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Almost sure central limit theorem for products of sums of partial sums
Journal of Inequalities and Applications volume 2016, Article number: 49 (2016)
Abstract
Considering a sequence of i.i.d. positive random variables, for products of sums of partial sums we establish an almost sure central limit theorem, which holds for some class of unbounded measurable functions.
1 Introduction and main results
Let \(\{X_{n};n\geq1\}\) be a sequence of random variables and define \(S_{n}=\sum_{i=1}^{n} X_{i}\). Some results as regards the limit theorem of products \(\prod_{j=1}^{n}S_{j}\) were obtained in recent years. Rempala and Wesolowski [1] obtained the following asymptotics for products of sums for a sequence of i.i.d. random variables.
Theorem A
Let \(\{X_{n};n\geq1\}\) be a sequence of i.i.d. positive square integrable random variables with \(\mathbb{E}X_{1}=\mu\), the coefficient of variation \(\gamma=\sigma/\mu\), where \(\sigma ^{2}=\operatorname{Var}(X_{1})\). Then
Here and in the sequel, \(\mathcal{N}\) is a standard normal random variable and \(\stackrel{d}{\rightarrow}\) denotes the convergence in distribution.
Gonchigdanzan and Rempala [2] discussed the almost sure central limit theorem (ASCLT) for the products of partial sums and obtained the following result.
Theorem B
Let \(\{X_{n};n\geq1\}\) be a sequence of i.i.d. positive random variables with \(\mathbb{E}X_{1}=\mu\), \(\operatorname {Var}(X_{1})=\sigma^{2}\) the coefficient of variation \(\gamma=\sigma/\mu \). Then
where F is the distribution function of the random variable \(e^{\sqrt {2}\mathcal{N}}\). Here and in the sequel, \(I \{\cdot \}\) denotes the indicator function.
Tan and Peng [3] proved the result of Theorem B still holds for some class of unbounded measurable functions and obtained the following result.
Theorem C
Let \(\{X_{n};n\geq1\}\) be a sequence of i.i.d. positive random variables with \(\mathbb{E}X_{1}=\mu\), \(\operatorname {Var}(X_{1})=\sigma^{2}\), \(\mathbb{E}|X_{1}|^{3}<\infty\), the coefficient of variation \(\gamma =\sigma/\mu\). Let \(g(x)\) be a real valued almost everywhere continuous function on \(\mathbb{R}\) such that \(|g(e^{x})\phi(x)|\leq c(1+|x|)^{-\alpha}\) with some \(c>0\) and \(\alpha>5\). Then
where \(F(\cdot)\) is the distribution function of the random variable \(e^{\sqrt {2}\mathcal{N}}\) and \(\phi(x)\) is the density function of the standard normal random variable.
Zhang et al. [4] discussed the almost sure central limit theory for products of sums of partial sums and obtained the following result.
Theorem D
Let \(\{X,X_{n};n\geq1\}\) be a sequence of i.i.d. positive square integrable random variables with \(\mathbb{E}X=\mu\), \(\operatorname{Var}(X)=\sigma^{2}<\infty\), the coefficient of variation \(\gamma =\sigma/\mu\). Denote \(S_{n}=\sum_{i=1}^{n} X_{i}\), \(T_{k}=\sum_{i=1}^{k} S_{i}\). Then
where \(F(\cdot)\) is the distribution function of the random variable \(e^{\sqrt{10/3}\mathcal{N}}\).
The purpose of this article is to establish that Theorem D holds for some class of unbounded measurable functions.
Our main result is the following theorem.
Theorem 1.1
Let \(\{X_{n};n\geq1\}\) be a sequence of i.i.d. positive random variables with \(\mathbb{E}X_{1}=\mu\), \(\operatorname {Var}(X_{1})=\sigma^{2}\), \(\mathbb{E}|X_{1}|^{3}<\infty\), the coefficient of variation \(\gamma =\sigma/\mu\). Let \(g(x)\) be a real valued almost everywhere continuous function on \(\mathbb{R}\) such that \(|g(e^{\sqrt{10/3} x})\phi(x)|\leq c(1+|x|)^{-\alpha}\) with some \(c>0\) and \(\alpha>5\). Denote \(S_{n}=\sum_{i=1}^{n} X_{i}\), \(T_{k}=\sum_{i=1}^{k} S_{i}\). Then
where \(F(\cdot)\) is the distribution function of the random variable \(e^{\sqrt{10/3}\mathcal{N}}\). Here and in the sequel, \(\phi(x)\) is the density function of the standard normal random variable.
