An Edgeworth expansion for a sum of m-dependent random variables
Given a sequence \(X_ 1,...,X_ n\) of m-dependent random variables, put \(S=X_ 1+...+X_ n,\) \(\sigma^ 2=ES^ 2\). An Edgeworth expansion for the distribution of S/\(\sigma\) is established, under the assumption of the existence of moments of order \(3+\alpha\) \((0<\alpha \leq 1)\). To be precise, let \(\Phi\) (t) and \(\Psi\) (t) be the distribution and density function, respectively, of a standard normal distribution. Let \(\mu =E(S/\sigma)^ 3\). Define L as \((m+1)^ 2\sum_{j\leq n}E| X_ j|^ 3/\sigma^ 3,\) M as \((m+1)^{2+\alpha}\sum_{j\leq n}E| X_ j|^{3+\alpha}/\sigma^{3+\alpha},\) and N as \(\sum_{| j- k| \leq 12m}E| X_ j|^ 3E(X_ k)^ 2/\sigma^ 5.\) It is shown that there is a universal constant K such that \[ \sup_{t\in {\mathbb{R}}}| P(S/\sigma \leq t)-\Phi (t)-(\mu^ 3/6)(1- t)^ 2\Psi (t)| \leq K(A+\delta A^{-1}), \] where A is given as \(M+L^ 2+N \log^{3/2}L^{-1}+\exp (-1/MK)+L \exp (-(LK)^{-2})\), and \(\delta\) as \(\sup \{| E \exp (it\quad S/\sigma)|,(KL)^{- 1}<| t| <A^{-1}\}.\) It is also noted that the result is of the best possible order, provided \(\delta\) is small enough.
scientific article; zbMATH DE number 3938147