Geodesic
Statistical independence of the Boolean function superposition
Prikladnaâ diskretnaâ matematika, no. 13 (2011), pp. 11-12

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It is proved here that if a Boolean function $f(x,y)$ is statistically independent on the variables in $x$, then the same is true for any Boolean function $g(f(x,y),z)$, but this may not be so for a superposition $g(f_1(x,y),\dots,f_s(x,y),z)$ where $s\geq2$ and every function $f_1(x,y),\dots,f_s(x,y)$ is statistically independent on $x$. 
@article{PDM_2011_13_a3,
     author = {O. L. Kolcheva and I. A. Pankratova},
     title = {Statistical independence of the {Boolean} function superposition},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {11--12},
     publisher = {mathdoc},
     number = {13},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_13_a3/}
}
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O. L. Kolcheva; I. A. Pankratova. Statistical independence of the Boolean function superposition. Prikladnaâ diskretnaâ matematika, no. 13 (2011), pp. 11-12. http://geodesic.mathdoc.fr/item/PDM_2011_13_a3/