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The almost sure local central limit theorem for products of partial sums under negative association
Journal of Inequalities and Applications volume 2018, Article number: 275 (2018)
Abstract
Let \(\{X_{n}, n\geq1\}\) be a strictly stationary negatively associated sequence of positive random variables with \(\mathrm{E}X_{1}=\mu>0\) and \(\operatorname{Var}(X_{1})=\sigma^{2}<\infty\). Denote \(S_{n}=\sum_{i=1}^{n}X_{i}, p_{k}=\mathrm{P}(a_{k}\leq ({\prod}_{j=1}^{k}S_{j}/(k!\mu^{k}) )^{1/(\gamma\sigma_{1} \sqrt{k})}< b_{k})\) and \(\gamma=\sigma/\mu\) the coefficient of variation. Under some suitable conditions, we derive the almost sure local central limit theorem
where \(\sigma_{1}^{2}=1+\frac{1}{\sigma^{2}}\sum_{j=2}^{\infty}\operatorname{Cov}(X_{1},X_{j})>0\).
1 Introduction
Definition 1.1
([1])
A finite family of random variables \(X_{1},X_{2},\ldots,X_{n},n\geq2\), is said to be negatively associated (NA) if, for every pair of disjoint subsets A and B of \(\{1,2,\ldots,n\}\), we have
where \(f_{1}\) and \(f_{2}\) are coordinatewise increasing and the covariance exists. An infinite family of random variables (r.v.) is NA if every finite subfamily is NA.
Obviously, if \(\{X_{i}, i\geq1\}\) is NA, and \(\{f_{i}, i\geq1\}\) is a sequence of nondecreasing (or nonincreasing) functions, then \(\{f_{i}(X_{i}), i\geq1\}\) is also NA. We refer to Roussas [2] for NA’s fundamental properties and applications in several fields, Shao [3] for the moment inequalities, Jing and Liang [4] and Cai [5] for the strong limit theorems, Chen et al. [6] and Sung [7] for the complete convergence.
Let \(S_{n}:=\sum_{i=1}^{n}X_{i}\) denote the partial sum of \(\{X_{i}, i\geq1\} \) and \(\prod_{j=1}^{n}S_{j}\) is known as a product of partial sum \(S_{j}\), the study on partial sum has received extensive attention. Such well-known classic laws as the central limit theorem (CLT), the almost sure central limit theorem (ASCLT), and law of the iterated logarithm (LIL) are known for characterizing the asymptotic behavior of \(S_{n}\). However, the study of asymptotic behavior for product of partial sum is not so far, it was initiated by Arnold and Villaseñor [8]. This paper intends to study the limit behavior of product \(\prod_{j=1}^{n}S_{j}\) under negative association.
Let \(\{X_{n}, n\geq1\}\) be a strictly stationary NA sequence of positive r.v. with \(\mathrm{E}X_{1}=\mu>0\), \(\operatorname{Var}(X_{1})=\sigma^{2}<\infty\), and the coefficient of variance \(\gamma=\sigma/\mu\). Assume that
-
1.
Li and Wang [9] obtained the following version of the CLT:
$$ \biggl(\frac{\prod_{j=1}^{n}S_{j}}{n!\mu^{n}} \biggr)^{1/(\gamma\sigma_{1} \sqrt{n})} \stackrel{\mathrm{d}}{ \rightarrow} \exp(\sqrt{2}\mathcal {N}), \quad \mbox{as } n\rightarrow \infty, $$(1.3)where \(\mathcal{N}\) is a standard normal distribution random variable.
-
2.
Li and Wang [10] proved the following ASCLT:
$$ \lim_{n\rightarrow\infty}\frac{1}{\log n}\sum_{k=1}^{n} \frac{1}{k}\mathrm{I} \biggl\{ \biggl(\frac{\prod_{j=1}^{k}S_{j}}{k!\mu^{k}} \biggr)^{1/(\gamma\sigma_{1} \sqrt{k})}\leq x \biggr\} =F(x)\quad \mbox{a.s. for all }x\in \mathbb{R}, $$(1.4)
here and elsewhere, \(\mathrm{I}\{A\}\) represents the indicative function of the event A and \(F(\cdot)\) is the distribution function of the log-normal random variable \(\exp(\sqrt {2}\mathcal{N})\).
