A Bivariate Invariance Principle

A notable result from analysis of Boolean functions is the Basic Invariance Principle (BIP), a quantitative nonlinear generalization of the Central Limit Theorem for multilinear polynomials. We present a generalization of the BIP for bivariate multilinear polynomials, i.e., polynomials over two n-length sequences of random variables. This bivariate invariance principle arises from an iterative application of the BIP to bound the error in replacing each of the two input sequences. In order to prove this invariance principle, we first derive a version of the BIP for random multilinear polynomials, i.e., polynomials whose coefficients are random variables. As a benchmark, we also state a naive bivariate invariance principle which treats the two input sequences as one and directly applies the BIP. Neither principle is universally stronger than the other, but we do show that for a notable class of bivariate functions, which we term separable functions, our subtler principle is exponentially tighter than the naive benchmark.