On the Sum of Unitary Divisors Maximum Function

Bhabesh Das; Helen K. Saikia; Bhabesh Das; Helen K. Saikia

doi:10.3934/Math.2017.1.96

AIMS Mathematics

2017, Volume 2, Issue 1: 96-101. doi: 10.3934/Math.2017.1.96

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Research article

On the Sum of Unitary Divisors Maximum Function

Bhabesh Das ^{1
,
,},
Helen K. Saikia ²

1.
Department of Mathematics, B.P.Chaliha College, Assam-781127, India
2.
Department of Mathematics, Gauhati University, Assam-781014, India

Received: 03 October 2016 Accepted: 10 December 2016 Published: 10 February 2017

It is well-known that a positive integer

$d$ is called a unitary divisor of an integer

$n$ if

$d|n$ and gcd

$\left(d,\frac{n}{d}\right)=1$ . Divisor function

$\sigma^{*}(n)$ denote the sum of all such unitary divisors of

$n$ . In this paper we consider the maximum function

$U^{*}(n)=\max\{k\in\mathbb{N}:\sigma^{*}(k)|n\}$ and study the function

$U^{*}(n)$ for

$n=p^{m}$ , where

$p$ is a prime and

$m\geq 1$ .

Keywords:

Citation: Bhabesh Das, Helen K. Saikia. On the Sum of Unitary Divisors Maximum Function[J]. AIMS Mathematics, 2017, 2(1): 96-101. doi: 10.3934/Math.2017.1.96

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Abstract

It is well-known that a positive integer

$d$ is called a unitary divisor of an integer

$n$ if

$d|n$ and gcd

$\left(d,\frac{n}{d}\right)=1$ . Divisor function

$\sigma^{*}(n)$ denote the sum of all such unitary divisors of

$n$ . In this paper we consider the maximum function

$U^{*}(n)=\max\{k\in\mathbb{N}:\sigma^{*}(k)|n\}$ and study the function

$U^{*}(n)$ for

$n=p^{m}$ , where

$p$ is a prime and

$m\geq 1$ .

1. Introduction

Notation: For two sequences of positive constants $\{a_n, n\geq1\}$ and $\{b_n, n\geq1\}$ , symbols $a_n\sim b_n$ , $a_n = O(b_n)$ and $a_n = o(b_n)$ stand for $\lim a_n/b_n = 1$ , $\lim a_n/b_n\in(0, \infty)$ and $\lim a_n/b_n = 0$ , respectively. For simplicity, we shall write $\stackrel{\mathbb{P}}{\longrightarrow}$ , $\stackrel{a.s.}{\longrightarrow}$ and $\stackrel{L^p}{\longrightarrow}$ to express the convergence in probability, the almost certain convergence and $p$ -mean convergence, respectively.

The following concept of superadditive function was introduced in ^[1].

Definition 1.1. A function $\phi: R^n\rightarrow R$ is called superadditive if $\phi(\boldsymbol{x}\vee\boldsymbol{y}) +\phi(\boldsymbol{x}\wedge\boldsymbol{y}) \geq\phi(\boldsymbol{x})+\phi(\boldsymbol{y})$ for all $\boldsymbol{x}, \boldsymbol{y}\in R^n$ , where $\vee$ is for componentwise maximum and $\wedge$ is for componentwise minimum.

Hu ^[2] introduced the concept of negatively superadditive-dependent (NSD) based on the above concept of superadditive function.

Definition 1.2. A random vector $\boldsymbol{X} = (X_1, X_2, \cdots, X_n)$ is said to be NSD if

$\mathbb{E}\phi(X_1, X_2, \cdots, X_n)\leq \mathbb{E}\phi(X_1^*, X_2^*, \cdots, X_n^*),$

where $X_1^*, X_2^*, \cdots, X_n^*$ are independent such that $X_i^*$ and $X_i$ have the same distribution for each $i$ and $\phi$ is a superadditive function such that the expectations in the above equation exists. A sequence $\{X_n, \, n\geq1\}$ of random variables is said to be NSD if for each $n\geq1$ , $(X_1, X_2, \cdots, X_n)$ is NSD.

Hu ^[2] established some basic properties and three structural theorems of NSD random variables. An interesting example was also presented in ^[2], which illustrated that NSD is not necessarily negatively associated (NA, ^[3]). Christofides and Vaggelatou ^[4] showed that NA is NSD. Eghbal et al. ^[5] derived two maximal inequalities and strong law of large numbers of quadratic forms of NSD random variables. Shen et al. ^[6] studied almost sure convergence and strong stability for weighted sums of NSD random variables. Wang et al. ^[7] studied complete convergence for arrays of rowwise NSD random variables, with applications to nonparametric regression. For more research of the limit theory for NSD random variables, the author can refer the reader to ^{[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]}.

NA random variable has been studied many times and attracted extensive attention, so it is very significant to investigate the limit theorems of this wider NSD class, which is highly desirable and of considerable significance in theory and application.