The almost sure central limit theorem was proposed by Brosamler [11] and Schatte [12]. In recent years, the ASCLT has been extensively studied, and an attractive research direction is to prove it under associated or dependent situations. There are some literature works for \(\alpha, \rho, \phi\)-mixing and associated random variables, we refer to Matuła [13], Lin [14], Zhang et al. [15], Matuła and Stȩpień [16], Hwang [17], Li [18], Miao and Xu [19], Wu and Jiang [20].
A more general version of ASCLT for products of partial sums was proved by Weng et al. [21]. The following theorem is due to them.
Theorem A
Let \(\{X_{n},n\geq1\}\) be a sequence of independent and identically distributed positive random variables with \(\mathrm{E}X_{1}^{3}<\infty, \mathrm{E}X_{1}=\mu,\operatorname{Var}(X_{1})=\sigma^{2}, \gamma=\sigma/\mu\). \(a_{k}, b_{k}\) satisfy
Let
and assume for sufficiently large k, \(p_{k}\geq1/(\log k)^{\delta_{1}}\) for some \(\delta_{1}>0\). Then we have
This result may be called almost sure local central limit theorem (ASLCLT) for the product \(\prod_{j=1}^{n}S_{j}\) of independent and identically distributed positive r.v., while (1.4) may be called almost sure global central limit theorem (ASGCLT).
The ASLCLT for partial sums of independent and identically distributed r.v. was stimulated by Csáki et al. [22], and Khurelbaatar [23] extended it to the case of ρ-mixing sequences, Jiang and Wu [24] extended it to the case of NA sequences. Zang [25] obtained the ASLCLT for a sample range.
In this paper, our concern is to give a common generalization of (1.7) to the case of NA sequences. The remainder of the paper is organized as follows. Section 2 provides our main result. Section 3 gives some auxiliary lemmas. The proofs of the theorem and some lemmas are in Sect. 4.
2 Main results
In the following, we assume that \(\{X_{n},n\geq1\}\) is a strictly stationary negatively associated sequence of positive r.v.’s with \(\mathrm{E}X_{1}=\mu>0,\operatorname{Var}(X_{1})=\sigma^{2}<\infty,\mathrm {E}X_{1}^{3}<\infty\), the coefficient of variation \(\gamma=\sigma/\mu\). \(a_{k}, b_{k}\) satisfy
and
Then we study the asymptotic behavior of the logarithmic average
where the expression in the sum above is defined to be one if the denominator is zero. That is, let \(\{a_{n},n\geq1\}\) and \(\{b_{n},n\geq1\}\) be two sequences of real numbers and
Therefore, we should study the asymptotic limit properties of \(\frac {1}{\log n}\sum_{k=1}^{n}\frac{\alpha_{k}}{k}\) under suitable conditions.
In the following discussion, we shall use the definition of the Cox–Grimmett coefficient
and we can verify that the formula
is correct for a stationary sequence of negatively associated random variables.
In the following, \(\xi_{n}\sim\eta_{n}\) denotes \(\xi_{n}/\eta_{n}\rightarrow1\), \(n\rightarrow\infty\). \(\xi_{n}=O(\eta_{n})\) denotes that there exists a constant \(c>0\) such that \(\xi_{n}\leq c\eta_{n}\) for sufficiently large n. The symbols \(c, c_{1}, c_{2}, \ldots \) represent generic positive constants.
Theorem 2.1
Let \(\{X_{n},n\geq1\}\) be a strictly stationary negatively associated sequence of positive r.v. with \(\mathrm {E}X_{1}=\mu>0,\operatorname{Var}(X_{1})=\sigma^{2}<\infty,\mathrm {E}X_{1}^{3}<\infty, \gamma=\sigma/\mu\). \(a_{k}, b_{k}\) satisfy (2.1), assume that (1.1) and (1.2) hold, and
and
for sufficiently large k and some \(0<\delta_{1}<1/4\). Then we have
where \(\alpha_{k}\) is defined by (2.5).
