The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences ^††thanks: The paper was supported by NAFOSTED, Grant No. 101.03-2015.11.

The motivation of this paper is an open question raised recently by Chen and Sung ChenSung14 . Let $1<\alpha\leq 2$, $\gamma>0$ and let $\{X,X_{n},n\geq 1\}$ be a sequence of negatively associated and identically distributed random variables with $E(X)=0$. Sung Sung11 proved that if

then

where $\{a_{ni},n\geq 1,1\leq i\leq n\}$ are constants satisfying

Here and thereafter, $\log$ denotes the logarithm to the base $2$. Chen and Sung ChenSung14 proved that for the case where $\gamma>\alpha$, the condition $E|X|^{\gamma}<\infty$ is optimal. They raised an open question about finding the optimal condition for (1.2) when $\gamma\leq\alpha$. For the case where $\gamma<\alpha$, Chen and Sung (ChenSung14, , Corollary 2.2) proved that (1.2) holds under an almost optimal condition that

In this note, by using some results related to regularly varying functions, we provide the necessary and sufficient conditions for

where $\tilde{L}(\cdot)$ is the de Bruijn conjugate of a slowly varying function $L(\cdot)$, defined on $[A,\infty)$ for some $A>0$. This result is new even when the random variables are i.i.d. By letting $L(x)\equiv\log^{-1/\gamma}(x),\ x\geq 2$, we obtain optimal moment condition for (1.2).

Weak laws of large numbers with the norming constants are of the form $n^{1/\alpha}\tilde{L}(n^{1/\alpha})$ were studied by Gut Gut04 , and Matsumoto and Nakata MN13 . The Marcinkiewicz–Zygmund strong law of large numbers has been extended and generalized in many directions by a number of authors, see BaKa ; DedeckerMerlevede ; GutStadmueller ; HechnerHeinkel ; Rio95a ; Rio95b ; Szewczak and references therein. To our best knowledge, there is not any result in the literature that considers strong law of large numbers with general normalizing constants $n^{1/\alpha}\tilde{L}(n^{1/\alpha})$ except Gut and Stadmüller GutStadmueller who studied the Kolmogorov strong law of large numbers, but for delay sums. The main result of this paper fills this gap. Recently, Miao et al. MMX have studied the Marcinkiewicz–Zygmund-type strong law of large numbers where the norming constants are of the form $n^{1/\alpha}\log^{\beta/\alpha}n$ for some $\beta\geq 0$, which is a special case of our result.

The concept of negative association of random variables was introduced by Joag-Dev and Proschan JoPr . A collection $\{X_{1},\dots,X_{n}\}$ of random variables is said to be negatively associated if for any disjoint subsets $A,B$ of $\{1,\dots,n\}$ and any real coordinatewise nondecreasing functions $f$ on ${\mathbb{R}}^{|A|}$ and $g$ on ${\mathbb{R}}^{|B|}$,

whenever the covariance exists, where $|A|$ denotes the cardinality of $A$. A sequence $\{X_{n},n\geq 1\}$ of random variables is said to be negatively associated if every finite subfamily is negatively associated.

There is a weaker concept of dependence called negative dependence, which was introduced by Lehmann Lehmann66 and further investigated by Ebrahimi and Ghosh EG81 and Block et al. BSS . A collection of random variables $\{X_{1},\dots,X_{n}\}$ is said to be negatively dependent if for all $x_{1},\dots,x_{n}\in\mathbb{R}$,

and

A sequence of random variables $\{X_{i},i\geq 1\}$ is said to be negatively dependent if for any $n\geq 1$, the collection $\{X_{1},\dots,X_{n}\}$ is negatively dependent. A sequence of random variables $\{X_{i},i\geq 1\}$ is said to be pairwise negatively dependent if for all $x,y\in\mathbb{R}$ and for all $i\not=j$,

It is well known and easy to prove that $\{X_{i},i\geq 1\}$ is pairwise negatively dependent if and only if for all $x,y\in\mathbb{R}$ and for all $i\not=j$,

By Joag-Dev and Proschan (JoPr, , Property P3), negative association implies negative dependence. For examples about negatively dependent random variables which are not negatively associated, see (JoPr, , p. 289). Of course, pairwise independence implies pairwise negative dependence, but pairwise independence and negative dependence do not imply each other. Joag-Dev and Proschan JoPr pointed out that many useful distributions enjoy the negative association properties (and therefore, they are negatively dependent) including multinomial distribution, multivariate hypergeometric distribution, Dirichlet distribution, strongly Rayleigh distribution and distribution of random sampling without replacement. Limit theorems for negatively associated and negatively dependent random variables have received extensive attention. We refer to JiLi ; Matula92 ; Shao00 and references therein. These concepts of dependence can be extended to the Hilbert space-valued random variables; see, e.g., BDD ; HTV ; KKH ; Thanh13 , among others.

The rest of the paper is arranged as follows. Section 2 presents some results on slowly varying functions needed in proving the main results. Section 3 focuses on complete convergence for weighted sums of negatively associated and identically distributed random variables. In Sect. 4, we apply a result concerning slowly varying functions developed in Sect. 2 to give a counterpart of Martikainen’s strong law of large numbers (see Martikainen ) for sequences of pairwise negatively dependent and identically distributed random variables. As an application, we prove a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

Some technical results concerning slowly varying functions will be presented in this section.

