Abstract
In this paper, the single index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers and the double index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers are investigated successively for a class of random variables, which extends the classical results for independent and identically distributed random variables. As applications of the results, we further study the strong consistency for the weighted estimator in the nonparametric regression model and the least square estimators in the simple linear errors-in-variables model. Moreover, we also present some numerical study to verify the validity of our results.
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The authors are most grateful to the Editor-in-Chief, Associate Editor and two anonymous referees for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper.
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This work was supported by the National Natural Science Foundation of China (11671012, 11501004, 11501005), the Natural Science Foundation of Anhui Province (1508085J06) and The Key Projects for Academic Talent of Anhui Province (gxbjZD2016005).
Appendix
Appendix
Proof of Lemma 2.4
Without loss of generality, we may assume that \(a_{ni}\ge 0\) for all \(1\le i\le n,n\ge 1\) and \(\sum _{i=1}^{n}a_{ni}^{\alpha }\le n\). Note that \(\sum _{i=1}^{n}a_{ni}^{s}\le n\) for any \(0<s<\alpha \) according to Hölder’s inequality. First let us consider the case that \(0<p<1\).
For fixed \(n\ge 1\), denote for \(1\le i\le n\) that
Consequently, to prove (3), it suffices to prove
and
For (44), taking \(\zeta =\min (1,\alpha )\), according to Markov’s inequality, \(C_{r}\)-inequality and Lemma 2.3, we can derive that
Noting that \(\sum _{n=1}^{\infty }n^{\gamma p-1}P(|X|>n^{\gamma })<\infty \) is equivalent to \(E|X|^{p}<\infty \), we need only to prove that \(\sum _{n=1}^{\infty }n^{\gamma p-1-\gamma \zeta }E|X|^{\zeta }I(|X|\le n^{\gamma })<\infty .\) Actually,
Therefore, (44) holds. It retains to prove (45). According to Markov’s inequality, \(C_{r}\)-inequality and Lemma 2.3, we can also obtain that
which implies (45).
For \(p\ge 1\), we decompose the coefficients \(a_{ni}\) into \(a_{ni}^{(1)}=a_{ni}I(0\le a_{ni}\le 1)\) and \(a_{ni}^{(2)}=a_{ni}I(a_{ni}>1)\) for any \(n\ge 1\) and \(1\le i\le n\). Thus to prove (3), we need to prove
and
For fixed \(n\ge 1\), denote for \(1\le i\le n\) that
Noting that \(E|X|^{p}<\infty \) and \(EX_{ni}=0\), we derive from Lemma 2.3 (further from the Dominated Convergence Theorem if \(\gamma p=1\)) that
Similarly, noting that \(\max _{1\le i\le n}a_{ni}^{(2)}\le n^{1/\alpha }\) and \(\gamma \ge 1/p>1/\alpha \), we obtain that
Therefore, \(\max _{1\le k\le n}\left| \sum _{i=1}^{k}EY_{ni}^{(j)}\right| \le \varepsilon n^{\gamma }/2\) for all n large enough and \(j=1,2\). We first prove (47). Recall that \(\sum _{n=1}^{\infty }n^{\gamma p-1}P(|X|>n^{\gamma })<\infty \) is equivalent to \(E|X|^{p}<\infty \). If \(p\ge 2\), it follows from (1.2), Lemma 2.3, \(C_{r}\)-inequality and Jensen’s inequality that for any \(r>\max \{p,(\gamma p-1)/(\gamma -1/2)\}\),
For \(I_{1}\), since \(\sum _{n=1}^{\infty }n^{\gamma p-1}P(|X|>n^{\gamma })<\infty \) is equivalent to \(E|X|^{p}<\infty \), we derive from the proof of (46) that
For \(I_{2}\), noting that \(r>(\gamma p-1)/(\gamma -1/2)\), \(|Y_{ni}^{(1)}|\le |a_{ni}^{(1)}X_{ni}|\) and that \(E|X|^{p}<\infty \) implies \(EX^{2}<\infty \) according to Hölder’s inequality, we have that
If \(1\le p<2\), the proof of the desired result (47) is similar to that of (49) and \(I_{1}\) with \(r=2\); hence, the detail is omitted. (48) is yet to be proved. We also deal with the case \(p\ge 2\) first. It follows from (1.2), Lemma 2.1, \(C_{r}\)-inequality and Jensen’s inequality that for \(r>\max \{\alpha ,(\gamma p-1)/(\gamma -1/2)\}\),
It follows from Lemmas 2.1 and 2.2 that
On the other hand, noting that \(|Y_{ni}^{(2)}|\le |a_{ni}^{(2)}X_{ni}|\), we have
Thus, (48) follows from (50)–(52)immediately.
If \(1\le p<2\), we can assume without loss of generality that \(p<\alpha <2\), then we can choose \(r=2\) such that \(\alpha <r\). The proof of (48) is similar to that of (50) and \(J_{1}\) with \(r=2\). This completes the proof of the lemma. \(\square \)
Proof of Lemma 2.5
Assume without loss of generality that \(a_{ni}\ge 0\) for all \(1\le i\le n\) and \(n\ge 1\) and \(\sum _{i=1}^{n}|a_{ni}|^{\alpha }\le n\). It follows from Hölder’s inequality that \(\sum _{i=1}^{n}|a_{ni}|^{s}\le n\) and \(\sum _{i=1}^{n}|a_{ni}|^{r}\le n^{r/\alpha }\) for any \(0<s<\alpha <r\). To prove (4), we consider the following three cases.
Case 1 \(0< p<\alpha \le 2\)
Noting that \(\max _{1\le i\le n}|X_{i}|\le Cn^{\delta }\) \(\mathrm{a.s.}\), we derive from (1) that for \(r>\max \{2,1/(1/p-1/\alpha -\delta )\}\),
Case 2 \(2<\alpha \le 2p/(2-p)\)
Noting that \(\alpha p/(\alpha -p)\ge 2\) and \(\sup _{n\ge 1}E|X_{n}|^{\alpha p/(\alpha -p)}<\infty \), we can obtain that \(\sup _{n\ge 1}EX_{n}^{2}<\infty \). For \(r>\max \{\alpha ,1/(1/p-1/\alpha -\delta ),1/(1/p-1/2)\}\), we obtain that
Case 3 \(\alpha >2p/(2-p)\)
Let \(r>\max \{\alpha ,1/(1/p-1/\alpha -\delta )\}\). Noting that \(\alpha p/(\alpha -p)<2\) and \(0\le \delta <(1/p-1/\alpha )\), we obtain that
Consequently, the desired result (4) follows from the Borel–Cantelli lemma. The proof is completed. \(\square \)
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Wu, Y., Wang, X., Hu, S. et al. Weighted version of strong law of large numbers for a class of random variables and its applications. TEST 27, 379–406 (2018). https://doi.org/10.1007/s11749-017-0550-6
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DOI: https://doi.org/10.1007/s11749-017-0550-6
Keywords
- Strong law of large numbers
- Rosenthal-type inequality
- Double index weight
- Nonparametric regression model
- Simple linear errors-in-variables model
- Strong consistency