arXiv subjects

Deli Li

Publications and source records attributed to Deli Li.

17 recordsLinked to original sources

arXiv2026

Some Results on the Central Limit Theorem for Subsequences in Banach Spaces

Let $\{X, X_{n}; n \geq 1 \}$ be a sequence of i.i.d. $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. This note is devoted to study the classical central limitr theorem for subsequences of sums of i.i.d. $\mathbf{B}$-valued random variables. We show that, under the assumption that $\mathbf{B}$ is of cotype $2$ space, $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ converges weakly if and only if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. We conjecture that this result is false if $\mathbf{B}$ is not of cotype $2$ space. In addition, we show that, if $\left(\frac{S_{m_{n}}}{\sqrt{{m_{n}}}} \right)_{n \geq 1}$ converges weakly for a subsequence $\{m_{n}; ~n \geq 1\}$ of positive integers. and $\left(\frac{S_{n}}{\sqrt{n}} \right)_{n \geq 1}$ does not converge weakly, then $\displaystyle \left(S_{n}/a_{n} \right)_{n \geq 1}$ does not converge weakly to a non-degenerate probability measure for any sequence $\{a_{n}; n \geq 1 \}$ of positive real numbers.

math.PR↗

arXiv2022

Some results on probabilities of moderate deviations

Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}X^{2} < \infty$. Let $S_{n} = \sum_{i=1}^{n} X_{i}$, $n \geq 1$. Let $g(\cdot): ~[0, \infty) \rightarrow [0, \infty)$ be a nondecreasing regularly varying function with index $\rho \geq 0$ and $\lim_{t \rightarrow \infty} g(t) = \infty$. Let $\mu = \mathbb{E}X$ and $\sigma^{2} = \mathbb{E}(X - \mu)^{2}$. In this paper, on the scale $g(\log n)$, we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form $\displaystyle \log \mathbb{P}\left(S_{n} - n \mu > x \sqrt{ng(\log n)} \right)$, $\displaystyle \log \mathbb{P}\left(S_{n} - n \mu < -x \sqrt{ng(\log n)} \right)$, and $\displaystyle \log \mathbb{P}\left(\left|S_{n} - n \mu \right| > x \sqrt{ng(\log n)} \right)$ for all $x > 0$. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of $X$.

math.PR↗

arXiv2021

Pearson's goodness-of-fit tests for sparse distributions

Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say $k$, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with $k-1$ degrees of freedom when the sample size $n$ goes to infinity. In real applications, the number $k$ often changes with $n$ and may be even much larger than $n$. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.

stat.ME↗

arXiv2020

A supplement to the laws of large numbers and the large deviations

Let $0 < p < 2$. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independent and identically distributed $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. In this paper, a supplement to the classical laws of large numbers and the classical large deviations is provided. We show that if $S_{n}/n^{1/p} \rightarrow_{\mathbb{P}} 0$, then, for all $s > 0$, \[ \limsup_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = - (\bar{\beta} - p)/p \] and \[ \liminf_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = -(\underline{\beta} - p)/p, \] where \[ \bar{\beta} = - \limsup_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t} ~~\mbox{and}~~\underline{\beta} = - \liminf_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t}. \] The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-J{\o}rgensen (1974), de Acosta (1981), and Ledoux and Talagrand (1991), respectively. As a special case of this result, the main results of Hu and Nyrhinen (2004) are not only improved, but also extended.

math.PR↗

arXiv2018

Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The spectral radius is defined as the maximum absolute value of the $n$ eigenvalues of the product matrix. When $m=1$, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When $m$ is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When $m=m_n$ tends to infinity as $n$ goes to infinity, we show that the logarithmic spectral radii have a normal limit.

math.ST↗

arXiv2017

A Simple Point Estimator of the Power of Moments

Let $X$ be an observable random variable with unknown distribution function $F(x) = \mathbb{P}(X \leq x), - \infty < x < \infty$, and let \[\ \theta = \sup\left \{ r \geq 0:~ \mathbb{E}|X|^{r} < \infty \right \}. \] We call $\theta$ the power of moments of the random variable $X$. Let $X_{1}, X_{2}, ..., X_{n}$ be a random sample of size $n$ drawn from $F(\cdot)$. In this paper we propose the following simple point estimator of $\theta$ and investigate its asymptotic properties: \[ \hat{\theta}_{n} = \frac{\log n}{\log \max_{1 \leq k \leq n} |X_{k}|}, \] where $\log x = \ln(e \vee x), ~- \infty < x < \infty$. In particular, we show that \[ \hat{\theta}_{n} \rightarrow_{\mathbb{P}} \theta~~\mbox{if and only if}~~ \lim_{x \rightarrow \infty} x^{r} \mathbb{P}(|X| > x) = \infty ~~\forall~r > \theta. \] This means that, under very reasonable conditions on $F(\cdot)$, $\hat{\theta}_{n}$ is actually a consistent estimator of $\theta$. Hypothesis testing for the power of moments is conducted and, as an application of our main results, the formula for finding the p-value of the test is given. In addition, a theoretical application of our main results is provided together with three illustrative examples.

