Geodesic
On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 156-160

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The following theorem is proved. If $\xi_1,\xi_2,\dots$ is a sequence of non-degenerate identically distributed independent random variables with values in $Z^2$, then $$ \sup_{m\in Z^2}\mathbf P(\xi_1+\dots+\xi_n=m)\le Cn^{-1}\Delta^{-1/2}, $$ where $C$ is an absolute constant, $\Delta=(P_L-P_0)(1-P_L)$, $$ P_0=\max_{m\in Z^2}\mathbf P\{\xi=x\},\qquad P_L=\max_H\mathbf P\{\xi\in H\}, $$ $H$ is a set of points belonging to some straight line. 
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     author = {A. G. Postnikov and A. A. Judin},
     title = {On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {156--160},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/}
}
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A. G. Postnikov; A. A. Judin. On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 156-160. http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/