A random variable $X$ is called to be a two-tailed Pareto distribution whose density is

$\begin{align} f(x) = \left\{ {\begin{array}{*{20}{lrl}} {{ \frac{q}{x^2}\quad\mbox{if}\;x\leq-1}}, \\ {{ 0\quad\;\;\, \mbox{if}\;-1 < x < 1}}, \\ {{ \frac{p}{x^2}\quad\mbox{if}\;x\geq1, }} \end{array}} \right. \end{align}$

(1.1)

where $p+q = 1$ .

Let $\{X_n, \, n\geq1\}$ be independent Pareto-Zipf random variables satisfying $\mathbb{P}(X_n = 0) = 1-1/n$ ,

$\begin{align} \mathbb{P}(X_n\leq x) = 1-\frac{1}{x+n}\quad\mbox{for}\;\mbox{all}\;x > 0, \end{align}$

(1.2)

and $f_{X_n}(x) = \frac{1}{(x+n)^2}\mathbb{I}(x > 0)$ .

Obviously, if the random variable $X_n$ satisfies Eq (1.1) or (1.2), then $\mathbb{E}|X_n| = \infty$ , $n\geq1$ . Alder ^[25] considered independent and identically distributed (i.i.d.) random variables satisfying Eq (1.1) and studied the strong law of large numbers. Alder ^[26] obtained the weak law of large numbers for Pareto-Zipf random variables. For more research on laws of large numbers for i.i.d. random variables with infinite mean, the author can refer to works of Adler ^[27,28] and Matsumoto and Nakata ^[29,30,31].

Yang et al. ^[24] investigated the law of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means, and obtained the following theorems which extend and improve the corresponding ones in ^[25,26]:

Theorem 1.1. Let $\{X_n, \, n\geq1\}$ be a nonnegative sequence of NSD random variables whose distributions are defined by $\mathbb{P}(X_n = 0) = 1-1/c_n$ for $n\geq1$ and the tail probability

$\begin{align} \mathbb{P}(X_n > x) = \frac{1}{x+c_n}\quad\mathit{\mbox{for}}\;\mathit{\mbox{all}}\;x > 0\;\mathit{\mbox{and}}\;n\geq1, \end{align}$

(1.3)

where $\{c_n\}$ is a nondecreasing constant sequence with $c_n\geq1$ and

$\begin{align} C_n = \sum\limits_{j = 1}^{n}\frac{1}{c_j}\rightarrow \infty. \end{align}$

(1.4)

Then we have

$\begin{align} \frac{\sum_{j = 1}^{n}c_j^{-1}X_j}{C_n\log C_n}\stackrel{\mathbb{P}}{\longrightarrow}1. \end{align}$

(1.5)

Theorem 1.2. Let $\{X_n, \, n\geq1\}$ be a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1). Then for all $\beta > 0$ we have

$\begin{align} \frac{1}{\log^{\beta}n}\sum\limits_{j = 1}^{n}\frac{\log^{\beta-2}j}{j}X_j\stackrel{a.s.}{\longrightarrow}\frac{p-q}{\beta}. \end{align}$

(1.6)

In the current work, the author studies the weak and strong laws of large numbers for NSD random variables. The obtained results in this article extend and improve Theorems 1.1 and 1.2. Meanwhile, the author investigates $p$ -mean convergence for NSD random variables under some appropriate conditions, which was not considered in ^[24].

Throughout this paper, the symbol $C$ denotes a positive constant which may differ from one place to another. The symbol $\mathbb{I}(A)$ denotes the indicator function of the event $A$ .

2. Some lemmas and main results

To prove our main results, we first present some technical lemmas.

Lemma 2.1. (^[2]) If $(X_1, X_2, \cdots, X_n)$ is NSD and $f_1, f_2, \cdots, f_n$ are all non decreasing, then $(f_1(X_1), \; f_2(X_2), \cdots, f_n(X_n))$ is also NSD.

As we know, moment inequalities are very important tools in establishing the limit theorems for sequences of random variables. Shen et al. ^[6] presented the following Marcinkiewicz-Zygmund inequality with exponent $2$ .

Lemma 2.2. (^[6]) Let $\{X_n, \, n\geq1\}$ be a sequence of NSD random variables with $\mathbb{E}X_n = 0$ and $\mathbb{E}X_n^2 < \infty$ for $n\geq1$ . Then

$\mathbb{E}\Biggl(\max\limits_{1\leq k\leq n}\Biggl(\sum\limits_{i = 1}^kX_i\Biggr)^2\Biggl)\leq 2\sum\limits_{i = 1}^n\mathbb{E}X_i^2, \;\;n\geq1.$

By means of similar methods in Shao ^[32], Wang et al. ^[7] established the following Rosenthal-type maximal inequality, which is very useful in establishing the convergence properties for NSD random variables:

Lemma 2.3. (^[7]) Let $p > 1$ . Let $\{X_n, \, n\geq1\}$ be a sequence of NSD random variables with $\mathbb{E}|X_i|^p < \infty$ for each $i\geq1$ . Then for all $n\geq1$ ,