Remark 2.2
Let \(a_{k}=0\) and \(b_{k}=x\) in (2.3). By CLT (1.3), we have
Obviously (2.9) holds, then (2.10) becomes (1.4), which is the ASGCLT. Thus the ASLCLT is a general result which contains the ASGCLT.
3 Auxiliary lemmas
In order to prove the main theorem, we need to use the concept of a triangular array of random variables. Let \(b_{k,n}=\sum_{i=k}^{n}1/i\) and \(Y_{i}=(X_{i}-\mu)/\sigma\). We define a triangular array \(Z_{1,n},Z_{2,n},\ldots,Z_{n,n}\) as \(Z_{k,n}=b_{k,n}Y_{k}\) and put \(S_{k,n}=Z_{1,n}+Z_{2,n}+\cdots+Z_{k,n}\) for \(1\leq k\leq n\). Let
where
Note that, for \(l>k\), we have
So, by the property of NA sequences, \(S_{l,l}-S_{k,k}-b_{k+1,l}\widetilde{S}_{k}\) and \(U_{k}\) are negatively associated.
The following Lemma 3.1 is due to Liang et al. [26].
Lemma 3.1
Let \(\{X_{n},n\geq1\}\) be a sequence of NA random variables with \(\mathrm{E}X_{n}=0\) and \(\{a_{ni},1\leq i\leq n,n\geq1\}\) be an array of real numbers such that \(\sup_{n}\sum_{i=1}^{n}a_{ni}^{2}<\infty\) and \(\max_{1\leq i\leq n}|a_{ni}|\rightarrow0\) as \(n\rightarrow\infty\). Assume that \(\sum_{j:|k-j|\geq n}|\operatorname{Cov}(X_{k},X_{j})|\rightarrow0\) as \(n\rightarrow\infty\) uniformly for \(k\geq1\). If \(\operatorname{Var}(\sum_{i=1}^{n}a_{ni}X_{i})=1\) and \(\{X_{n}^{2},n\geq1 \}\) is a uniformly integrable family, then \(\sum_{i=1}^{n}a_{ni}X_{i} \stackrel{\mathrm {d}}{\rightarrow} \mathcal{N}\), where \(\mathcal{N}\) is a standard normal distribution random variable.
Now we obtain the CLT for triangular arrays.
Lemma 3.2
Let \(\{Y_{n},n\geq1\}\) be a strictly stationary sequence of negatively associated random variables with \(\mathrm{E}Y_{1}=0, \operatorname{Var}(Y_{1})=1\) and \(\sigma _{1}^{2}=1+\sum_{j=2}^{\infty}\operatorname{Cov}(Y_{1},Y_{j})>0\). Suppose that there exist constants \(\delta_{2}\) and \(\delta_{3}\) such that \(0 < \delta _{2}, \delta_{3} < 1\). Assume also that (1.1) and (1.2) hold. If
for sufficiently large n, then
The proof is quite long and will be left to Sect. 4.
The following Lemma 3.3 is a corollary to Corollary 2.2 in Matuła [27] under a strictly stationary condition.
Lemma 3.3
If the conditions of Lemma 3.2 and (2.8) hold, assume also \(\mathrm{E}|Y_{1}|^{3}<\infty\). Let
Then we have
and
Lemma 3.4
If the conditions of Theorem 2.1 hold, and assume that there exists \(\delta_{4}\) such that \(0<\delta_{1}<\delta_{4}<1/4\). Let \(\varepsilon_{l}=1/(\log l)^{\delta_{4}}\), where \(l=3,4,\ldots,n\), then the following asymptotic relations hold:
where \(\mathcal{H}:=\{(k,l):1\leq k< l\leq n, \log l>(\log n)^{\delta_{2}}\textit{ and }k< l/(\log l)^{2+\delta_{3}}\}\) and \(0<\epsilon<1-2(\delta_{1}+\delta_{4})\).