math.PR↗

arXiv2017

An Extension of Feller's Strong Law of Large Numbers

~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independent and identically distributed Banach space valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let $\{a_{n}; n \geq 1\}$ and $\{b_{n}; n \geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left\{b_{n}/a_{n};~ n \geq 1 \right\}$ is a nondecreasing sequence. We show that \[ \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0~~\mbox{almost surely} \] for every Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$ if $S_{n}/a_{n} \rightarrow 0$ almost surely for every symmetric Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.

math.PR↗

arXiv2017

A probability inequality for sums of independent Banach space valued random variables

Let $(\mathbf{B}, \|\cdot\|)$ be a real separable Banach space. Let $\varphi(\cdot)$ and $\psi(\cdot)$ be two continuous and increasing functions defined on $[0, \infty)$ such that $\varphi(0) = \psi(0) = 0$, $\lim_{t \rightarrow \infty} \varphi(t) = \infty$, and $\frac{\psi(\cdot)}{\varphi(\cdot)}$ is a nondecreasing function on $[0, \infty)$. Let $\{V_{n};~n \geq 1 \}$ be a sequence of independent and symmetric {\bf B}-valued random variables. In this note, we establish a probability inequality for sums of independent {\bf B}-valued random variables by showing that for every $n \geq 1$ and all $t \geq 0$, \[ \mathbb{P}\left(\left\|\sum_{i=1}^{n} V_{i} \right\| > t b_{n} \right) \leq 4 \mathbb{P} \left(\left\|\sum_{i=1}^{n} \varphi\left(\psi^{-1}(\|V_{i}\|)\right) \frac{V_{i}}{\|V_{i}\|} \right\| > t a_{n} \right) + \sum_{i=1}^{n}\mathbb{P}\left(\|V_{i}\| > b_{n} \right), \] where $a_{n} = \varphi(n)$ and $b_{n} = \psi(n)$, $n \geq 1$. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed ${\bf B}$-valued random variables.

math.PR↗

arXiv2015

A comparison theorem for the law of large numbers in Banach spaces

Let $(\mathbf{B}, \|\cdot\|)$ be a real separable Banach space. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. {\bf B}-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let $\{a_{n}; n \geq 1\}$ and $\{b_{n}; n \geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left\{b_{n}/a_{n};~ n \geq 1 \right\}$ is a nondecreasing sequence. In this paper, we provide a comparison theorem for the law of large numbers for i.i.d. {\bf B}-valued random variables. That is, we show that $\displaystyle \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0$ almost surely (resp. in probability) for every {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > b_{n}) = 0$) if $S_{n}/a_{n} \rightarrow 0$ almost surely (resp. in probability) for every symmetric {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > a_{n}) = 0$). To establish this comparison theorem for the law of large numbers, we invoke two tools: 1) a comparison theorem for sums of independent {\bf B}-valued random variables and, 2) a symmetrization procedure for the law of large numbers for sums of independent {\bf B}-valued random variables. A few consequences of our main results are provided.

math.PR↗

arXiv2014

A Characterization of Chover-Type Law of Iterated Logarithm

Let $0 < α\leq 2$ and $- \infty < β< \infty$. Let $\{X_{n}; n \geq 1 \}$ be a sequence of independent copies of a real-valued random variable $X$ and set $S_{n} = X_{1} + \cdots + X_{n}, ~n \geq 1$. We say $X$ satisfies the $(α, β)$-Chover-type law of the iterated logarithm (and write $X \in CTLIL(α, β)$) if $\limsup_{n \rightarrow \infty} \left| \frac{S_{n}}{n^{1/α}} \right|^{(\log \log n)^{-1}} = e^β$ almost surely. This paper is devoted to a characterization of $X \in CTLIL(α, β)$. We obtain sets of necessary and sufficient conditions for $X \in CTLIL(α, β)$ for the five cases: $α= 2$ and $0 < β< \infty$, $α= 2$ and $β= 0$, $1 < α< 2$ and $-\infty < β< \infty$, $α= 1$ and $- \infty < β< \infty$, and $0 < α< 1$ and $-\infty < β< \infty$. As for the case where $α= 2$ and $-\infty < β< 0$, it is shown that $X \notin CTLIL(2, β)$ for any real-valued random variable $X$. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm (i.e., $X \in CTLIL(α, 1/α)$) is given; that is, $X \in CTLIL(α, 1/α)$ if and only if $\inf \left \{b:~ \mathbb{E} \left(\frac{|X|^α}{(\log (e \vee |X|))^{bα}} \right) < \infty \right\} = 1/α$ where $\mathbb{E}X = 0$ whenever $1 < α\leq 2$.