$\mathbb{E}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{i = 1}^kX_i\Biggr|^p\Biggl)\leq 2^{3-p}\sum\limits_{i = 1}^n\mathbb{E}|X_i|^p\;\;for \;1 < p\leq2$

and

$\mathbb{E}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{i = 1}^kX_i\Biggr|^p\Biggl)\leq 2\Biggl(\frac{15p}{\ln p}\Biggr)^p \Biggl[\sum\limits_{i = 1}^n\mathbb{E}|X_i|^p+\Biggl(\mathbb{E}X_i^2\Biggr)^{p/2}\Biggr]\;\;for \;p > 2.$

Lemma 2.4. (^[6]) Let $\{X_n, \, n\geq1\}$ be a sequence of NSD random variables. If

$\sum\limits_{n = 1}^{\infty}\mathit{\mbox{Var}}(X_n) < \infty,$

then $\sum_{n = 1}^{\infty}(X_n-EX_n)$ almost certainly converges.

Now we state our main results and the proofs will be presented in next section.

Theorem 2.1. Let $\{X_n, \, n\geq1\}$ be a nonnegative sequence of NSD random variables whose distributions are defined by $\mathbb{P}(X_n = 0) = 1-1/c_n$ for $n\geq1$ and the tail probability

$\begin{align} \mathbb{P}(X_n > x) = \frac{1}{x+c_n}\quad\mathit{\mbox{for}}\;\mathit{\mbox{all}}\;x > 0\;\mathit{\mbox{and}}\;n\geq1, \end{align}$

(2.1)

where $\{c_n, n\geq1\}$ is a nondecreasing constant sequence with $c_n\geq1$ and

$\begin{align} C_n = \sum\limits_{j = 1}^{n}\frac{1}{c_j}\rightarrow \infty. \end{align}$

(2.2)

Let $\{D_n, n\geq1\}$ be a sequence of constants satisfying $D_n\rightarrow \infty$ and $C_n = o(D_n)$ . Then we have

$\begin{align} \frac{1}{D_n}\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_j-\mathbb{E}X_{nj})\Biggr|\stackrel{\mathbb{P}}{\longrightarrow}0, \end{align}$

(2.3)

where $X_{nj} = X_j\mathbb{I}(X_j\leq D_nc_j)+D_nc_j\mathbb{I}(X_j > D_nc_j)$ , $1\leq j\leq n$ .

Take $D_n = C_n\log C_n$ , then we can obtain the following corollary which extends Theorem 1.1.

Corollary 2.1. Let $\{X_n, \, n\geq1\}$ be a nonnegative sequence of NSD random variables whose distributions are defined by $\mathbb{P}(X_n = 0) = 1-1/c_n$ for $n\geq1$ and the tail probability Eq (2.1), where $\{c_n, n\geq1\}$ is a nondecreasing constant sequence satisfying $c_n\geq1$ and Eq (2.2). Then

$\begin{align} \frac{1}{C_n\log C_n}\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_j-\mathbb{E}X_{nj})\Biggr|\stackrel{\mathbb{P}}{\longrightarrow}0. \end{align}$

(2.4)

Remark 2.1. Yang et al. ^[24] proved that

$\frac{1}{C_n\log C_n}\sum\limits_{j = 1}^nc_j^{-1}\mathbb{E}X_j\mathbb{I}(X_j\leq c_jC_n\log C_n)\rightarrow1$

and

$\frac{1}{C_n\log C_n}\sum\limits_{j = 1}^nc_j^{-1}\mathbb{E}\bigl(c_jC_n\log C_n\mathbb{I}(X_j > c_jC_n\log C_n)\bigr) = \sum\limits_{j = 1}^n\mathbb{P}(X_j > c_jC_n\log C_n)\rightarrow0,$

which yields

$\frac{1}{C_n\log C_n}\sum\limits_{j = 1}^nc_j^{-1}EX_{nj}\rightarrow1.$

Then we can find that Theorem 1.1 is a special case of Corollary 2.1 for $k = n$ . Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem 1.1.

Theorem 2.2. Let $\{X_n, \, n\geq1\}$ be a sequence of identically distributed NSD random variables. Let $\{d_n, n\geq1\}$ be a sequence of positive constants satisfying $d_n\uparrow\infty$ , and $\{c_n, n\geq1\}$ be a sequence of positive constants such that $\varphi(n)\equiv c_nd_n$ satisfies $\varphi(n)\rightarrow \infty$ as $n\rightarrow \infty$ ,

$\begin{align} \sum\limits_{m = n}^{\infty}\frac{1}{\varphi^2(m)} = O\bigl(n\varphi^{-2}(n)\bigr) \end{align}$

(2.5)

and

$\begin{align} \sum\limits_{n = 1}^{\infty}\mathbb{P}(|X_1| > \varphi(n)) < \infty. \end{align}$

(2.6)

Then

$\begin{align} \frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}(X_j-\mathbb{E}\widetilde{X_{j}})\longrightarrow0\;\;\mathit{\mbox{a.s.}}, \end{align}$

(2.7)

where $\widetilde{X_{j}} = -\varphi(j)\mathbb{I}(X_j < -\varphi(j))+X_j\mathbb{I}(|X_j|\leq \varphi(j))+\varphi(j)\mathbb{I}(X_j > \varphi(j))$ , $1\leq j\leq n$ .

