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Research | Open | Published:

An improved version of a result of Chandra, Li, and Rosalsky

Abstract

For an array of rowwise pairwise negative quadrant dependent, mean 0 random variables, Chandra, Li, and Rosalsky provided conditions under which weighted averages converge in $\mathscr{L}_{1}$ to 0. The Chandra, Li, and Rosalsky result is extended to $\mathscr{L}_{r}$ convergence ($1\leq r<2$) and is shown to hold under weaker conditions by applying a mean convergence result of Sung and an inequality of Adler, Rosalsky, and Taylor.

Introduction

For an array of mean 0 random variables $\{X_{n,j}, 1 \leq j \leq k _{n}, n \geq 1 \}$ and an array of constants $\{a_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$, Chandra, Li, and Rosalsky [2, Theorem 3.1] recently provided conditions under which the weighted averages $\sum_{j=1}^{k_{n}} a_{n,j}X_{n,j}$ obey the degenerate mean convergence law

$$ \sum_{j=1}^{k_{n}} a_{n,j}X_{n,j} \stackrel{\mathscr{L}_{1}}{\longrightarrow } 0. $$

The random variables comprising the array are assumed to be (i) rowwise pairwise negative quadrant dependent and (ii) stochastically dominated by a random variable. (Technical definitions such as these will be reviewed in Sect. 2.) In this note, Theorem 3.1 of Chandra, Li, and Rosalsky [2] is extended to $\mathscr{L}_{r}$ convergence where $1 \leq r < 2$ and is shown to hold under weaker conditions. This is accomplished by applying a result of Sung [3] and an inequality of Adler, Rosalsky, and Taylor [1]. This note owes much to the work of Sung [3].

Preliminaries

In this section, some definitions will be reviewed and the needed results of Sung [3] and Adler, Rosalsky, and Taylor [1] will be stated.

Definition 2.1

The random variables comprising an array $\{X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ are said to be rowwise pairwise negative quadrant dependent (PNQD) if for all $n \geq 1$ and all $i, j \in \{1,\ldots, k_{n}\}$ ($i \neq j$),

$$ \mathbb{P} (X_{n,i} \leq x, X_{n,j} \leq y ) \leq \mathbb{P} (X_{n,i} \leq x ) \mathbb{P} (X_{n,j} \leq y ) \quad \text{for all } x, y \in \mathbb{R}. $$

Definition 2.2

The random variables comprising an array $\{Y_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ are said to be stochastically dominated by a random variable Y if there exists a constant D such that

$$ \mathbb{P} \bigl( \vert Y_{n,j} \vert > y \bigr) \leq D \mathbb{P} \bigl( \vert DY \vert > y \bigr), \quad y \geq 0, 1 \leq j \leq k_{n}, n \geq 1. $$
(2.1)

Lemma 2.1

(Adler, Rosalsky, and Taylor [1, Lemma 2.3])

If the random variables in the array $\{Y_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ are stochastically dominated by a random variable Y, then for all $n \geq 1$ and $j \in \{1,\ldots, k_{n} \}$,

$$ \mathbb{E} \bigl( \vert Y_{n,j} \vert I \bigl( \vert Y_{n,j} \vert > y \bigr) \bigr) \leq D^{2} \mathbb{E}\bigl( \vert Y \vert I\bigl( \vert DY \vert > y\bigr)\bigr) \quad \textit{for all } y \geq 0, $$

where D is as in (2.1).

Proposition 2.1

(Sung [3, Theorem 2.1])

Let $\{X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ be an array of rowwise PNQD random variables and let $r \in [1, 2)$. Let $\{a_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ be an array of constants. Suppose that

$$ \sup_{n \geq 1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \mathbb{E} \vert X_{n,j} \vert ^{r} < \infty $$
(2.2)

and

$$ \lim_{n \rightarrow \infty } \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \mathbb{E} \bigl( \vert X_{n,j} \vert ^{r} I \bigl( \vert a_{n,j} \vert ^{r} \vert X_{n,j} \vert ^{r} > \varepsilon \bigr) \bigr) = 0 \quad \textit{for all } \varepsilon > 0. $$
(2.3)

Then

$$ \sum_{j=1}^{k_{n}} a_{n,j} (X_{n,j} - \mathbb{E}X_{n,j} ) \stackrel{\mathscr{L}_{r}}{\longrightarrow } 0 $$

and, a fortiori,

$$ \sum_{j=1}^{k_{n}} a_{n,j} (X_{n,j} - \mathbb{E}X_{n,j} ) \stackrel{\mathbb{P}}{ \longrightarrow } 0. $$

Improved version of the Chandra, Li, and Rosalsky [2] result

We will now use Lemma 2.1 and Proposition 2.1 to present the following improved version of Theorem 3.1 of Chandra, Li, and Rosalsky [2].

