Geodesic
On edgeworth expansions for dependency-neighborhoods chain structures with strong mixing characteristics
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 21-40

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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 $W$. 
@article{TVP_2007_52_1_a1,
     author = {V. I. Rotar'},
     title = {On edgeworth expansions for dependency-neighborhoods chain structures with strong mixing characteristics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {21--40},
     publisher = {mathdoc},
     volume = {52},
     number = {1},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2007_52_1_a1/}
}
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V. I. Rotar'. On edgeworth expansions for dependency-neighborhoods chain structures with strong mixing characteristics. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 21-40. http://geodesic.mathdoc.fr/item/TVP_2007_52_1_a1/