Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications

Abstract

In the paper, the Marcinkiewicz–Zygmund type moment inequality for extended negatively dependent (END, in short) random variables is established. Under some suitable conditions of uniform integrability, the $L_r$ convergence, weak law of large numbers and strong law of large numbers for usual normed sums and weighted sums of arrays of rowwise END random variables are investigated by using the Marcinkiewicz–Zygmund type moment inequality. In addition, some applications of the $L_r$ convergence, weak and strong laws of large numbers to nonparametric regression models based on END errors are provided. The results obtained in the paper generalize or improve some corresponding ones for negatively associated random variables and negatively orthant dependent random variables.

Appendix

Proof of Lemma 2.3

If $r=1$ or $r=2$, then we can see that (2.2) holds trivially by $C_r$-inequality and Lemma 2.2 with $p=2$, respectively. So in the following, we only need to consider the case $1<r<2$.

For fixed $n\ge 1$, denote $M_n=\sum _{i=1}^nE|X_i|^r$. Without loss of generality, we assume that $M_n>0$. For any $\varepsilon >1$, it is easily seen that

$$\begin{aligned} E\left| \sum _{i=1}^nX_i\right| ^r\le & {} (1+\varepsilon )M_n+\int _{(1+\varepsilon )M_n}^\infty P\left( \left| \sum _{i=1}^nX_i\right| >t^{1/r}\right) dt. \end{aligned}$$

(4.12)

For fixed $n\ge 1$ and $t\ge (1+\varepsilon )M_n$, denote for $1\le i\le n$ that

$$\begin{aligned} Y_i=-t^{1/r}I(X_i<-t^{1/r})+X_iI(|X_i|\le t^{1/r})+t^{1/r}I(X_i>t^{1/r}). \end{aligned}$$

It follows by (4.12) that

$$\begin{aligned} E\left| \sum _{i=1}^nX_i\right| ^r\le & {} (1+\varepsilon )M_n+\int _{(1+\varepsilon )M_n}^\infty \sum _{i=1}^nP\left( \left| X_i\right|>t^{1/r}\right) dt\nonumber \\&+\int _{(1+\varepsilon )M_n}^\infty P\left( \left| \sum _{i=1}^nY_i\right|>t^{1/r}\right) dt\nonumber \\\le & {} (1+\varepsilon )M_n+\int _{(1+\varepsilon )M_n}^\infty \sum _{i=1}^nP\left( \left| X_i\right|>t^{1/r}\right) dt\nonumber \\&+\int _{(1+\varepsilon )M_n}^\infty P\left( \left| \sum _{i=1}^n(Y_i-EY_i)\right| >t^{1/r}-\left| \sum _{i=1}^nEY_i\right| \right) dt\nonumber \\\doteq & {} (1+\varepsilon )M_n+I_1+I_2. \end{aligned}$$

(4.13)

For $I_1$, we have

$$\begin{aligned} I_1\le \sum _{i=1}^n\int _{0}^\infty P\left( \left| X_i\right| >t^{1/r}\right) dt=\sum _{i=1}^nE|X_i|^r=M_n. \end{aligned}$$

(4.14)

Note that

$$\begin{aligned} \sup _{t\ge (1+\varepsilon )M_n}t^{-1/r}\left| \sum _{i=1}^nEY_i\right|\le & {} 2\sup _{t\ge (1+\varepsilon )M_n}t^{-1/r}\cdot t^{1/r-1}\sum _{i=1}^n E|X_i|^rI(|X_i|> t^{1/r}) \nonumber \\\le & {} 2(1+\varepsilon )^{-1}. \end{aligned}$$

(4.15)

Hence, by (4.15), Markov’s inequality and Lemma 2.2 with $p=2$, we can get that

$$\begin{aligned} I_2\le & {} \int _{(1+\varepsilon )M_n}^\infty P\left( \left| \sum _{i=1}^n(Y_i-EY_i)\right|>\left[ 1-2(1+\varepsilon )^{-1}\right] t^{1/r}\right) dt\nonumber \\\le & {} \left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\int _{(1+\varepsilon )M_n}^\infty t^{-2/r} E\left| \sum _{i=1}^n(Y_i-EY_i)\right| ^2dt\nonumber \\\le & {} C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\sum _{i=1}^n\int _{(1+\varepsilon )M_n}^\infty t^{-2/r} EX_i^2I(|X_i|\le t^{1/r})dt\nonumber \\&+ \, C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\sum _{i=1}^n\int _{(1+\varepsilon )M_n}^\infty P(|X_i|> t^{1/r})dt\nonumber \\\doteq & {} I_{21}+I_{22}. \end{aligned}$$

