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Let $mathbf{X}$ be a random variable uniformly distributed on the discrete cube $left{ -1,1right} ^{n}$, and let $T_{rho}$ be the noise operator acting on Boolean functions $f:left{ -1,1right} ^{n}toleft{ 0,1right} $, where $rhoin[0,1]$ is the noise parameter, representing the correlation coefficient between each coordination of $mathbf{X}$ and its noise-corrupted version. Given a convex function $Phi$ and the mean $mathbb{E}f(mathbf{X})=ain[0,1]$, which Boolean function $f$ maximizes the $Phi$-stability $mathbb{E}left[Phileft(T_{rho}f(mathbf{X})right)right]$ of $f$? Special cases of this problem include the (symmetric and asymmetric) $alpha$-stability problems and the Most Informative Boolean Function problem. In this paper, we provide several upper bounds for the maximal $Phi$-stability. Considering specific $Phi$s, we partially resolve Mossel and ODonnells conjecture on $alpha$-stability with $alpha>2$, Li and Medards conjecture on $alpha$-stability with $1<alpha<2$, and Courtade and Kumars conjecture on the Most Informative Boolean Function which corresponds to a conjecture on $alpha$-stability with $alpha=1$. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut--Kalai--Naor (FKN) theorem. Our improvements of the FKN Theorem are sharp or asymptotically sharp for certain cases.
We investigate the Renyi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Renyi entropy, for Poisson-Bernoulli variables. As applications we prove
We derive a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit i.i.d. sequence. We use this coupling to verify a conjecture of Darling and Erdos (1956).
Let $T_{epsilon}$ be the noise operator acting on Boolean functions $f:{0, 1}^nto {0, 1}$, where $epsilonin[0, 1/2]$ is the noise parameter. Given $alpha>1$ and fixed mean $mathbb{E} f$, which Boolean function $f$ has the largest $alpha$-th moment $m
The purpose of this note is to share some observations and speculations concerning the asymptotic behavior of Gromov-Witten invariants. They may be indicative of some deep phenomena in symplectic topology that in full generality are outside of the re
Trace reconstruction considers the task of recovering an unknown string $x in {0,1}^n$ given a number of independent traces, i.e., subsequences of $x$ obtained by randomly and independently deleting every symbol of $x$ with some probability $p$. The