Geodesic
Integro-local theorems for sums of independent random vectors in the series scheme
Matematičeskie zametki, Tome 79 (2006) no. 4, pp. 505-521

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Let $S(n)=\xi(1)+\dots+\xi(n)$ be a sum of independent random vectors $\xi(i)=\xi_{(n)}(i)$ with general distribution depending on a parameter $n$. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability $\mathsf P(S(n)\in\Delta[x))$, where $\Delta[x)$ is a cube with edge $\Delta$ and vertex at a point $x$. 
@article{MZM_2006_79_4_a2,
     author = {A. A. Borovkov and A. A. Mogul'skii},
     title = {Integro-local theorems for sums of independent random vectors in the series scheme},
     journal = {Matemati\v{c}eskie zametki},
     pages = {505--521},
     publisher = {mathdoc},
     volume = {79},
     number = {4},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_4_a2/}
}
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A. A. Borovkov; A. A. Mogul'skii. Integro-local theorems for sums of independent random vectors in the series scheme. Matematičeskie zametki, Tome 79 (2006) no. 4, pp. 505-521. http://geodesic.mathdoc.fr/item/MZM_2006_79_4_a2/