Weighted version of strong law of large numbers for a class of random variables and its applications

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.

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

$$\begin{aligned}&Y_{ni}=a_{ni}X_{ni}I(|X_{ni}|\le n^{\gamma }),\\&Z_{ni}=a_{ni}X_{ni}-Y_{ni}=a_{ni}X_{ni}I(|X_{ni}|>n^{\gamma }). \end{aligned}$$

Consequently, to prove (3), it suffices to prove

$$\begin{aligned} \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}Y_{ni}\right| >\varepsilon n^{\gamma }/2\right) <\infty \end{aligned}$$

(44)

and

$$\begin{aligned} \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}Z_{ni}\right| >\varepsilon n^{\gamma }/2\right) <\infty . \end{aligned}$$

(45)

For (44), taking $\zeta =\min (1,\alpha )$, according to Markov’s inequality, $C_{r}$-inequality and Lemma 2.3, we can derive that

$$\begin{aligned}&\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}Y_{ni}\right|>\varepsilon n^{\gamma }/2\right) \\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma \zeta }E\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}X_{ni}I(|X_{ni}|\le n^{\gamma })\right| \right) ^{\zeta }\\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma \zeta }\sum _{i=1}^{n}a_{ni}^{\zeta }E|X_{ni}|^{\zeta }I(|X_{ni}|\le n^{\gamma })\\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-1-\gamma \zeta }E|X|^{\zeta }I(|X|\le n^{\gamma })+C\sum _{n=1}^{\infty }n^{\gamma p-1}P(|X|>n^{\gamma }). \end{aligned}$$

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,

$$\begin{aligned}&\sum _{n=1}^{\infty }n^{\gamma p-1-\gamma \zeta }E|X|^{\zeta }I(|X|\le n^{\gamma })\nonumber \\&\quad =\sum _{j=1}^{\infty }E|X|^{\zeta }I((j-1)^{\gamma }<|X|\le j^{\gamma })\sum _{n=j}^{\infty }n^{\gamma p-1-\gamma \zeta }\nonumber \\&\quad \le C\sum _{j=1}^{\infty }j^{\gamma p-\gamma \zeta }E|X|^{\zeta }I((j-1)^{\gamma }<|X|\le j^{\gamma })\le CE|X|^{p}<\infty . \end{aligned}$$

(46)

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

$$\begin{aligned}&\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}Z_{ni}\right|>\varepsilon n^{\gamma }/2\right) \\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p/2-2}E\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}X_{ni}I(|X_{ni}|>n^{\gamma })\right| \right) ^{p/2}\\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p/2-2}\sum _{i=1}^{n}a_{ni}^{p/2}E|X_{ni}|^{p/2}I(|X_{ni}|>n^{\gamma })\\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p/2-1}E|X|^{p/2}I(|X|>n^{\gamma })\\&\quad =C\sum _{j=1}^{\infty }E|X|^{p/2}I(j^{\gamma }<|X|\le (j+1)^{\gamma })\sum _{n=1}^{j}n^{\gamma p/2-1}\\&\quad \le C\sum _{j=1}^{\infty }j^{\gamma p/2}E|X|^{p/2}I(j^{\gamma }<|X|\le (j+1)^{\gamma })\le CE|X|^{p}<\infty , \end{aligned}$$

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

$$\begin{aligned} \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}^{(1)}X_{ni}\right| >\varepsilon n^{\gamma }\right) <\infty \end{aligned}$$

(47)

and

$$\begin{aligned} \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}^{(2)}X_{ni}\right| >\varepsilon n^{\gamma }\right) <\infty . \end{aligned}$$

(48)

For fixed $n\ge 1$, denote for $1\le i\le n$ that

$$\begin{aligned}&Y_{ni}^{(1)}=a_{ni}^{(1)}[-n^{\gamma }I(X_{ni}<-n^{\gamma })+X_{ni}I(|X_{ni}|\le n^{\gamma })+n^{\gamma }I(X_{ni}>n^{\gamma })],\\&Z_{ni}^{(1)}=a_{ni}^{(1)}X_{ni}-Y_{ni}^{(1)}=a_{ni}^{(1)}[(X_{ni}+n^{\gamma })I(X_{ni}<-n^{\gamma })+(X_{ni}-n^{\gamma })I(X_{ni}>n^{\gamma })];\\&Y_{ni}^{(2)}=-n^{\gamma }I(a_{ni}^{(2)}X_{ni}<-n^{\gamma })+a_{ni}^{(2)}X_{ni}I(|a_{ni}^{(2)}X_{ni}|\le n^{\gamma })+n^{\gamma }I(a_{ni}^{(2)}X_{ni}>n^{\gamma }),\\&Z_{ni}^{(2)}=a_{ni}^{(2)}X_{ni}-Y_{ni}^{(2)}=(a_{ni}^{(2)}X_{ni}+n^{\gamma })I(a_{ni}^{(2)}X_{ni}<-n^{\gamma })\\&\qquad \quad +\, (a_{ni}^{(2)}X_{ni}-n^{\gamma })I(a_{ni}^{(2)}X_{ni}>n^{\gamma }). \end{aligned}$$

