A Characterization of Chover-Type Law of Iterated Logarithm

Let $0 < \alpha \leq 2$ and $- \infty < \beta < \infty$. 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 $(\alpha, \beta)$-Chover-type law of the iterated logarithm (and write $X \in CTLIL(\alpha, \beta)$) if $\limsup_{n \rightarrow \infty} \left| \frac{S_{n}}{n^{1/\alpha}} \right|^{(\log \log n)^{-1}} = e^{\beta}$ almost surely. This paper is devoted to a characterization of $X \in CTLIL(\alpha, \beta)$. We obtain sets of necessary and sufficient conditions for $X \in CTLIL(\alpha, \beta)$ for the five cases: $\alpha = 2$ and $0 < \beta < \infty$, $\alpha = 2$ and $\beta = 0$, $1 < \alpha < 2$ and $-\infty < \beta < \infty$, $\alpha = 1$ and $- \infty < \beta < \infty$, and $0 < \alpha < 1$ and $-\infty < \beta < \infty$. As for the case where $\alpha = 2$ and $-\infty < \beta < 0$, it is shown that $X \notin CTLIL(2, \beta)$ 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., $X \in CTLIL(\alpha, 1/\alpha)$) is given; that is, $X \in CTLIL(\alpha, 1/\alpha)$ if and only if $\inf \left \{b:~ \mathbb{E} \left(\frac{|X|^{\alpha}}{(\log (e \vee |X|))^{b\alpha}} \right) < \infty \right\} = 1/\alpha$ where $\mathbb{E}X = 0$ whenever $1 < \alpha \leq 2$.Comment: 11 page

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

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<lim sup_{n\to \infty}|S_n|/\sqrtnh(n)<\infty, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(\log \log n)^p, where p>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.Comment: Published at http://dx.doi.org/10.1214/009117905000000198 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org