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Register a new userAn inequality for correlated measurable functions
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Authors
Fabio Zucca
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A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory.
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The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy theory, approximation theory and probability theory. In this article, we concentrate on a special case of the conjecture, and give the best known lower bound in dimension 3, using a conditional expectation argument.
Let h_R denote an L ^{infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{infty} norm of the `hyperbolic sums $$ sum _{|R|=2 ^{-n}} alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{eta} for some positive eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A particular minimization problem for a non-periodic (Heisenberg) uncertainty constant is studied.
We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.
Suppose $f : [0,1]^2 rightarrow mathbb{R}$ is a $(c,alpha)$-mixed Holder function that we sample at $l$ points $X_1,ldots,X_l$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_1),ldots,f(X_l)$ be given. If $l ge c_1 n log^2 n$, then we can compute an approximation $tilde{f}$ such that $$ |f - tilde{f} |_{L^2} = mathcal{O}(n^{-alpha} log^{3/2} n), $$ with probability at least $1 - n^{2 -c_1}$, where the implicit constant only depends on the constants $c > 0$ and $c_1 > 0$.
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