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 $L_{n} = \max_{1 \leq i < j \leq p_{n}} \left | \hat{\rho}^{(n)}_{i,j} \right |$ of the sample correlation matrix ${\bf \Gamma}_{n} = \left ( \hat{\rho}_{i,j}^{(n)} \right )_{1 \leq i, j \leq p_{n}}$ where $\hat{\rho}^{(n)}_{i,j}$ 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 \pi}} 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.Comment: 16 page

A generalization of the global limit theorems of R. P. Agnew

For distribution functions {Fn,n≥0}, the relationship between the weak convergence of Fn to F0 and the convergence of ∫Rϕ(|Fn−F0|)dx to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew wherein ϕ(t)=|t|r. The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line

The Davis-Gut law for independent and identically distributed Banach space valued random elements

An analog of the Davis-Gut law for a sequence of independent and identically distributed Banach space valued random elements is obtained, which extends the result of Li and Rosalsky (A supplement to the Davis-Gut law. J. Math. Anal. Appl. 330 (2007), 1488-1493)

On the Weak Law with Random Indices for Arrays of Banach Space Valued Random Elements

Abstract For a sequence of constants {an, n ≥ 1}, an array of rowwise independent and stochastically dominated random elements {Vnj, j ≥ 1, n ≥ 1} in a real separable Rademacher type p Banach space for some p ∈ [1, 2], and a sequence of positive integer-valued random variables {Tn, n ≥ 1}, a general weak law of large numbers of the for