math.PR↗

arXiv2012

A Characterization of a New Type of Strong Law of Large Numbers

By applying results obtained from the new versions of the classical Levy, Ottaviani, and Hoffmann-Jorgensen (1974) inequalities proved by Li and Rosalsky(2013) and by using techniques developed by Hechner and Heinkel (2010), we provide a characterization of a new type of strong law of large numbers for independent and identically distributed real-valued random variables. Versions of this strong law of large numbers are also presented in a Banach space setting.

math.PR↗

arXiv2010

On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

Let $ \{X, X_{k,i}; i \geq 1, k \geq 1 \}$ be a double array of nondegenerate i.i.d. random variables and let $\{p_{n}; n \geq 1 \}$ be a sequence of positive integers such that $n/p_{n}$ is bounded away from $0$ and $\infty$. This paper is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry $L_{n} = \max_{1 \leq i < j \leq p_{n}} \left | \hatρ^{(n)}_{i,j} \right |$ of the sample correlation matrix ${\bf Γ}_{n} = \left ( \hatρ_{i,j}^{(n)} \right )_{1 \leq i, j \leq p_{n}}$ where $\hatρ^{(n)}_{i,j}$ denotes the Pearson correlation coefficient between $(X_{1, i},..., X_{n,i})'$ and $(X_{1, j},..., X_{n,j})'$. We show under the assumption $\mathbb{E}X^{2} < \infty$ that the following three statements are equivalent: \begin{align*} & {\bf (1)} \quad \lim_{n \to \infty} n^{2} \int_{(n \log n)^{1/4}}^{\infty} \left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x} \right) \right) dF(x) = 0, \\ & {\bf (2)} \quad \left ( \frac{n}{\log n} \right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2, \\ & {\bf (3)} \quad \lim_{n \rightarrow \infty} \mathbb{P} \left (n L_{n}^{2} - a_{n} \leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 π}} e^{-t/2} \right \}, - \infty < t < \infty \end{align*} where $F(x) = \mathbb{P}(|X| \leq x), x \geq 0$ and $a_{n} = 4 \log p_{n} - \log \log p_{n}$, $n \geq 2$. To establish this result, we present six interesting new lemmas which may be beneficial to the further study of the sample correlation matrix.

math.PR↗

arXiv2010

A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers

For the partial sums formed from a sequence of i.i.d. random variables having a finite absolute p'th moment for some p in (0,2), we extend the recent and striking discovery of Hechner and Heinkel (Journal of Theoretical Probability (2010)) concerning "complete moment convergence" to the two cases 0<p<1 and p=1. Moreover, for 0<p<2, we obtain "almost sure convergence" analogues of these "complete moment convergence" results and these "almost sure convergence" analogues may be regarded as being a refinement of the celebrated Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. Versions of the above results in a Banach space setting are also presented.

math.PR↗

arXiv2008

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n\geq 1$, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form $$ \sum_{n \ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k|^{\frac{1}{q}} - εb_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$ and either $n_0=1, 0<p<2, a_n=1, b_n=n$ or $n_0=3, p=2, a_n=(\log n)^{-\frac{1}{2q}}, b_n=n\log n$. These results extend results of Chow (1988) and Li and Spătaru (2005) from the independent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.

math.PR↗

arXiv2007

Characterization of LIL behavior in Banach space

In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper, we show that there are analogous results in the Banach space setting. One of our main new tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.

math.PR↗

arXiv2006

Some strong limit theorems for the largest entries of sample correlation matrices

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i 1/2)$, (ii) $\lim_{n\to \infty}n^{1-α}L_n=0$ a.s. $(1/2<α\leq 1)$, (iii) $\lim_{n\to \infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\to \infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s. are shown to hold under optimal sets of conditions. These results follow from some general theorems proved for arrays of i.i.d. two-dimensional random vectors. The converses of the limit laws (i) and (iii) are also established. The current work was inspired by Jiang's study of the asymptotic behavior of the largest entries of sample correlation matrices.

math.PR↗

arXiv2005

Some results on two-sided LIL behavior

Let {X,X_n;n\geq 1} be a sequence of i.i.d. mean-zero random variables, and let S_n=\sum_{i=1}^nX_i,n\geq 1. We establish necessary and sufficient conditions for having with probability 1, 0 1 and to h(n)=(\log n)^r, r>0, we obtain analogues of the Hartman-Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {S_n/c_n;n\ge 1}, where c_n is a sufficiently regular normalizing sequence.

math.PR↗