The proof will be left to Sect. 4.
The following result is due to Khurelbaatar [23].
Lemma 3.5
Assume that \(\{\xi_{n}, n\geq1\}\) is a non-negative random sequence such that \(\mathrm{E}\xi_{k}=1, k=1,2,\ldots \) , and
for some \(\epsilon>0\) and positive constant c, then
The following Lemma 3.6 is obvious.
Lemma 3.6
Assume that the non-negative random sequence \(\{\xi_{n}, n\geq1\}\) satisfies (3.14) and the sequence \(\{\eta_{n}, n\geq1\}\) is such that, for any \(\varepsilon>0\), there exists \(k_{0}=k_{0}(\varepsilon,\omega)\) for which
Then we also have
4 Proofs of the main result and lemmas
The main aspect of our proof of Theorem 2.1 is verification condition (3.13) for \(\alpha_{k}\), where \(\alpha_{k}\) is defined by (2.5). We use ASCLT (1.4) with remainders and the following elementary inequalities:
with some constant c. Moreover, for each \(k > 0\), there exists \(c_{1}= c_{1}(k)\) such that
Proof of Theorem 2.1
Let
Thus, \(-\infty\leq\hat{a}_{k}\leq0\leq\hat{b}_{k}\leq\infty\) by (2.1). By the definition of \(U_{k}\) in (3.1), we have \(p_{k}=\mathrm{P}(\hat{a}_{k}\leq U_{k}<\hat{b}_{k})\) and
First assume that
with some constant c. Note that
where \(\delta_{2},\delta_{3}\) are defined by Lemma 3.2. Note also that \(\operatorname{Var}(\alpha_{k})=0\) if \(p_{k}= 0\) and
And by the condition of (2.9), we have
If either \(p_{k}=0\) or \(p_{l}=0\), then obviously \(\operatorname{Cov}(\alpha _{k},\alpha_{l})=0\), so we may assume that \(p_{k}p_{l}\neq0\), by (2.1), we have
for \(\delta_{1}<1/4\) and \(\delta_{2}<7/8\). Now we estimate the bound of ∑3. Let \(A_{n}\) be an integer such that \(\log A_{n} \sim(\log n)^{\delta_{2}}\) for sufficiently large n. Then
So, it remains to estimate the bound of ∑4. Let \(1\leq k< l\) and \(\varepsilon_{l}=1/(\log l)^{\delta_{4}}\), where \(0<\delta_{1}<\delta _{4}<1/4\), we have
where
and
So by (3.3), Lemma 3.3, and (4.1), we obtain
So, by using Lemma 3.4, we have
Combining (4.7)–(4.10) implies that
Hence applying Lemma 3.5, our theorem is proved under the restricting condition (4.5).
Then, we remove the restricting condition (4.5). Fix \(x>0\) and define
Clearly \(\widetilde{b}_{k}-\widetilde{a}_{k}\leq\min(2x,c)\) and \(\widetilde {p}_{k}\leq p_{k}\), so assuming \(\widetilde{p}_{k}\neq0\), then we also have \(p_{k}\neq0\), thus
By the law of large numbers, we get \((\frac{S_{i}}{i\mu}-1)\stackrel {\mathrm{P}}{\rightarrow}0\). Noting that \(x^{2}/(1+\theta x)^{2}\leq4x^{2}\) for \(|x|<1/2\) and \(\theta\in(0,1)\), and by using Markov’s inequality, \(\forall\varepsilon>0\), we have
Then we have \(T_{k}\stackrel{\mathrm{P}}{\rightarrow} 0 \) by (4.12) and \(S_{k,k}/(\sigma_{1}\sqrt{2k})\stackrel{\mathrm {d}}{\rightarrow} \mathcal{N}\) by Lemma 2.4 of Li and Wang [10]. So, by Slutsky’s theorem, we have
Thus, we obtain
and
Applying ASCLT (1.4), i.e.,
and Lemma 3.6, (4.14), and (4.15), we obtain
and
Since \(\widetilde{a}_{k}\) and \(\widetilde{b}_{k}\) satisfy (4.5), we get
where
Equations (4.11) and (4.17)–(4.19) together imply that
On the other hand, if \(\widetilde{p}_{k}\neq0\), then we have
Applying Lemma 3.6, (4.19), and (4.20) implies that
Hence
By the arbitrariness of x, let \(x\rightarrow\infty\) in (4.21), we have
Thus
This completes the proof of Theorem 2.1. □
Proof of Lemma 3.2
Let \(\sigma_{k,l}^{2}:=\operatorname{Var}(\sum_{j=k+1}^{l}b_{j,l}Y_{j})\). First, we prove that
where k and l satisfy (3.3). Note that \(\{Y_{n},n\geq1\}\) is a strictly stationary NA sequence with \(\mathrm{E}(Y_{1})=0\) and \(\operatorname{Var}(Y_{1})=1\), we have
By elementary calculations, under condition (3.3), we obtain
Thus, by (4.23) and (4.24), we get
By the condition of (1.1), for some \(\varepsilon>0\), we have
And
where
and
Hence, by (4.23), we get
Equation (4.22) immediately follows from (4.25), (4.26), (4.27), and (4.29).
