OAI.IndependentProducts.main_gaussian_12

The theorem states that, for every real t>0, the averaged asymptotic midpoint modulus at t of the closed real L1 span of the path products, taken for the product law of independent copies of the square of a standard real Gaussian, is at least a positive explicit quantity. Here the product law puts one independent copy of the law of x² (x standard Gaussian) on each nonroot vertex of the tree of finite sequences of positive integers, the path product of a vertex is the product of the coordinates along its nonroot prefixes (the empty product 1 at the root), and the space is the topological closure in L1 of the real span of these products, equipped with its own norm. The averaged midpoint modulus at t is the infimum, over unit vectors x, of the supremum, over closed finite-codimension subspaces F, of the infimum over y in F with norm at least 1 of (‖x+ty‖+‖x−ty‖)/2 − 1. The theorem asserts that this modulus is at least gaussianTail(8√6/t)/4, where gaussianTail(r) is the standard Gaussian probability that |x| ≥ r, and also that gaussianTail(8√6/t)/4 is strictly positive. The multiplier hypotheses are not assumed in the statement.