Theorem 3.1

Let $\{X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ be an array of rowwise PNQD mean 0 random variables which are stochastically dominated by a random variable X with $\mathbb{E}\vert X\vert ^{r} < \infty $ for some $r \in [1, 2)$. Let $\{a_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ be an array of constants such that

$$ \sup_{n \geq 1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} < \infty $$
(3.1)

and

$$ \lim_{n \rightarrow \infty } \sup_{1 \leq j \leq k_{n}} \vert a_{n,j} \vert = 0. $$
(3.2)

Then

$$ \sum_{j=1}^{k_{n}} a_{n,j} X_{n,j} \stackrel{\mathscr{L}_{r}}{\longrightarrow } 0 $$
(3.3)

and, a fortiori,

$$ \sum_{j=1}^{k_{n}} a_{n,j} X_{n,j} \stackrel{\mathbb{P}}{\longrightarrow } 0. $$

Remark 3.1

Before proving Theorem 3.1, we point out that Theorem 3.1 of Chandra, Li, and Rosalsky [2]

  1. (i)

    only treated the case $r = 1$,

  2. (ii)

    had the additional condition

    $$ \text{for each } n \geq 1, \text{either } \min_{1 \leq j \leq k_{n}} a _{n,j} \geq 0 \text{ or } \max_{1 \leq j \leq k_{n}} a_{n,j} \leq 0, $$
  3. (iii)

    had the condition

    $$ \sup_{n \geq 1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert < \infty \quad \text{and} \quad \lim_{n \rightarrow \infty } \sum_{j=1}^{k_{n}} a_{n,j}^{2} = 0, $$

the second half of which is clearly stronger than (3.2).

Proof of Theorem 3.1

Letting D be as in (2.1) with $Y_{n,j}$ replaced by $X_{n,j}$, $1 \leq j \leq k_{n}$, $n \geq 1$ and Y replaced by X, it follows that

$$ \mathbb{E} \vert X_{n,j} \vert ^{r} \leq D^{r+1} \mathbb{E} \vert X \vert ^{r}, \quad 1 \leq j \leq k_{n}, n \geq 1. $$

Thus

$$ \sup_{n \geq 1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \mathbb{E} \vert X_{n,j} \vert ^{r} \leq D^{r+1} \Biggl(\sup_{n \geq 1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \Biggr) \mathbb{E} \vert X \vert ^{r} < \infty $$

by (3.1) and $\mathbb{E}\vert X\vert ^{r} < \infty $, thereby verifying (2.2).

Next, we show that (2.3) holds. Let

$$ \lambda _{n} = D \sup_{1 \leq j \leq k_{n}} \vert a_{n,j} \vert , \quad n \geq 1. $$

Then $\lim_{n \rightarrow \infty } \lambda _{n} = 0$ by (3.2). Now the stochastic domination hypothesis ensures that

$$ \mathbb{P} \bigl( \vert X_{n,j} \vert ^{r} > x \bigr) \leq D \mathbb{P} \bigl( \vert DX \vert ^{r} > x \bigr) = D \mathbb{P} \bigl(D \bigl(D ^{r-1} \vert X \vert ^{r} \bigr) > x \bigr), \quad x \geq 0, 1 \leq j \leq k_{n}, n \geq 1 $$

and so by Lemma 2.1 with $Y_{n,j}$ replaced by $\vert X_{n,j}\vert ^{r}$, $1 \leq j \leq k_{n}$, $n \geq 1$ and Y replaced by $D^{r-1} \vert X\vert ^{r}$,

$$ \begin{aligned}[b] & \mathbb{E} \bigl( \vert X_{n,j} \vert ^{r} I \bigl( \vert X_{n,j} \vert ^{r} > x \bigr) \bigr) \\ &\quad \leq D^{2} \mathbb{E} \bigl(D^{r-1} \vert X \vert ^{r} I \bigl(D^{r} \vert X \vert ^{r} > x \bigr) \bigr) \\ &\quad = D^{r+1} \mathbb{E} \bigl( \vert X \vert ^{r} I \bigl(D^{r} \vert X \vert ^{r} > x \bigr) \bigr), \quad x \geq 0, 1 \leq j \leq k_{n}, n \geq 1. \end{aligned} $$
(3.4)

