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Balanced And/Or Trees and Linear Threshold Functions

We consider random balanced Boolean formulas, built on the two connectives and and or, and a fixed number of variables. The probability distribution induced on Boolean functions is shown to have a limit when letting the depth of these formulas grow to infinity. By investigating how this limiting distribution depends on the two underlying probability distributions, over the connectives and over the Boolean variables, we prove that its support is made of linear threshold functions, and give the speed of convergence towards this limiting distribution.