arXiv subjects
Publications and source records attributed to Tsung-Lin Cheng.
In this paper, we obtain the upper and lower bounds for two inequalities related to the range statistics. The first one is concerning the one-variable case and the second one is about the bivariate case.
In this paper, we propose a new proof of the Jensen formula in 1895. We also derive some formulas similar to those in Pitman and Yor, 2003. Besides, a new formula of the generalized Bernoulli function is also derived. At the end of the paper, the probability density functions of sinh and tanh are studied briefly for general cases.
Let $X_n=\sum_{i=1}^{\infty}a_iε_{n-i}$, where the $ε_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-β})$ with $β>1/2$. When $1/2<β<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bounds for the normalized sum $Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d}))$ are obtained when $Q_N/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.