On weak laws of large numbers for maximal partial sums of pairwise independent random variables

A real-valued function $L(\cdot)$ is said to be slowly varying (at infinity) if it is a positive and measurable function on $[A,\infty)$ for some $A\geq 0$, and for each $\lambda>0$,

In [6], de Bruijn proved that if $L(\cdot)$ is a slowly varying function, then there exists a slowly varying function $\tilde{L}(\cdot)$, unique up to asymptotic equivalence, satisfying

The function $\tilde{L}(\cdot)$ is called the de Bruijn conjugate of $L(\cdot)$ ([2, p. 29]). Bojanić and Seneta [3] showed that for most of “nice” slowly varying functions, we can choose $\tilde{L}(x)=1/L(x)$. Especially, if $L(x)=\log^{\gamma}(x)$ or $L(x)=\log^{\gamma}(\log(x))$ for some $\gamma\in\mathbb{R}$, then $\tilde{L}(x)=1/L(x)$. Here and thereafter, for a real number $x$, $\log(x)$ denotes the natural logarithm (base $\mathrm{e}$) of $\max\{x,\mathrm{e}\}$.

Let $L(\cdot)$ be a slowly varying function and let $r>0$. By using a suitable asymptotic equivalence version (see Lemma 2.2 and Lemma 2.3 (i) in Anh et al. [1]), we can assume that $L(\cdot)$ is positive and differentiable on $[a,\infty)$, and $x^{r}L(x)$ is strictly increasing on $[a,\infty)$ for some large $a$. Next, let $L_{1}(\cdot)$ be a slowly varying function with $L_{1}(0)=0$ with a linear growth to $L(a)$ over $[0,a)$, and $L_{1}(x)\equiv L(x)$ on $[a,\infty)$. Then (i) $L_{1}(x)$ is continuous on $[0,\infty)$ and differentiable on $[a,\infty)$, and (ii) $x^{r}L_{1}(x)$ is strictly increasing on $[0,\infty)$. In this paper, we will assume, without loss of generality, that properties (i) and (ii) are fulfilled for all slowly varying functions.

The starting point of the current research is the following weak law of large numbers (WLLN) which was proved by Gut [8]. Hereafter, $\mathbf{1}(A)$ denotes the indicator function of a set $A$.

The above WLLN has been extended in several directions, see [12, 13] for WLLNs with random indices for arrays of independent random variables taking values in Banach spaces, and see [4, 5, 9, 10] and the references therein for WLLNs for dependent random variables and dependent random vectors. Boukhari [4, Theorem 1.2] showed that for $0<p<1$, condition (1.2) implies

irrespective of the joint distribution of the $X_{n}$’s. Boukhari [4] presented an example showing that his result does not hold when $p=1$. The proof of the sufficient part of Theorem 1 in [8] works well with pairwise independent random variables since we do not involve the maximal partial sums. Krulov [10] and Chandra [5] established WLLNs for maximal partial sums for the case where the summands are negatively associated and asymptotically almost negatively associated, respectively. The authors in [5, 10] considered general normalizing constants, which showed that the sufficient part of Theorem 1 also holds for $1\leq p<2$. However, the method used in [5, 10] requires a Kolmogorov-type maximal inequality (see Lemma 1.2 in [5]) which does not hold for pairwise independent random variables.

The aim of this paper is to establish WLLNs for maximal partial sums of pairwise independent random variables thereby extending the sufficient part of Theorem 1 for the case $p=1$ to WLLN for maximal partial sums from sequences of pairwise independent random variables. We use a technique developed by Rio [11] to avoid using the Kolmogorov maximal inequality. In addition, we also establish a WLLN for maximal partial sums of pairwise independent random variables under a uniform integrability condition, and present an example to show that this result does not hold in general if the uniform integrability assumption is weakened to the uniform boundedness of the moments.

Let $\Lambda$ be a nonempty index set. A family of random variables $\{X_{\lambda},\lambda\in\Lambda\}$ is said to be stochastically dominated by a random variable $X$ if

Some authors use an apparently weaker definition of $\{X_{\lambda},\lambda\in\Lambda\}$ being stochastically dominated by a random variable $Y$, namely that

for some constants $C_{1},C_{2}\in(0,\infty)$. It is shown recently by Rosalsky and Thành [14] that (1.4) and (1.5) are indeed equivalent. We note that if (1.4) is satisfied, then for all $t>0$ and $r>0$

and

The following theorem is the main result of this paper.

We postpone the proof of Theorem 2 to Section 2. From Theorem 3.2 of Boukhari [4], we have that if $\{X_{n},n\geq 1\}$ is a sequence of pairwise independent random variables, and $\{b_{n},n\geq 1\}$ is a sequence of positive constants, then

By using Theorem 2 and (1.8), we obtain the following corollary.

Theorem 2 also enables us to establish a WLLN for maximal partial sums of pairwise independent random variables under a uniform integrability condition. After this paper was submitted, Thành [16, Corollary 4.10] established a similar WLLN for triangular arrays of random variables satisfying a Kolmogorov-type maximal inequality. Theorem 4 and Corollary 4.10 of Thành [16] do not imply each other.

Hereafter, we denote the de Bruijn conjugate of a slowly varying function $L(\cdot)$ by $\tilde{L}(\cdot)$.

The following example shows that in Theorem 4, the assumption that $\{|X_{n}|^{p}L(|X_{n}|^{p}),n\geq 1\}$ is uniformly integrable cannot be weakened to

The following lemma plays an important role in the proof of Theorem 2. It gives a general approach to the WLLN. In this lemma, we do not require any dependence structure. Throughout this section, we use the symbol $C$ to denote a universal positive constant which is not necessarily the same in each appearance.

This paper establishes WLLNs for maximal partial sums of pairwise independent random variables without using the Kolmogorov maximal inequality. The method can be easily adapted to dependent random variables. We have the following result: