On asymptotic expansions in the central limit theorem

F. N. Galstyan

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 3, pp. 535-540

Résumé

Let $\{X_k\}$ be a sequence of independent identically distributed random variables with $\mathbf EX_1=0$, $\mathbf EX_1^2=1$ and $\mathbf E|X_1|^{m+2}\infty$ for some integer $m\ge1$. Put $$ S_n=\sum_{k=1}^nX_k,\quad F_n(x)=\mathbf P\{S_n\sqrt n\},\quad f(t)=\mathbf Ee^{itX_1}. $$ Suppose that Cramér's condition (c): $\varlimsup\limits_{|t|\to\infty}|f(t)|1$ is satisfied. It is known that, in this case, $F_n(x)=G(x)+o(n^{-m/2})$ where $$ G(x)=\Phi(x)+\frac{e^{-x^2/2}}{\sqrt{2\pi}}\sum_{k=1}^mQ_k(x)n^{-k/2}, $$ $\Phi(x)$ is the normal distribution function, $Q_k(x)$ is a polynomial whose coefficients depend only on the cumulants of $X_1$. Theorem 1 contains a sufficient condition for convergence of the series $$ \sum_{n=1}^\infty n^{-1+\frac{m+\delta}2}\sup_x|F_n(x)-G(x)|,\quad0\le\delta1. $$ Theorem 2 indicates a necessary and sufficient condition for this convergence in the special case of symmetric random variables.