arXiv subjects

Andrew Rosalsky

Publications and source records attributed to Andrew Rosalsky.

8 recordsLinked to original sources

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 \[\ θ= \sup\left \{ r \geq 0:~ \mathbb{E}|X|^{r} < \infty \right \}. \] We call $θ$ 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 $θ$ and investigate its asymptotic properties: \[ \hatθ_{n} = \frac{\log n}{\log \max_{1 \leq k \leq n} |X_{k}|}, \] where $\log x = \ln(e \vee x), ~- \infty < x < \infty$. In particular, we show that \[ \hatθ_{n} \rightarrow_{\mathbb{P}} θ~~\mbox{if and only if}~~ \lim_{x \rightarrow \infty} x^{r} \mathbb{P}(|X| > x) = \infty ~~\forall~r > θ. \] This means that, under very reasonable conditions on $F(\cdot)$, $\hatθ_{n}$ is actually a consistent estimator of $θ$. 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 $φ(\cdot)$ and $ψ(\cdot)$ be two continuous and increasing functions defined on $[0, \infty)$ such that $φ(0) = ψ(0) = 0$, $\lim_{t \rightarrow \infty} φ(t) = \infty$, and $\frac{ψ(\cdot)}{φ(\cdot)}$ 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 $t \geq 0$, \[ \mathbb{P}\left(\left\|\sum_{i=1}^{n} V_{i} \right\| > t b_{n} \right) \leq 4 \mathbb{P} \left(\left\|\sum_{i=1}^{n} φ\left(ψ^{-1}(\|V_{i}\|)\right) \frac{V_{i}}{\|V_{i}\|} \right\| > t a_{n} \right) + \sum_{i=1}^{n}\mathbb{P}\left(\|V_{i}\| > b_{n} \right), \] where $a_{n} = φ(n)$ and $b_{n} = ψ(n)$, $n \geq 1$. 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↗

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↗

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↗