Remark 2.2. We will show that Theorem 1.2 is a special case of Theorem 2.2. In fact, if we assume that $\{X_n, \, n\geq1\}$ is a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1), and take $c_n = n\log^{2-\beta}n$ and $d_n = \log^{\beta}n$ ( $\beta > 0$ ), then $\varphi(n) = c_nd_n = n\log^{2}n$ . We can verify that $\varphi(n) = n\log^{2}n$ satisfies the conditions stated in Theorem 2.2.

First, it is clear that $\varphi(n) = n\log^{2}n$ satisfies $\varphi(n)\rightarrow \infty$ as $n\rightarrow \infty$ .

Second, we have by standard calculations that

$\sum\limits_{m = n}^{\infty}\frac{1}{\varphi^2(m)} \sim\int_{n}^{\infty}\frac{1}{x^2\log^{4}x}{\mathrm{d}}x = O\bigl(n^{-1}\log^{-4}(n)\bigr) = O\bigl(n\varphi^{-2}(n)\bigr),$

which shows that Eq (2.5) is verified.

Next, we have by Eq (1.1) and $\varphi(n) = n\log^{2}n$ that

$\begin{eqnarray*} \sum\limits_{n = 1}^{\infty}\mathbb{P}(|X_1| > \varphi(n))& = &\sum\limits_{n = 1}^{\infty}\mathbb{P}(|X_1| > n\log^{2}n)\\ & = &\sum\limits_{n = 1}^{\infty}\Biggl(\int_{-\infty}^{-n\log^{2}n}qx^{-2}{\mathrm{d}}x+\int_{n\log^{2}n}^{\infty}px^{-2}{\mathrm{d}}x\Biggr)\\ & = &\sum\limits_{n = 1}^{\infty}\frac{p+q}{n\log^{2}n} = \sum\limits_{n = 1}^{\infty}\frac{1}{n\log^{2}n} < \infty \end{eqnarray*}$

and then Eq (2.6) is verified.

Finally, we also obtain by Eq (1.1) and $\varphi(j) = j\log^{2}j$ that

$\begin{eqnarray*} \frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}\mathbb{E}\widetilde{X_{j}} & = &\frac{1}{d_n}\sum\limits_{j = 1}^n\bigl(-d_j\mathbb{P}(X_j < -\varphi(j)) +c_j^{-1}\mathbb{E}X_{j}\mathbb{I}(|X_j| < \varphi(j))+d_j\mathbb{P}(X_j > \varphi(j))\bigr)\\ & = &\frac{p-q}{\log^{\beta}n}\sum\limits_{j = 1}^n\frac{\log^{\beta-2}j}{j}+\frac{p-q}{\log^{\beta}n}\sum\limits_{j = 1}^n\frac{\log^{\beta-1}j}{j}\\ & = \;:&J_1+J_2. \end{eqnarray*}$

By similar argument as in the proof of $H\rightarrow0$ in ^[24], we can obtain $J_1\rightarrow0$ . By similar argument as in the proof of Eq (3.5) in ^[24], we can prove $J_2\rightarrow \frac{p-q}{\beta}$ . Then we obtain by Eq (2.7) that

$\frac{1}{\log^{\beta}n}\sum\limits_{j = 1}^{n}\frac{\log^{\beta-2}j}{j}X_j\stackrel{a.s.}{\longrightarrow}\frac{p-q}{\beta}.$

To sum up, Theorem 1.2 is a special case of Theorem 2.2 and then Theorem 2.2 extends Theorem 1.2.

Next, we present a new theorem of $p$ -mean convergence for NSD random variables under some appropriate conditions, which was not considered in ^[24,25,26].

Theorem 2.3. Let $\{X_n, \, n\geq1\}$ be a sequence of NSD random variables satisfying

$\begin{align} \lim\limits_{x\to\infty}\sup\limits_{j\geq1}x^{\alpha}\mathbb{P}(|X_{j}| > x) < \infty, \;\;\alpha\in(1, 2). \end{align}$

(2.8)

Let $\{d_n, n\geq1\}$ be a sequence of positive constants satisfying $d_n\uparrow\infty$ , and $\{c_n, n\geq1\}$ be a sequence of positive constants such that $c_j\geq1$ and

$\begin{align} \sum\limits_{j = 1}^nc_j^{-\alpha} = o(d_n^{\alpha}). \end{align}$

(2.9)

Then for $p\in(1, \alpha)$ ,

$\begin{align} \frac{1}{d_n}\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_j-\mathbb{E}\widehat{X_{nj}})\Biggr|\stackrel{L^p}{\longrightarrow}0, \end{align}$

(2.10)

where $\widehat{X_{nj}} = -d_nc_j\mathbb{I}(X_j < -d_nc_j)+X_j\mathbb{I}(|X_j|\leq d_nc_j)+d_nc_j\mathbb{I}(X_j > d_nc_j)$ , $1\leq j\leq n$ .

