Natural Exponential Families: Resolution of A Conjecture and Existence of Reduction Functions

One-parameter natural exponential family (NEF) plays fundamental roles in probability and statistics. This article contains two independent results: (a) A conjecture of Bar-Lev, Bshouty and Enis states that a polynomial with a simple root at $0$ and a complex root with positive imaginary part is the variance function of some NEF with mean domain $\left(0,\infty\right)$ if and only if the real part of the complex root is not positive. This conjecture is resolved. The positive answer to this conjecture enlarges existing family of polynomials that are able to generate NEFs, and it helps prevent practitioners from choosing incompatible functions as variance functions for statistical modeling using NEFs. (b) if a random variable $\xi$ has parametric distributions that form a infinitely divisible NEF whose induced measure is absolutely continuous with respect to its basis measure, then there exists a deterministic function $h$, called "reduction function", such that $\mathbb{E} \left(h\left(\xi\right)\right)=\mathbb{V}\left(\xi\right)$, i.e., $h\left(\xi\right)$ is an unbiased estimator of the variance of $\xi$. The reduction function has applications to estimating latent, low-dimensional structures and to dimension reduction in the first and/or second moments in high-dimensional data.Comment: 14 pages and 1 figure, in this version, the proof of the conjecture is much more concise, and the proof of the existence of redunction functions uses a different approac

The variance conjecture on projections of the cube

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in S^{n-1}}{\mathbb E}_\mu\langle x,\theta\rangle^2{\mathbb E}_\mu|x|^2,$$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces

Entropic uncertainty relations in multidimensional position and momentum spaces

Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. The lower bound in the new relation is optimal, and the new entropic uncertainty relation implies the famous variance-based uncertainty principle for multimode states. The article concludes with an open conjecture

The distribution of the variance of primes in arithmetic progressions

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is known unconditionally in the range x/(log x)^A \leq Q \leq x; this last range can be improved to x^{\frac 12+\epsilon} \leq Q \leq x under the Generalized Riemann Hypothesis (GRH). We argue that Hooley's conjecture should hold down to (loglog x)^{1+o(1)} \leq q \leq x for all values of q, and that this range is best possible. We show under GRH and a linear independence hypothesis on the zeros of Dirichlet L-functions that for moderate values of q, \phi(q)e^{-y}V(e^y;q) has the same distribution as that of a certain random variable of mean asymptotically \phi(q) log q and of variance asymptotically 2\phi(q)(log q)^2. Our estimates on the large deviations of this random variable allow us to predict the range of validity of Hooley's Conjecture.Comment: 26 pages; Modified Definition 2.1, the error term for the variance in Theorem 1.2 and its proo

More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ and $K$, $L$ are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.Comment: Minor corrections, some additional references, agnowledgemen