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

$$\begin{aligned}&n^{-\gamma }\max _{1\le k\le n}\left| \sum _{i=1}^{k}EY_{ni}^{(1)}\right| =n^{-\gamma }\max _{1\le k\le n}\left| \sum _{i=1}^{k}EZ_{ni}^{(1)}\right| \\&\quad \le n^{-\gamma }\sum _{i=1}^{n}a_{ni}E|X_{ni}|I(|X_{ni}|>n^{\gamma })\le Cn^{1-\gamma }E|X|I(|X|>n^{\gamma })\\&\quad \le Cn^{1-\gamma p} E|X|^{p}I(|X|>n^{\gamma })\rightarrow 0,~ n\rightarrow \infty . \end{aligned}$$

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

$$\begin{aligned}&n^{-\gamma }\max _{1\le k\le n}\left| \sum _{i=1}^{k}EY_{ni}^{(2)}\right| =n^{-\gamma }\max _{1\le k\le n}\left| \sum _{i=1}^{k}EZ_{ni}^{(2)}\right| \\&\quad \le n^{-\gamma }\sum _{i=1}^{n}a_{ni}E|X_{ni}|I(|a_{ni}X_{ni}|>n^{\gamma })\le Cn^{-\gamma p}\sum _{i=1}^{n}a_{ni}^{p}E|X|^{p}I(|X|>n^{\gamma -1/\alpha })\\&\quad \le Cn^{1-\gamma p} E|X|^{p}I(|X|>n^{\gamma -1/\alpha })\rightarrow 0,~ n\rightarrow \infty . \end{aligned}$$

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 $r>\max \{p,(\gamma p-1)/(\gamma -1/2)\}$,

$$\begin{aligned}&\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}^{(1)}X_{ni}\right|>\varepsilon n^{\gamma }\right) \nonumber \\&\quad \le \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le i\le n}|X_{ni}|>n^{\gamma }\right) \nonumber \\&\qquad +\,C\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}(Y_{ni}^{(1)}-EY_{ni}^{(1)})\right|>\varepsilon n^{\gamma }/2\right) \nonumber \\&\quad \le \sum _{n=1}^{\infty }n^{\gamma p-2}\sum _{i=1}^{n}P(|X_{ni}|>n^{\gamma })+C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}E\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}(Y_{ni}^{(1)}-EY_{ni}^{(1)})\right| \right) ^{r}\nonumber \\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-1}P(|X|>n^{\gamma })\nonumber \\&\qquad +\,C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left\{ \sum _{i=1}^{n}E|Y_{ni}^{(1)}|^{r}+\left( \sum _{i=1}^{n}E|Y_{ni}^{(1)}|^{2}\right) ^{r/2}\right\} \nonumber \\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\sum _{i=1}^{n}E|Y_{ni}^{(1)}|^{r}+C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}E|Y_{ni}^{(1)}|^{2}\right) ^{r/2}=:I_{1}+I_{2}.\nonumber \\ \end{aligned}$$

(49)

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

$$\begin{aligned} I_{1}\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\sum _{i=1}^{n}|a_{ni}^{(1)}|^{r}[E|X_{ni}|^{r}I(|X_{ni}|\le n^{\gamma })+n^{\gamma r}P(|X_{ni}|> n^{\gamma })]\\\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-1-\gamma r}[E|X|^{r}I(|X|\le n^{\gamma })+n^{\gamma r}P(|X|> n^{\gamma })]<\infty . \end{aligned}$$

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

$$\begin{aligned} I_{2}\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}E|Y_{ni}^{(1)}|^{2}\right) ^{r/2}\le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}|a_{ni}^{(1)}|^{2}E|X_{ni}|^{2}\right) ^{r/2}\\\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}|a_{ni}^{(1)}|^{2}EX^{2}\right) ^{r/2}\le C\sum _{n=1}^{\infty }n^{-1+\gamma p-1-r(\gamma -1/2)}<\infty . \end{aligned}$$