3. The proofs

Proof of Theorem 2.1. We first observe that for every $\varepsilon > 0$ ,

$\begin{eqnarray*} & &\mathbb{P}\Biggl(\frac{1}{D_n}\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_j-\mathbb{E}X_{nj})\Biggr| > 2\varepsilon\Biggr)\\ &\leq&\mathbb{P}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_j-X_{nj})\Biggr| > D_n\varepsilon\Biggr) +\mathbb{P}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_{nj}-\mathbb{E}X_{nj})\Biggr| > D_n\varepsilon\Biggr)\\ & = \;:&H_1+H_2. \end{eqnarray*}$

To prove Eq (2.3), we need only to show that $H_i\rightarrow0$ as $n\rightarrow \infty$ , $i = 1, 2$ . For $H_1$ , we have by the definition of $X_{nj}$ , $C_n = o(D_n)$ , Eqs (2.1) and (2.2) that

$\begin{eqnarray*} H_1&\leq&\mathbb{P}\Biggl(\bigcup\limits_{j = 1}^n(X_j\neq X_{nj})\Biggr)\, \leq\, \sum\limits_{j = 1}^n\mathbb{P}(X_j > D_nc_j)\\ & = &\sum\limits_{j = 1}^n\frac{1}{D_nc_j+c_j} = \frac{1}{D_n+1}\sum\limits_{j = 1}^nc_j^{-1} = \frac{C_n}{D_n+1}\rightarrow0.\\ \end{eqnarray*}$

For fixed $n\geq1$ , $X_{nj}$ is the nondecreasing function of $X_{j}$ . Hence, it follows by Lemma 2.1 that $\{X_{nj}, \, 1\leq j\leq n\}$ is a sequence of NSD random variables. Hence we have by Markov's inequality and Lemma 2.3 with $1 < p\leq2$ ,

$\begin{eqnarray*} H_2&\leq&\frac{C}{D_n^p}\mathbb{E}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}(X_{nj}-\mathbb{E}X_{nj})\Biggr|\Biggr)^p\, \leq\, \frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|X_{nj}|^p\\ & = &\frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|X_{j}|^p\mathbb{I}(X_j\leq D_nc_j)+C\sum\limits_{j = 1}^n\mathbb{P}(X_j > D_nc_j)\\ & = &\frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{0}^{(D_nc_j)^p}\mathbb{P}(|X_{j}|^p\mathbb{I}(X_j\leq D_nc_j)\geq t){\mathrm{d}}t+C\sum\limits_{j = 1}^n\frac{1}{D_nc_j+c_j}\\ & = &\frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{0}^{(D_nc_j)^p}\mathbb{P}(|X_{j}|^p\geq t){\mathrm{d}}t+C\frac{C_n}{D_n+1}\\ & = &\frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{0}^{(D_nc_j)^p}\frac{1}{t^{1/p}+c_j}{\mathrm{d}}t+C\frac{C_n}{D_n+1} \qquad\qquad\, (\mbox{by}\;(2.1)\;)\\ &\leq&\frac{C}{D_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{0}^{(D_nc_j)^p}\frac{1}{t^{1/p}}{\mathrm{d}}t+C\frac{C_n}{D_n+1}\\ & = &C\frac{C_n}{D_n}+C\frac{C_n}{D_n+1}\rightarrow0. \end{eqnarray*}$

The proof is completed.

Proof of Theorem 2.2. Obviously, to prove Eq (2.7), we need only to show

$\begin{align} \frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}\bigl(X_j-\widetilde{X_{j}}\bigr)\longrightarrow0\;\;\mbox{a.s.} \end{align}$

(3.1)

and

$\begin{align} \frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}\bigl(\widetilde{X_{j}}-\mathbb{E}\widetilde{X_{j}}\bigr)\longrightarrow0\;\;\mbox{a.s.}. \end{align}$

(3.2)

By Eq (2.6), $d_n\uparrow\infty$ and the Borel-Cantelli lemma, we obtain

$\frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}|X_j|\mathbb{I}(|X_j| > \varphi(j))\longrightarrow0\;\;\mbox{a.s.}.$

Noting that

$|X_j+\varphi(j)|\mathbb{I}(X_j < -\varphi(j))+|X_j-\varphi(j)|\mathbb{I}(X_j > \varphi(j))\leq |X_j|\mathbb{I}(|X_j| > \varphi(j)).$

Then

$\begin{eqnarray*} &&\Biggl|\frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}(X_j-\widetilde{X_{j}})\Biggr|\\ & = &\Biggl|\frac{1}{d_n}\sum\limits_{j = 1}^nc_j^{-1}X_j\mathbb{I}(|X_j| > \varphi(j)) +(X_j+\varphi(j))\mathbb{I}(X_j < -\varphi(j))+(X_j-\varphi(j))\mathbb{I}(X_j > \varphi(j))\Biggr|\\ &\leq&\frac{2}{d_n}\sum\limits_{j = 1}^nc_j^{-1}|X_j|\mathbb{I}(|X_j| > \varphi(j))\longrightarrow0\;\;\mbox{a.s.}, \end{eqnarray*}$

which yields Eq (3.1).