Remark 1
Let \(f(x)=g(e^{\sqrt{10/3} x})\), \(t=e^{\sqrt{10/3} x}\). Then
Since \(F(x)\) is the distribution function of the random variable \(e^{\sqrt{10/3}\mathcal{N}}\), we can get \(F(x)=\Phi (\sqrt{\frac{3}{10}}\log x )\), where \(\Phi (x)\) is the distribution function of the standard normal random variable. Hence we have the following: Let \(f(x)=g(e^{\sqrt{10/3} x})\) and \(f(x)\) be a real valued almost everywhere continuous function on \(\mathbb{R}\) such that \(|f(x)\phi(x)|\leq c(1+|x|)^{-\alpha}\) with some \(c>0\) and \(\alpha>5\), then (1.5) is equivalent to
Remark 2
By the proof of Theorem 2 of Berkes et al. [5], in order to prove (1.5), it suffices to show (1.6) holds true for \(f(x)\phi(x)=(1+|x|)^{-\alpha}\) with \(\alpha>5\). Here and in the sequel, \(f(x)\) satisfies \(f(x)\phi(x)=(1+|x|)^{-\alpha}\) with \(\alpha>5\).
2 Preliminaries
In the following, the notation \(a_{n}\sim b_{n}\) means that \(\lim_{n \to\infty}a_{n}/ b_{n}= 1\) and \(a_{n}\ll b_{n}\) means that \(\limsup_{n \to\infty}|a_{n}/ b_{n}|<+\infty\). We denote \(b_{k,n}=\sum_{j=k}^{n} \frac {1}{j}\), \(c_{k,n}=2\sum_{j=k}^{n} \frac{j+1-k}{j(j+1)}\), \(d_{k,n}=\frac {n+1-k}{n+1}\), \(\widetilde{X}_{i}=\frac{{X}_{i}-\mu}{\sigma}\), \(\widetilde{S}_{k}=\sum_{i=1}^{k}\widetilde{X}_{i}\), \({S}_{k,n}=\sum_{i=1}^{k} c_{i,n}\widetilde{X}_{i}\). By Lemma 2.1 of Wu [6], we can get
Let
Note that
By the fact that \(\log (1+x)=x+\frac{\delta}{2}x^{2}\), where \(|x|<1\), \(\delta\in(-1,0)\), thus we have
By the fact that \(\mathbb{E}|X_{1}|^{2}<\infty\), using the Marcinkiewicz-Zygmund strong large number law, we have
Thus
In order to prove Theorem 1.1, we introduce the following lemmas.
Lemma 2.1
Let X and Y be random variables. Set \(F(x)=P(X< x)\), \(G(x)=P(X+Y< x)\), then for any \(\varepsilon>0\) and \(x\in\mathbb{R}\),
Proof
See Lemma 3 on p.16 of Petrov [7]. □
Lemma 2.2
Let \(\{X_{n};n\geq1\}\) be a sequence of i.i.d. positive random variables. Denote \(S_{n}=\sum_{i=1}^{n} X_{i}\), \(F^{s}\) denotes the distribution function obtained from F by symmetrization and choose \(L >0\) so large that \(\int_{|x|\leq L}x^{2}\,\mathrm{d}F^{s}(x)\geq1\). Then, for any \(n\geq1\), \(\lambda>0\), there exists a \(c>0\) such that
holds for \(\lambda\sqrt{n}\geq L\).
Proof
See (20) on p.73 of Berkes et al. [5]. □
Let
where \(1<\beta<(\alpha-3)/2\).
Lemma 2.3
Under the conditions of Theorem 1.1, we get
Proof
It is easy to get
Since \(|R_{i}|\ll\frac{\log i}{\sqrt{i}}\) a.s.; see (2.1). By the law of iterated logarithm (Feller [8], Theorem 2), we get
We complete the proof of Lemma 2.3. □
Let \(G_{i}\), \(F_{i}\), F denote the distribution functions of \(Y_{i}\), \(\frac {\widetilde{S}_{i}}{\sqrt{i}}\), \(\widetilde{X}_{1}\), respectively. Φ denotes the distribution function of the standard normal distribution function. Set
Obviously \(\sigma_{i}\leq1\), \(\lim_{i \to\infty}\sigma_{i}= 1\).
