Geodesic
Poisson approximation for the distribution of the number of “parallelograms” in a random sample from $\mathbb Z_N^q$
Matematičeskie voprosy kriptografii, Tome 3 (2012) no. 2, pp. 63-78 
V. I. Kruglov. Poisson approximation for the distribution of the number of “parallelograms” in a random sample from $\mathbb Z_N^q$. Matematičeskie voprosy kriptografii, Tome 3 (2012) no. 2, pp. 63-78. http://geodesic.mathdoc.fr/item/MVK_2012_3_2_a2/
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     title = {Poisson approximation for the distribution of the number of {\textquotedblleft}parallelograms{\textquotedblright} in a~random sample from $\mathbb Z_N^q$},
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     url = {http://geodesic.mathdoc.fr/item/MVK_2012_3_2_a2/}
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For a random sample with replacement $\xi_1,\dots,\xi_T$ from a group $\mathbb Z_N^q$, $N\geq4$, we consider the distribution of the number $\zeta$ of 4-element subsets satisfying the relation of type $\xi_{i_1}-\xi_{i_2}=\xi_{i_3}-\xi_{i_4}$ and additional condition given in terms of a metric on this group. Estimates of the accuracy of Poisson approximation for the distribution of $\zeta$ are obtained and conditions of the weak convergence of $\zeta$ to the Poisson law are established.

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