It follows by the definition of $\widetilde{X_{j}}$ that

$\begin{eqnarray*} \sum\limits_{j = 1}^{\infty}\frac{1}{\varphi^2(j)}\mathbb{E}\bigl(\widetilde{X_{j}}-\mathbb{E}\widetilde{X_{j}}\bigr)^2 &\leq&C\sum\limits_{j = 1}^{\infty}\frac{1}{\varphi^2(j)}\mathbb{E}X_{j}^2\mathbb{I}(|X_j|\leq\varphi(j))+C\sum\limits_{j = 1}^{\infty}\mathbb{P}(|X_j| > \varphi(j))\\ & = \;:&I_1+I_2. \end{eqnarray*}$

We obtain directly by Eq (2.6) that $I_2 < \infty$ . Let $F(x)$ be the distribution of $X_1$ , then

$\begin{eqnarray*} I_1& = &C\sum\limits_{j = 1}^{\infty}\frac{1}{\varphi^2(j)}\, \mathbb{E}X_{1}^2\mathbb{I}(|X_1|\leq\varphi(j))\\ & = &C\sum\limits_{j = 1}^{\infty}\frac{1}{\varphi^2(j)}\int_{-\infty}^{\infty}x^2\mathbb{I}(|X_1|\leq\varphi(j)){\mathrm{d}}F(x) \end{eqnarray*}$

$\begin{align} = \, \, \;C\int_{-\infty}^{\infty}x^2\sum\limits_{j:\, \varphi(j)\geq|x|}\frac{1}{\varphi^2(j)}\, {\mathrm{d}}F(x).\qquad \end{align}$

(3.3)

Define $N(|x|) = \sharp\{j:\, \varphi(j) < |x|\}$ and $j_* = \inf\{j:\, \varphi(j)\geq|x|\}$ . Hence we can obtain $N(|x|)\geq j_*-1$ and

$\begin{eqnarray*} \sum\limits_{j:\, \varphi(j)\geq|x|}\frac{1}{\varphi^2(j)}&\leq&\sum\limits_{j = j_*}^{\infty}\frac{1}{\varphi^2(j)}\\ &\leq&C\frac{j_*}{\varphi^2(j_*)}\qquad\qquad(\mbox{by Eq (2.5)})\\ &\leq&C\frac{j_*}{x^2} \end{eqnarray*}$

$\begin{align} \quad\;\, \leq\, \, \;C\frac{\;N(|x|)+1\;}{x^2}. \end{align}$

(3.4)

It follows by Eqs (2.6), (3.3) and (3.4) that

$\begin{eqnarray*} I_1&\leq&C\int_{-\infty}^{\infty} (N(|x|)+1)\, {\mathrm{d}}F(x)\, = \, C\mathbb{E}N(|X_1|)+C\\ & = &CE\Biggl[\sum\limits_{j = 1}^{\infty}\mathbb{I}(|X_1| > \varphi(j))\Biggr]+C\\ & = &C\sum\limits_{j = 1}^{\infty}\mathbb{P}(|X_1| > \varphi(j))+C < \infty. \end{eqnarray*}$

Now we obtain by $I_1 < \infty$ and $I_2 < \infty$ that

$\begin{align} \sum\limits_{j = 1}^{\infty}\frac{1}{\varphi^2(j)}\mathbb{E}\bigl(\widetilde{X_{j}}-\mathbb{E}\widetilde{X_{j}}\bigr)^2 < \infty. \end{align}$

(3.5)

Consequently, by Lemma 2.4 and Eq (3.5), we get

$\sum\limits_{j = 1}^{\infty}\frac{1}{\varphi(j)}\bigl(\widetilde{X_{j}}-\mathbb{E}\widetilde{X_{j}}\bigr)\quad \mbox{converges}\;\;\mbox{a.s.},$

which implies Eq (3.2) by Kronecker's lemma, together with the condition $d_n\uparrow\infty$ .

The proof is completed.

Proof of Theorem 2.3. Noting that

$\begin{eqnarray*} &&\mathbb{E}\Biggl\{\frac{1}{d_n}\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(X_j-\mathbb{E}\widehat{X_{nj}}\bigr)\Biggr|\Biggr\}^p\\ &\leq&\frac{1}{d_n^p}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(\widehat{X_{nj}}-\mathbb{E}\widehat{X_{nj}}\bigr)\Biggr|\Biggr\}^p +\frac{1}{d_n^p}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(X_{j}-\widehat{X_{nj}}\bigr)\Biggr|\Biggr\}^p\\ &\leq&\frac{1}{d_n^p}\Biggl\{\mathbb{E}\Biggl(\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(\widehat{X_{nj}}-\mathbb{E}\widehat{X_{nj}}\bigr)\Biggr|\Biggr)^2\Biggr\}^{p/2} +\frac{1}{d_n^p}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(X_{j}-\widehat{X_{nj}}\bigr)\Biggr|\Biggr\}^p\\ & = \;:&J_1+J_2. \end{eqnarray*}$

To prove Eq (2.10), it is sufficient to prove $J_1\rightarrow0$ and $J_2\rightarrow0$ . By Lemma 2.1 and the fact that $\widehat{X_{nj}}$ is the nondecreasing function of $X_{j}$ , $\{\widehat{X_{nj}}, \, 1\leq j\leq n\}$ is also a sequence of NSD random variables.