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 $r>\max \{\alpha ,(\gamma p-1)/(\gamma -1/2)\}$,

$$\begin{aligned}&\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}a_{ni}^{(2)}X_{ni}\right|>\varepsilon n^{\gamma }\right) \nonumber \\&\quad \le \sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le i\le n}|a_{ni}^{(2)}X_{ni}|>n^{\gamma }\right) \nonumber \\&\qquad +\sum _{n=1}^{\infty }n^{\gamma p-2}P\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}(Y_{ni}^{(2)}-EY_{ni}^{(2)})\right|>\varepsilon n^{\gamma }/2\right) \nonumber \\&\quad \le \sum _{n=1}^{\infty }n^{\gamma p-2}\sum _{i=1}^{n}P(|a_{ni}^{(2)}X_{ni}|>n^{\gamma })\nonumber \\&\qquad +\,C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}E\left( \max _{1\le k\le n}\left| \sum _{i=1}^{k}(Y_{ni}^{(2)}-EY_{ni}^{(2)})\right| \right) ^{r}\nonumber \\&\quad \le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left\{ \sum _{i=1}^{n}E|Y_{ni}^{(2)}|^{r}+\left( \sum _{i=1}^{n}E|Y_{ni}^{(2)}|^{2}\right) ^{r/2}\right\} =:J_{1}+J_{2}.\qquad \end{aligned}$$

(50)

It follows from Lemmas 2.1 and 2.2 that

$$\begin{aligned} J_{1}= & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\sum _{i=1}^{n}\left\{ E|a_{ni}^{(2)}X_{ni}|^{r}I(|a_{ni}^{(2)}X_{ni}|\le n^{\gamma })+n^{\gamma r}P(|a_{ni}^{(2)}X_{ni}|>n^{\gamma })\right\} \nonumber \\\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\sum _{i=1}^{n}\left\{ E|a_{ni}^{(2)}X|^{r}I(|a_{ni}^{(2)}X|\le n^{\gamma })+n^{\gamma r}P(|a_{ni}^{(2)}X|>n^{\gamma })\right\} <\infty .\nonumber \\ \end{aligned}$$

(51)

On the other hand, noting that $|Y_{ni}^{(2)}|\le |a_{ni}^{(2)}X_{ni}|$, we have

$$\begin{aligned} J_{2}\le & {} C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}|a_{ni}^{(2)}|^{2}EX_{ni}^{2}\right) ^{r/2}\le C\sum _{n=1}^{\infty }n^{\gamma p-2-\gamma r}\left( \sum _{i=1}^{n}|a_{ni}^{(2)}|^{2}EX^{2}\right) ^{r/2}\nonumber \\\le & {} C\sum _{n=1}^{\infty }n^{-1+\gamma p-1-r(\gamma -1/2)}<\infty . \end{aligned}$$

(52)

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 $r>\max \{2,1/(1/p-1/\alpha -\delta )\}$,

$$\begin{aligned}&\sum _{n=1}^{\infty }P\left( \left| \sum _{i=1}^{n}a_{ni}X_{i}\right| >\varepsilon n^{\frac{1}{p}}\right) \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}E\left| \sum _{i=1}^{n}a_{ni}X_{i}\right| ^{r}\\&\quad \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left\{ \sum _{i=1}^{n}E|a_{ni}X_{i}|^{r}+\left( \sum _{i=1}^{n}E|a_{ni}X_{i}|^{2}\right) ^{\frac{r}{2}}\right\} \\&\quad \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left[ n^{\frac{r}{\alpha }+\delta r}+n^{(\frac{2}{\alpha }+2\delta )\cdot \frac{r}{2}}\right] \le C\sum _{n=1}^{\infty }n^{-r(\frac{1}{p}-\frac{1}{\alpha }-\delta )}<\infty . \end{aligned}$$

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

$$\begin{aligned} \sum _{n=1}^{\infty }P\left( \left| \sum _{i=1}^{n}a_{ni}X_{i}\right| >\varepsilon n^{\frac{1}{p}}\right)\le & {} C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left\{ \sum _{i=1}^{n}E|a_{ni}X_{i}|^{r}+\left( \sum _{i=1}^{n}E|a_{ni}X_{i}|^{2}\right) ^{\frac{r}{2}}\right\} \\\le & {} C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left[ n^{\frac{r}{\alpha }+\delta r}+n^{\frac{r}{2}}\right] <\infty . \end{aligned}$$

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

$$\begin{aligned}&\sum _{n=1}^{\infty }P\left( \left| \sum _{i=1}^{n}a_{ni}X_{i}\right| >\varepsilon n^{\frac{1}{p}}\right) \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left\{ \sum _{i=1}^{n}E|a_{ni}X_{i}|^{r}+\left( \sum _{i=1}^{n}E|a_{ni}X_{i}|^{2}\right) ^{\frac{r}{2}}\right\} \\&\quad \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left[ n^{\frac{r}{\alpha }+\delta r}+n^{\frac{r}{2}[1+(2-\alpha p/(\alpha -p))\delta ]}\right] \le C\sum _{n=1}^{\infty }n^{-\frac{r}{p}}\left[ n^{\frac{r}{\alpha }+\delta r}+n^{\frac{r}{\alpha }}\right] <\infty . \end{aligned}$$

Consequently, the desired result (4) follows from the Borel–Cantelli lemma. The proof is completed. $\square$