Geodesic
Statistical independence of the Boolean function superposition.~II
Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 14-15

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Let $x,y,z$ be sets of different Boolean variables, $f(x,y)$, $f_1(x,y)$, $f_2(x,y)$, $f_1(x,y)\oplus f_2(x,y)$ are Boolean functions being statistically independent on the variables in $x$, and $h(x_1,x_2,z)$, $g(x)$ are any Boolean functions. Then the function $h(f_1(x,y),f_2(x,y),z)$ is statistically independent on the variables in $x$; and the same is true for the function $f(x,y)\oplus g(x)$ iff $f$ is balanced or $g=\mathrm{const}$. 
@article{PDMA_2012_5_a5,
     author = {O. L. Kolcheva and I. A. Pankratova},
     title = {Statistical independence of the {Boolean} function {superposition.~II}},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {14--15},
     publisher = {mathdoc},
     number = {5},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2012_5_a5/}
}
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O. L. Kolcheva; I. A. Pankratova. Statistical independence of the Boolean function superposition.~II. Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 14-15. http://geodesic.mathdoc.fr/item/PDMA_2012_5_a5/