We have by Lemma 2.2 that

$\begin{eqnarray*} J_1^{2/p}& = &\frac{1}{d_n^2}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(\widehat{X_{nj}}-\mathbb{E}\widehat{X_{nj}}\bigr)\Biggr|\Biggr\}^2\\ &\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\mathbb{E}\bigl(\widehat{X_{nj}}-\mathbb{E}\widehat{X_{nj}}\bigr)^2\\ &\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\mathbb{E}X_j^2\mathbb{I}(|X_j|\leq d_nc_j)+C\sum\limits_{j = 1}^n\mathbb{P}(|X_j| > d_nc_j)\\ & = \;:&J_3+J_4. \end{eqnarray*}$

By $d_n\uparrow\infty$ , Eqs (2.8) and (2.9), we have

$\begin{align} J_4\, \leq\, C\frac{1}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{align}$

(3.6)

Now we will show $J_3\rightarrow0$ . Observing

$\begin{eqnarray*} J_3& = &\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{(d_nc_j)^2}\mathbb{P}(X_j^2\mathbb{I}(|X_j|\leq d_nc_j)\geq t) {\mathrm{d}}t\\ &\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{(d_nc_j)^2}\mathbb{P}(X_j^2\geq t) {\mathrm{d}}t. \end{eqnarray*}$

Let $t = u^2$ , then

$J_3\leq \frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{d_nc_j}u\mathbb{P}(|X_j|\geq u) {\mathrm{d}}u.$

From Eq (2.8), we know that, there exists $M > 0$ and $N_0\in {\mathrm{\textbf{N}}}$ such that

$\begin{align} \mathbb{P}(|X_j|\geq u)\leq M u^{-\alpha}\quad\mbox{for}\;\;u > N_0. \end{align}$

(3.7)

Since $d_n\uparrow\infty$ and $c_j\geq1$ , while $n$ is sufficiently large, we can obtain $d_nc_j > N_0$ . Hence

$\begin{eqnarray*} J_3&\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{N_0}u\mathbb{P}(|X_j|\geq u) {\mathrm{d}}u +\frac{CM}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_{N_0}^{d_nc_j}u^{1-\alpha} {\mathrm{d}}u\\ & = \;:&J_3'+J_3''. \end{eqnarray*}$

By $\alpha < 2$ , $c_j\geq1$ and Eq (2.9), we have

$\begin{eqnarray*} J_3'&\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\int_0^{N_0}u{\mathrm{d}}u\;\leq\;\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\\ &\leq&\frac{C}{d_n^{2-\alpha}}\frac{1}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty \end{eqnarray*}$

and

$\begin{eqnarray*} J_3''&\leq&\frac{C}{d_n^2}\sum\limits_{j = 1}^nc_j^{-2}\bigl[(d_nc_j)^{2-\alpha}-N_0^{2-\alpha}\bigr]\\ &\leq&\frac{C}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{eqnarray*}$

Finally, we need only to show $J_2\rightarrow0$ as $n\rightarrow \infty$ . Let

$Z_{nj} = X_{j}-\widehat{X_{nj}} = (X_{j}+d_nc_j)\mathbb{I}(X_j < - d_nc_j)+(X_{j}-d_nc_j)\mathbb{I}(X_j > d_nc_j).$

We first prove that

$\begin{align} \mathbb{E}Z_{nj}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{align}$

(3.8)

Observing

$\begin{eqnarray*} |\mathbb{E}Z_{nj}|&\leq&\mathbb{E}|Z_{nj}|\, \leq\, \mathbb{E}|X_{j}|\mathbb{I}(|X_{j}| > d_nc_j)\\ & = &\Biggl(\int_0^{d_nc_j}+\int_{d_nc_j}^{\infty}\Biggr)\mathbb{P}(|X_{j}|\mathbb{I}(|X_{j}| > d_nc_j)\geq t) {\mathrm{d}}t\\ & = &\int_0^{d_nc_j}\mathbb{P}(|X_{j}| > d_nc_j) {\mathrm{d}}t+\int_{d_nc_j}^{\infty}\mathbb{P}(|X_{j}|\geq t) {\mathrm{d}}t\\ & = &d_nc_j\mathbb{P}(|X_{j}| > d_nc_j)+\int_{d_nc_j}^{\infty}\mathbb{P}(|X_{j}|\geq t) {\mathrm{d}}t\\ & = \;:&J_5+J_6. \end{eqnarray*}$

