_𝑖+𝐶^{𝑚−𝑛−1}_𝑘=1𝜌_𝑘𝑘𝑘log(1+𝑛≤)),byLemma2.1,_{∞𝑖=𝑛+1}Var𝑋_𝑖𝑏²_𝑖+𝐶⁸_𝑖=1𝜌_𝑘𝑘𝑘log(1+𝑛)+𝐶^𝑛_𝑘=9𝜌_𝑘𝑘(log𝑘)log𝑛²+^∞_𝑘=𝑛+1𝜌_𝑘𝑘log(𝑘)→0as𝑛→∞. (2.10) Therefore there exists a random variable 𝑆∈𝐿₂ such that 𝑆_𝑛=^𝑛_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖^𝑝−→𝑆. (2.11)
Next we will show that 𝑆₂^𝑛→𝑆 a.s. Let 𝜀>0 be arbitrary. Note ^∞_𝑛=1𝑃{|^2𝑛_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝑏_𝑖≤1−𝑆|>𝜀}𝜀^2∞_𝑛=1(^∞_𝑖=2𝑛+1Var𝑋_𝑖𝑏²_𝑖+𝐶^∞_𝑘=1𝜌_𝑘𝑘𝑘log(1+2^𝑛)),byLemma2.1,=𝐶^∞_𝑖=3^[log2𝑖]_𝑛=1Var𝑋_𝑖𝑏²_𝑖+𝐶^∞_𝑛=1(^2𝑛_𝑘=1𝜌_𝑘𝑘𝑘log(1+2^𝑛)+^∞_𝑘=2𝑛+1𝜌_𝑘𝑘𝑘log(1+2^𝑛))≤𝐶^∞_𝑖=2^[log2𝑖]_𝑛=1Var𝑋_𝑖𝑏²_𝑖+𝐶^∞_𝑘=1(^∞_{𝑛=[log2𝑘]}𝜌_𝑘𝑘(log𝑘)²(log2^𝑛)²+^[log2𝑘]_𝑛=1𝜌_𝑘𝑘(1+log𝑘)),byLemma2.2,≤𝐶^∞_𝑖=2^[log2𝑖]_𝑛=1Var𝑋_𝑖𝑏²_𝑖+𝐶^∞_𝑘=1𝜌_𝑘𝑘log𝑘+𝐶^∞_𝑘=1𝜌_𝑘𝑘(log𝑘)²<∞, (2.12) where the last line follows by using (1.4) and (1.5). Thus by the Borel Cantelli lemma 𝑆₂^𝑛→𝑆 almost surely. To finish the proof we utilize the generalization of the Rademacher-Menchoff maximal inequality given by Serfling [7] and argue as in Hu et al. [5]. It is sufficient to show that, for any 𝜀>0, ^∞_𝑛=1𝑃{max_{2𝑛−1<𝑘≤2𝑛}|𝑆_𝑘−𝑆₂^𝑛−1|>𝜀}<∞. (2.13) Using Serfling's inequality and (3.8) from Hu et al. [5] ^∞_𝑛=1𝑃{max_{2𝑛−1<𝑘≤2𝑛}|𝑆_𝑘−𝑆₂^𝑛−1|>𝜀}≤1+𝐶^∞2_𝑛=2^𝑛_{𝑖=2𝑛−1+1}(Var𝑋_𝑖)(log𝑖)²𝑏²_𝑖+𝐶^∞_𝑛=2𝑛^{22𝑛−1−1}_𝑘=1𝜌_𝑘𝑘𝑘log(1+2^𝑛)≤1+𝐶^∞_𝑖=1(Var𝑋_𝑖)(log𝑖)²𝑏²_𝑖+𝐶^∞∞_{𝑘=1𝑛=1+[log𝑘]}𝑛²(𝜌_𝑘𝑘𝑘2^𝑛)≤1+𝐶^∞_𝑖=1(Var𝑋_𝑖)(log𝑖)²𝑏²_𝑖+𝐶^∞∞_{𝑘=1𝑛=1+[log𝑘]}(𝑛²2^𝑛)𝜌_𝑘≤1+𝐶^∞_𝑖=1(Var𝑋_𝑖)(log𝑖)²𝑏²_𝑖+𝐶^∞_𝑘=1𝜌_𝑘𝑘(log𝑘)²,byLemma2.3,<∞. (2.14)