The notion of regularly varying function can be found in (Seneta76, , Chapter 1). A real-valued function $R(\cdot)$ is said to be regularly varying with index of regular variation $\rho$ ($\rho\in\mathbb{R}$) if it is a positive and measurable function on $[A,\infty)$ for some $A>0$, and for each $\lambda>0$,

A regularly varying function with the index of regular variation $\rho=0$ is called slowly varying. It is well known that a function $R(\cdot)$ is regularly varying with the index of regular variation $\rho$ if and only if it can be written in the form

where $L(\cdot)$ is a slowly varying function (see, e.g., (Seneta76, , p. 2)). On the regularly varying functions and their important role in probability, we refer to Seneta Seneta76 , Bingham, Goldie and Teugels BGT , and more recent survey paper by Jessen and Mikosch JeMi . Regular variation is also one of the key notions for modeling the behavior of large telecommunications networks; see, e.g., Heath et al. HRS , Mikosch et al. MRRS .

The basic result in the theory of slowly varying functions is the representation theorem (see, e.g., (BGT, , Theorem 1.3.1)) which states that for a positive and measurable function $L(\cdot)$ defined on $[A,\infty)$ for some $A>0$, $L(\cdot)$ is slowly varying if and only if it can be written in the form

for some $B\geq A$ and for all $x\geq B$, where $c(\cdot)$ is a positive bounded measurable function defined on $[B,\infty)$ satisfying $\lim_{x\to\infty}c(x)=c\in(0,\infty)$ and $\varepsilon(\cdot)$ is a continuous function defined on $[B,\infty)$ satisfying $\lim_{x\to\infty}\varepsilon(x)=0$. Seneta Seneta73 (see also in (BGT, , Lemma 1.3.2)) proved that if $L(\cdot)$ is a slowly varying function defined on $[A,\infty)$ for some $A>0$, then there exists $B\geq A$ such that $L(x)$ is bounded on every finite closed interval $[a,b]\subset[B,\infty)$.

Let $L(\cdot)$ be a slowly varying function. Then by (BGT, , Theorem 1.5.13), there exists a slowly varying function $\tilde{L}(\cdot)$, unique up to asymptotic equivalence, satisfying

The function $\tilde{L}$ is called the de Bruijn conjugate of $L$, and $\left(L,\tilde{L}\right)$ is called a (slowly varying) conjugate pair (see, e.g., (BGT, , p. 29)). By (BGT, , Proposition 1.5.14), if $\left(L,\tilde{L}\right)$ is a conjugate pair, then for $a,b,\alpha>0$, each of $\left(L(ax),\tilde{L}(bx)\right)$, $\left(aL(x),a^{-1}\tilde{L}(x)\right),$ $\left(\left(L(x^{\alpha})\right)^{1/\alpha},(\tilde{L}(x^{\alpha}))^{1/\alpha}\right)$ is a conjugate pair. Bojanić, R. and Seneta Bojanic71 (see also Theorem 2.3.3 and Corollary 2.3.4 in BGT ) proved that if $L(\cdot)$ is a slowly varying function satisfying

for some $\lambda_{0}>1$, then for every $\alpha\in\mathbb{R}$,

and therefore, we can choose (up to aymptotic equivalence) $\tilde{L}(x)=1/L(x)$. In particular, if $L(x)=\log(x)$ then $\tilde{L}(x)=1/\log(x)$.

The following lemma follows from Theorem 1.5.12 and Proposition 1.5.15 in BGT . Here and thereafter, for a slowly varying function $L(\cdot)$ defined on $[A,\infty)$ for some $A>0$, we denote the Brujin conjugate of $L(\cdot)$ by $\tilde{L}(\cdot)$. Without loss of generality, we assume that $\tilde{L}(\cdot)$ is also defined on $[A,\infty)$, and that $L(x)$ and $\tilde{L}(x)$ are both bounded on finite closed intervals.

The second lemma shows that we can approximate a slowly varying function $L(\cdot)$ by a differentiable slowly varying function $L_{1}(\cdot)$. See Galambos and Seneta (GS73, , p. 111) for a proof.

Because of Lemma 2.2, we can work with differentiable slowly varying functions $L(\cdot)$ that satisfy (2.7) in our setting.

The proof of Lemma 2.3 (i) follows from direct calculations (by taking the derivative). Lemma 2.3 (ii) is an easy consequence of the representation theorem stated above.

The following lemma is a direct consequence of Karamata’s theorem (see (BGT, , Theorem 1.5.10)) as was so kindly pointed out to us by the referee.

The following proposition gives a criterion for $E\left(|X|^{\alpha}L^{\alpha}(|X|+A)\right)<\infty$.

In the following theorem, we establish complete convergence for weighted sums of negatively associated and identically distributed random variables. Theorem 3.1 is new even when the random variables are i.i.d. A special case of this result comes close to a solution of an open question of Chen and Sung ChenSung14 . In subsequent derivations, the symbol $C$ denotes a generic positive constant whose value may be different for each appearance.