Geodesic
The estimation of proximity of distribution of sequential sums of independent identically distributed random vectors
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 83-87

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Let $F$ be a distribution on $\mathbb R^k$, $F_k^n$ – times convolution of $F$ with itself, $\mathscr L^k=\{B\in\mathbb R^k,B=[a_1,b_1]\times\dots\times[a_k,b_k]\}$. It is proved that $$ \sup_{B\in\mathscr L^k}|F^{n+1}\{B\}-F^n\{B\}|\le\frac{c(F)}{\sqrt n}, $$ where $c(F)$ depends on some characteristics of $F$. 
@article{ZNSL_1980_97_a8,
     author = {A. Yu. Zaitsev},
     title = {The estimation of proximity of distribution of sequential sums of independent identically distributed random vectors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--87},
     publisher = {mathdoc},
     volume = {97},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a8/}
}
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A. Yu. Zaitsev. The estimation of proximity of distribution of sequential sums of independent identically distributed random vectors. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 83-87. http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a8/