On the absolute constants in the Berry-Esseen-type inequalities
Let \(X_1,X_2,\dots,X_n\) be a sequence of independent random variables with mean zero and \(\beta_{2+\delta,j}=\operatorname{E}|X_j|^{2+\delta}<\infty\) for all \(j\) and some \(\delta\in(0,1]\). Define \(\sigma_j^2=\operatorname{E}X_j^2\), for \(j=1,\dots,n\), and \[ B_n^2=\sum_{j=1}^n\sigma_j^2,\quad l_n=\frac{1}{B_n^{2+\delta}}\sum_{j=1}^n\beta_{2+\delta,j},\quad\text{ and }\quad\tau_n=\frac{1}{B_n^{2+\delta}}\sum_{j=1}^n\sigma_j^{2+\delta}. \] The author considers the Berry-Esseen-type inequality \[ \sup_x\left|\operatorname{P}\left(\frac{X_1+\dots+X_n}{B_n}<x\right)-\Phi(x)\right|\leq C_s(\delta)(l_n+s\tau_n), \] for \(0\leq s\leq1\), where \(\Phi\) is the distribution function of a standard normal random variable. Proving bounds for the constants \(C_s(\delta)\) appearing in such inequalities is a problem with a long and rich history. Noting that \(\tau_n\leq l_n\), one has that \(C_0(\delta)\leq(1+s)C_s(\delta)\) for any \(s\) and \(\delta\); the case \(s=0\) being the classical Berry-Esseen inequality. The author provides numerical upper bounds for \(C_s(\delta)\) when \(s=s_0(\delta)\) or \(s=s_1(\delta)\) given by \[ s_0(\delta)=\max\left\{s\in[0,1]:(1+s)C_s(\delta)=C_0(\delta)\right\} \] (the maximum value of \(s\) minimizing the sum of the coefficients of \(l_n\) and \(\tau_n\) in the Berry-Esseen-type inequality), and \[ s_1(\delta)=\min\left\{s\in[0,1]:C_s(\delta)=C_1(\delta)\right\} \] (the smallest value of \(s\) minimizing the coefficient of \(l_n\)), respectively. Space limitations prevent the author from giving an analytic expression or an algorithm to accompany the numerical upper bounds. Corresponding lower bounds on \(\inf_sC_s(\delta)\) are also given, and the special case of a Berry-Esseen-type inequality for compound Poisson distributions is discussed.
scientific article; zbMATH DE number 6369729