Then for arbitrary $\varepsilon > 0$,

$$\begin{aligned} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \mathbb{E} \bigl( \vert X_{n,j} \vert ^{r} I \bigl( \vert a_{n,j} \vert ^{r} \vert X_{n,j} \vert ^{r} > \varepsilon \bigr) \bigr) \leq & D^{r+1} \sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \mathbb{E} \biggl( \vert X \vert ^{r} I \biggl(D^{r} \vert X \vert ^{r} > \frac{\varepsilon }{ \vert a_{n,j} \vert ^{r}} \biggr) \biggr) \\ \leq & D^{r+1} \Biggl(\sum_{j=1}^{k_{n}} \vert a_{n,j} \vert ^{r} \Biggr) \mathbb{E} \biggl( \vert X \vert ^{r} I \biggl( \vert X \vert ^{r} > \frac{\varepsilon }{\lambda _{n} ^{r}} \biggr) \biggr) \\ \leq & D^{r+1} \Biggl(\sup_{m \geq 1} \sum _{j=1}^{k_{m}} \vert a_{m,j} \vert ^{r} \Biggr) \mathbb{E} \biggl( \vert X \vert ^{r} I \biggl( \vert X \vert ^{r} > \frac{\varepsilon }{ \lambda _{n}^{r}} \biggr) \biggr) \\ \rightarrow & 0 \quad \text{as } n \rightarrow \infty \end{aligned}$$

by (3.1), $\lambda _{n} \rightarrow 0$, and $\mathbb{E}\vert X\vert ^{r} < \infty $. Thus (2.3) holds, and conclusion (3.3) follows from Proposition 2.1. □

Remark 3.2

See Chandra, Li, and Rosalsky [2] for examples

  1. (i)

    showing that Theorem 3.1 can fail if the PNQD hypothesis is dispensed with,

  2. (ii)

    showing that $\sum_{j=1}^{k_{n}} a_{n,j}X_{n,j} \rightarrow 0$ almost surely does not necessarily hold under the hypotheses of Theorem 3.1.

Conclusions

For an array of rowwise PNQD random variables $\{X_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$, conditions are provided under which the following degenerate mean convergence law holds:

$$ \sum_{j=1}^{k_{n}}a_{n,j}X_{n,j} \stackrel{\mathscr{L}_{r}}{\longrightarrow } 0, $$

where $1 \leq r < 2$, $\mathbb{E}X_{n,j} = 0$, $1 \leq j \leq k_{n}$, $n \geq 1$, and $\{a_{n,j}, 1 \leq j \leq k_{n}, n \geq 1 \}$ is an array of constants. This result is an improved version of Theorem 3.1 of Chandra, Li, and Rosalsky [2] in that $\mathscr{L}_{1}$ convergence is extended to $\mathscr{L}_{r}$ convergence and the hypotheses are weakened. The result is obtained by applying a result of Sung [3] and an inequality of Adler, Rosalsky, and Taylor [1].

References

  1. 1.

    Adler, A., Rosalsky, A., Taylor, R.L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci. 12, 507–529 (1989)

  2. 2.

    Chandra, T.K., Li, D., Rosalsky, A.: Some mean convergence theorems for arrays of rowwise pairwise negative quadrant dependent random variables. J. Inequal. Appl. 2018, 221 (2018). https://doi.org/10.1186/s13660-018-1811-y

  3. 3.

    Sung, S.H.: Convergence in r-mean of weighted sums of NQD random variables. Appl. Math. Lett. 26, 18–24 (2013)

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Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Correspondence to Andrew Rosalsky.

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MSC

  • 60F25
  • 60F05

Keywords

  • Array of rowwise pairwise negative quadrant dependent random variables
  • Weighted averages
  • Degenerate mean convergence
  • Stochastic domination