The rate of convergence for quadratic forms
Let \(Y_1, Y_2, \dots\) be independent identically distributed random variables with \(EY_1=0\) and \(EY_1^2=1\). Let \(A=(a_{ij})\) be a real symmetric \(n\times n\) matrix with \(a_{ii}=0, i=1,2,\dots,n\). Put \[ \beta_s =E| Y_1|^s\;(s>0), \quad b_2^2 =\max_{1\leq i\leq n}\sum_{j=1}^n| a_{ij}| ^2, \quad d_s^2 =\sum_{1\leq i,j\leq n}| a_{ij}|^s\;(s>0), \] \[ \alpha =\frac{d_2}{b_2}\max_{1\leq j\leq n}\frac{\max_{1\leq i\leq n}| a_{ij}| } {\sqrt{\sum_{i=1}^na_{ij}^2}},\quad D_2 =\sum_{1\leq i<j\leq n}(A_{ii}A_{jj}-A_{ij}^2), \] where \(A_{ij}=\sum_{k=1}^na_{ik}a_{jk}\). Let \(\xi_1,\dots,\xi_n\) be independent standard normal random variables and assume that \(Y=(Y_1,\dots,Y_n)\) and \(\xi=(\xi_1,\dots,\xi_n)\) are also independent. Define quadratic forms \(\eta(Y)=YAY^T\) and \(\eta(\xi)=\xi A\xi^T\), and put \(\Delta_n=\sup_{x\in R}| P\{\eta(Y)<x\}-P\{\eta(\xi)<x\}| \). It is known that under the condition \(\max_{1\leq i\leq n}\sum_{j=1}^na_{ij}^2/\sum_{1\leq i,j\leq n}a_{ij}^2\to 0\) as \(n\to\infty\) the distribution of \(\eta(Y)\) is approximated by that of \(\eta(\xi)\) [see \textit{V. I. Rotar}', Theory Probab. Appl. 18, 499-507 (1973), translation from Teor. Veroyatn. Primen. 18, 527-534 (1973; Zbl 0304.60037) and ibid. 20, 512-532 (1975), resp. ibid. 20, 527-546 (1975; Zbl 0372.60030)], and in this paper the author estimates the rate of this convergence. Theorem. Let \(Y_1,Y_2,\dots,Y_n\) be independent identically distributed random variables with \(EY_1=0\), \(EY_1^2=1\) and \(EY_1^4<\infty\). Assume that \(0<D_2<d_2^4/4\). Then there exist absolute constants \(C_1>0\), \(C_2>0\) such that \[ \begin{aligned} \Delta_n &\leq C_1\left\{\frac{(\beta_3b_2d_2^2)^{1/3}}{D_2^{1/4}} (1+\alpha)\left(\ln\frac{d_2}{b_2}\right)^{5/3} +(\beta_3+2)\left(\frac{b_2}{d_2}\right)^{1/3}\right\}\\ &+ C_2\left\{(\beta_4+3)\left(\frac{b_2}{d_2}\right)^{2/3} +(\beta_3+2)\left(\frac{d_3^2}{d_2^2b_2}+\frac{d_4}{d_2b_2}\right)\right\}. \end{aligned} \] A similar estimate is also obtained for the case \(D_2\geq d_2^4/4\).
independent identically distributed random variables
scientific article; zbMATH DE number 1151884