On edgeworth expansions for dependency-neighborhoods chain structures with strong mixing characteristics

Let $W$ be the sum of dependent random variables, and let $h(x)$ be a function. This paper modifies results of [Y. Rinott and V. I. Rotar, Probab. Theory Related Fields, 126 (2003), pp. 528–570] concerning Edgeworth expansions for $\mathbf E\{h(W)\}$ in the case of the so-called dependency-neighborhoods structures. The most typical example of such structures is mixing on graphs. The main improvement consists in the replacement of $\phi$-mixing dependency characteristics by those of strong (or $\alpha$-) mixing, which essentially widens the possibilities of applications. Another improvement concerns the smoothness conditions. We show how conditions on the smoothness of $h$ may be replaced by conditions on the smoothness of the distribution of $W$.