(4.16)

Here, $C_2$ is defined by Lemma 2.2. For $I_{22}$, we have

$$\begin{aligned} I_{22}\le & {} C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\sum _{i=1}^n\int _0^\infty P(|X_i|> t^{1/r})dt\nonumber \\= & {} C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}M_n. \end{aligned}$$

(4.17)

For $I_{21}$, we can see that

$$\begin{aligned}&\int _{(1+\varepsilon )M_n}^\infty t^{-2/r} EX_i^2I(|X_i|\le t^{1/r})dt \le \int _{(1+\varepsilon )M_n}^\infty t^{-2/r} dt\int _{0}^{(1+\varepsilon )^{2/r}M_n^{2/r}}\nonumber \\&\qquad P(|X_i|>y^{1/2})dy +\int _{(1+\varepsilon )M_n}^\infty t^{-2/r} dt\int _{(1+\varepsilon )^{2/r}M_n^{2/r}}^{t^{2/r}} P(|X_i|>y^{1/2})dy\nonumber \\&\quad \doteq J_1+J_2. \end{aligned}$$

(4.18)

For $J_1$, it follows by Markov’s inequality that

$$\begin{aligned} J_1\le & {} \frac{r}{2-r}(1+\varepsilon )^{1-2/r}M_n^{1-2/r} \int _{0}^{(1+\varepsilon )^{2/r}M_n^{2/r}}E|X_i|^r y^{-r/2}dy\nonumber \\= & {} \frac{2r}{(2-r)^2}E|X_i|^r. \end{aligned}$$

(4.19)

For $J_2$, we have

$$\begin{aligned} J_2= & {} \int _{(1+\varepsilon )^{2/r}M_n^{2/r}}^{\infty }P(|X_i|>y^{1/2})dy \int _{y^{r/2}}^\infty t^{-2/r} dt\nonumber \\= & {} \frac{r}{2-r}\int _{(1+\varepsilon )^{2/r}M_n^{2/r}}^{\infty }y^{r/2-1}P(|X_i|>y^{1/2})dy\nonumber \\\le & {} \frac{r}{2-r}\int _{0}^{\infty }y^{r/2-1}P(|X_i|>y^{1/2})dy~=~\frac{2}{2-r}E|X_i|^r. \end{aligned}$$

(4.20)

Hence, by (4.16)-(4.20), we can get that

$$\begin{aligned} I_2\le & {} C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}M_n+C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\left[ \frac{2r}{(2-r)^2}+\frac{2}{2-r}\right] M_n\nonumber \\= & {} C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\left[ 1+\left( \frac{2}{2-r}\right) ^2\right] M_n. \end{aligned}$$

(4.21)

By (4.13), (4.14) and (4.21), we have

$$\begin{aligned} E\left| \sum _{i=1}^nX_i\right| ^r\le & {} \left\{ 2+\varepsilon +C_2\left[ 1-2(1+\varepsilon )^{-1}\right] ^{-2}\left[ 1+\left( \frac{2}{2-r}\right) ^2\right] \right\} M_n\nonumber \\\doteq & {} f(\varepsilon ) M_n. \end{aligned}$$

(4.22)

It is easily checked that $f(\varepsilon )$ is positive and continuous on $(1,\infty )$, and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 1^+}f(\varepsilon )=\lim _{\varepsilon \rightarrow \infty }f(\varepsilon )=\infty . \end{aligned}$$

Hence, $f(\varepsilon )$ has the minimum on $(1,\infty )$. Set $c_r=\inf _{1<\varepsilon <\infty }f(\varepsilon )$. It is obvious that $c_r>3$ does not depend on n, and thus (2.2) holds. This completes the proof of the lemma. $\square$