Let \(a_{l,j}=b_{j,l}/\sigma_{k,l}, k+1\leq j\leq l, l\geq1\). Obviously, \(\operatorname{Var}(\sum_{j=1}^{l}a_{l,j}Y_{j})=1\) and \(\sum_{j=l+1}^{\infty}|\operatorname{Cov}(Y_{1}, Y_{j})|\rightarrow0\) as \(l\rightarrow\infty\) by (1.1). Note that \(\sigma _{k,l}^{2}=2(l-k)\sigma_{1}^{2}(1+o(1))\), hence by (4.24) we have \(\sup_{l}\sum_{j=k+1}^{l}a_{nj}^{2}<\infty\) and \(\max_{k+1\leq j\leq l}|a_{lj}|\rightarrow0\) as \(l\rightarrow\infty\). Hence (3.4) is satisfied by applying Lemma 3.1.
This completes the proof of Lemma 3.2. □
Proof of Lemma 3.4
By the condition of (2.9), we have
It proves (3.7). The proofs of (3.8) and (3.9) are similar to the proof of (3.7). By using Markov’s inequality, (4.22), and \(\varepsilon _{l}=1/(\log l)^{\delta_{4}}\), we have
Noting the condition of \(0<\epsilon<1-2(\delta_{1}+\delta_{4})\), we get
It proves (3.10) and (3.11). By (4.12), we have
Thus
5 Conclusions
In this paper, we study the almost sure local central limit theorem (ASLCLT) for products of partial sums of negatively associated random variables. The obtained results extend the theorem of Weng et al. [21] for i.i.d. random variables to NA random variables, and it is a generalization of the result given by Jiang and Wu [24] from partial sums to products of partial sums under NA random variables. The main idea of the proofs relies on estimate of the covariance structure of the underlying NA sequence. It is a classic and effective technique for this kind of the problem.
Matuła and Stȩpień [16] provided a very mild assumption on the summability on covariances to obtain limit theorems (CLT and ASCLT). As we all know, the ASLCLT is a general result which contains the ASCLT. In this paper, the optimality of the assumptions of Theorem 2.1 is not discussed, in particular assumptions (1.1), (1.2), and (2.8). This will be another interesting topic of research, and we will leave this topic for the future.