Lemma 2.4
Under the conditions of Theorem 1.1, we have
Proof
Note that the estimation
holds for any bounded, measurable function \(\Psi(x)\) and the distribution functions \(H_{1}(x)\), \(H_{2}(x)\). Thus for \(2^{k}< i\leq 2^{k+1}\), we get
here and in the sequel \(a_{k}=f^{-1}(\frac{k}{(\log k)^{\beta }})\). Hence, by the Cauchy-Schwarz inequality and the fact that \(f(x)\phi(x)=(1+|x|)^{-\alpha}\), we obtain
By the same methods as that on p.72 of Berkes et al. [5], we get
Now we estimate \(\theta_{i}\). By Lemma 2.1, for any \(\varepsilon>0\), we have
By the Markov inequality and (2.1), we have
By Lemma 2.2, we have
By the Berry-Esseen inequality, we have
Let \(\varepsilon=i^{-1/3}\), then
Therefore, there exists \(\varepsilon_{0}>0\) such that
By Theorem 1 of Friedman et al. [9], we have
Hence
By the fact that \((\alpha+1)/2>\beta\), we have
We complete the proof of Lemma 2.4. □
Lemma 2.5
Under the conditions of Theorem 1.1, for \(l\geq l_{0}\), we have
where Ï„ is a constant \(0<\tau\leq1/8\).
Proof
For \(1\leq i \leq j/2\), \(j\geq j_{0}\) and any x, y, we first prove
Let \(\rho=\frac{i}{j}\). By the Chebyshev inequality, we have
where \(\tau_{1}\) is a constant \(0<\tau_{1}\leq1/8\).
By the Markov inequality and (2.1), for \(j\geq j_{0}\), we have
where \(\tau_{2}\) is a constant, \(0<\tau_{2}\leq1/8\).
By the Markov inequality, we have
where \(\tau_{3}\) is a constant that satisfies \(0<\tau_{3}\leq1/8\).
By Lemma 2.2 and the fact that \(\rho=\frac{i}{j}\), \(1\leq i \leq j/2\), we have
Set \(\tau=\min\{\tau_{1},\tau_{2},\tau_{3},1/8\}\), we get
We can get a similar upper estimate for \(P(Y_{i}\leq x,Y_{j}\leq y)\) in the same way. Thus there exists some constant M such that
A similar argument,
holds for some constant \(M'\). Thus we prove that (2.3) holds.
Let \(G_{i,j}(x,y)\) be the joint distribution function of \(Y_{i}\) and \(Y_{j}\). By (2.2) and (2.3), for \(2^{k}< i\leq2^{k+1}\), \(2^{l}< j\leq 2^{l+1}\), \(l-k\geq3\), \(l\geq l_{0}\), we can get
Thus we have
We complete the proof of Lemma 2.5. □
Lemma 2.6
Under the conditions of Theorem 1.1, denoting \(\eta _{k}=Z_{k}^{*}-\mathbb{E}Z_{k}^{*}\), we have
Proof
It follows from Lemma 2.4 and Lemma 2.5 that Lemma 2.6 also holds true. The proof is similar to that of Lemma 4 of Berkes et al. [5]. So we omit it here. □
3 Proof of theorem
By Lemma 2.6, we have
Letting \(N_{k}=[e^{k\lambda}]\), \((2\beta-1)^{-1}<\lambda<1\), we get
which implies
Note that for \(2^{k}< i\leq2^{k+1}\),
Set \(a=\int_{-\infty}^{\infty}f(x)\,\mathrm{d}\Phi(x)\). Noting that \(\sigma_{i}\leq1\), \(\lim_{i \to\infty}\sigma_{i}=1\), we have
Then by (3.2), (3.3), and (2.2) we get
Thus
Using \(\sum_{i=1}^{L}1/i=\log L+O(1)\) and \(\sum_{i=1}^{\infty }\frac{\theta_{i}}{i}<\infty\), we get
Thus by (3.1), we get
Then by Lemma 2.3, we have
The relation \(\lambda<1\) implies \(\lim_{k \to\infty}N_{k+1}/N_{k}=1\), thus (3.4) and the positivity of the \(Z_{k}\) yield
i.e. (1.6) holds for the subsequence \(N=2^{k}\). Using again the positivity of the terms, we get (1.6). We complete the proof of Theorem 1.1.
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Acknowledgements
The authors would like to thank the editor and the referees for their very valuable comments by which the quality of the paper has been improved. This research is supported by the National Natural Science Foundation of China (71271042) and the Guangxi China Science Foundation (2013GXNSFAA278003). It is also supported by the Research Project of Guangxi High Institution (YB2014150).
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FF conceived of the study and drafted and completed the manuscript. DW participated in the discussion of the manuscript. FF and DW read and approved the final manuscript.
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Feng, F., Wang, D. Almost sure central limit theorem for products of sums of partial sums. J Inequal Appl 2016, 49 (2016). https://doi.org/10.1186/s13660-016-0995-2
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DOI: https://doi.org/10.1186/s13660-016-0995-2