By Eq (3.7) and $\alpha > 1$ , we have

$J_5\leq \frac{M}{(d_nc_j)^{\alpha-1}}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty$

and

$J_6\leq M\int_{d_nc_j}^{\infty}t^{-\alpha} {\mathrm{d}}t\leq\frac{CM}{(d_nc_j)^{\alpha-1}}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty,$

which yields Eq (3.8). Therefore, we obtain by Lemma 2.3 that

$\begin{eqnarray*} J_2&\leq&\frac{1}{d_n^p}\mathbb{E}\Biggl\{\max\limits_{1\leq k\leq n}\Biggl|\sum\limits_{j = 1}^kc_j^{-1}\bigl(Z_{nj}-\mathbb{E}Z_{nj}\bigr)\Biggr|\Biggr\}^p\\ &\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|Z_{nj}|^p\\ &\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\mathbb{E}|X_{j}|^p\mathbb{I}(|X_{j}| > d_nc_j).\quad\quad(\mbox{by}\;\mbox{the}\, \mbox{definition}\, \mbox{of}\, Z_{nj}) \end{eqnarray*}$

By similar arguments as in the proof of Eq (3.8), we can obtain

$\mathbb{E}|X_{j}|^p\mathbb{I}(|X_{j}| > d_nc_j) = (d_nc_j)^p\mathbb{P}(|X_{j}| > d_nc_j)+\int_{(d_nc_j)^p}^{\infty}\mathbb{P}(|X_{j}|^p\geq t) {\mathrm{d}}t.$

Then

$\begin{eqnarray*} J_2&\leq&C\sum\limits_{j = 1}^n\mathbb{P}(|X_{j}| > d_nc_j)+\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{(d_nc_j)^p}^{\infty}\mathbb{P}(|X_{j}|^p\geq t) {\mathrm{d}}t\\ & = \;:&J_2'+J_2''. \end{eqnarray*}$

By similar arguments as the proof of $J_4\rightarrow0$ , we obtain $J_2'\rightarrow0$ . We also have by Eq (3.7), $p < \alpha$ and Eq (2.9) that

$\begin{eqnarray*} J_2''&\leq&\frac{C}{d_n^p}\sum\limits_{j = 1}^nc_j^{-p}\int_{(d_nc_j)^p}^{\infty}t^{-\alpha/p} {\mathrm{d}}t\\ &\leq&\frac{C}{d_n^{\alpha}}\sum\limits_{j = 1}^nc_j^{-\alpha}\rightarrow0\qquad\mbox{as}\quad n\rightarrow \infty. \end{eqnarray*}$

The proof is completed.

4. Conclusions

In this work the author investigated the limit theorems for negatively superadditive-dependent random variables, and obtained some new results on the law of large numbers and mean convergence under some appropriate conditions. As a future work, we propose to consider some other strong convergence for sequence of negatively superadditive-dependent random variables.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Social Sciences Planning Project of Anhui Province (AHSKY2018D98).

Conflict of interest

The author declares that he has no conflict of interest.

References

[1]	C. Ashbacker, An introduction to the Smarandache function, Erhus Univ. Press ,Vail, AZ, 1995.
[2]	David M. Burton, Elementary number theory, Tata McGraw-Hill Sixth Edition, 2007.
[3]	E. Cohen, Arithmetical functions associated with the unitary divisors of an integer. Math. Zeits., 74 (1960), 66-80.
[4]	P. Moree, H. Roskam, On an arithmetical function related to Euler’s totient and the discriminator. Fib. Quart., 33 (1995), 332-340.
[5]	J. Sandor, On certain generalizations of the Smarandnche function. Notes Num. Th. Discr. Math., 5 (1999), 41-51.
[6]	J. Sandor, A note on two arithmetic functions. Octogon Math. Mag., 8 (2000), 522-524.
[7]	J. Sandor, On a dual of the Pseudo-Smarandache function. Smarandnche Notition Journal, 13 (2002).
[8]	J. Sandor, A note on exponential divisors and related arithmetic functions. Sci. Magna, 1 (2005), 97-101.
[9]	J. Sandor, The unitary totient minimum and maximum functions. Sci. Studia Univ. ”Babes-Bolyai”, Mathematica, 2 (2005), 91-100.
[10]	J. Sandor, The product of divisors minimum and maximum function. Scientia Magna, 5 (2009), 13-18.
[11]	J. Sandor, On the Euler minimum and maximum functions. Notes Num. Th. Discr. Math., 15 (2009), 1-8.

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AIMS Mathematics

On the Sum of Unitary Divisors Maximum Function

Related Papers:

Abstract

1. Introduction

2. Some lemmas and main results

3. The proofs

4. Conclusions

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Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. Some lemmas and main results

3. The proofs

4. Conclusions

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Conflict of interest

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AIMS Mathematics

On the Sum of Unitary Divisors Maximum Function

Related Papers:

Abstract

1. Introduction

2. Some lemmas and main results

3. The proofs

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

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1. Introduction

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