References
Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11(1), 286–295 (1983)
Roussas, G.G.: Positive and negative dependence with some statistical application. In: Ghosh, S., Puri, M.L. (eds.) Asymptotics Nonparametrics and Time Series, pp. 757–788. Marcel Dekker, New York (1999)
Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343–356 (2000)
Jing, B.Y., Liang, H.Y.: Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21(4), 890–909 (2008)
Cai, G.H.: Strong laws for weighted sums of NA random variables. Metrika 68(3), 323–331 (2008)
Chen, P.Y., Hu, T.C., Liu, X., Volodin, A.: On complete convergence for arrays of row-wise negatively associated random variables. Theory Probab. Appl. 52(2), 323–328 (2008)
Sung, S.H.: On complete convergence for weighted sums of arrays of dependent random variables. Abstr. Appl. Anal. 2011, Article ID 630583 (2011)
Arnold, B.C., Villaseñor, J.A.: The asymptotic distribution of sums of records. Extremes 1(3), 351–363 (1999)
Li, Y.X., Wang, J.F.: Asymptotic distribution for products of sums under dependence. Metrika 66, 75–82 (2007)
Li, Y.X., Wang, J.F.: An almost sure central limit theorem for products of sums under association. Stat. Probab. Lett. 78(4), 367–375 (2008)
Brosamler, G.A.: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 104(3), 561–574 (1988)
Schatte, P.: On strong versions of the central limit theorem. Math. Nachr. 137(1), 249–256 (1988)
Matuła, P.: On almost sure limit theorems for positively dependent random variables. Stat. Probab. Lett. 74(1), 59–66 (2005)
Lin, F.M.: Almost sure limit theorem for the maxima of strongly dependent Gaussian sequences. Electron. Commun. Probab. 14, 224–231 (2009)
Zhang, Y., Yang, X.Y., Dong, Z.S.: An almost sure central limit theorem for products of sums of partial sums under association. J. Math. Anal. Appl. 355, 708–716 (2009)
Matuła, P., Stȩpień, I.: Weak and almost sure convergence for products of sums of associated random variables. ISRN Probab. Stat. 2012, Article ID 107096 (2012)
Hwang, K.S.: On the almost sure central limit theorem for self-normalized products of partial sums of ϕ-mixing random variables. J. Inequal. Appl. 2013, 155 (2013)
Li, Y.X.: An extension of the almost sure central limit theorem for products of sums under association. Commun. Stat., Theory Methods 42(3), 478–490 (2013)
Miao, Y., Xu, X.Y.: Almost sure central limit theorems for m-dependent random variables. Filomat 31(18), 5581–5590 (2017)
Wu, Q.Y., Jiang, Y.Y.: Almost sure central limit theorem for self-normalized partial sums of negatively associated random variables. Filomat 31(5), 1413–1422 (2017)
Weng, Z.C., Peng, Z.C., Nadarajah, S.: The almost sure local central limit theorem for the product of partial sums. Proc. Math. Sci. 121(2), 217–228 (2011)
Csáki, E., Földes, A., Révész, P.: On almost sure local and global central limit theorems. Probab. Theory Relat. Fields 97(3), 321–337 (1993)
Khurelbaatar, G.: On the almost sure local and global central limit theorem for weakly dependent random variables. Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica 38, 109–126 (1995)
Jiang, Y.Y., Wu, Q.Y.: The almost sure local central limit theorem for the negatively associated sequences. J. Appl. Math. 2013, Article ID 656257 (2013)
Zang, Q.P.: Almost sure local central limit theorem for sample range. Commun. Stat., Theory Methods 46(3), 1050–1055 (2017)
Liang, H.Y., Dong, X., Baek, J.: Convergence of weighted sums for dependent random variables. J. Korean Stat. Soc. 41(5), 883–894 (2004)
Matuła, P.: Some limit theorems for negatively dependent sequences. Yokohama Math. J. 41, 163–173 (1994)
Acknowledgements
The authors would like to thank the editor (Andrei I. Volodin) and three anonymous referees for careful reading of the paper and constructive feedback.
Funding
This work is jointly supported by the National Natural Science Foundation of China (71471173, 71873137, 11661029), the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (14JJD910002), and Research Project of Guangxi Distinguished Expert (2018).
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YJ carried out the design of the study and performed the analysis. QW participated in its design and coordination. All authors read and approved the final manuscript.
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Jiang, Y., Wu, Q. The almost sure local central limit theorem for products of partial sums under negative association. J Inequal Appl 2018, 275 (2018). https://doi.org/10.1186/s13660-018-1875-8
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DOI: https://doi.org/10.1186/s13660-018-1875-8