W1,+W_{1,+}-interpolation of probability measures on graphs

We generalize an equation introduced by Benamou and Brenier, characterizing Wasserstein W_p-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a final distributions f_0 and f_1, we prove the existence of a curve (f_t) satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem.Comment: 25 page

A proof of the Shepp-Olkin entropy monotonicity conjecture

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to conjecture that this distribution should be 'more random' when each coin is fairer. Indeed, Shepp and Olkin conjectured that the Shannon entropy of this distribution is monotonically increasing in this case. We resolve this conjecture, by proving that this intuition is correct. Our proof uses a construction which was previously developed by the authors to prove a related conjecture of Shepp and Olkin concerning concavity of entropy. We discuss whether this result can be generalized to $q$-R\'{e}nyi and $q$-Tsallis entropies, for a range of values of $q$.Comment: 16 page

W1,+W_{1,+}-interpolation of probability measures on graphs

We generalize an equation introduced by Benamou and Brenier in [BB00] and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a final distributions (f0(x))x∈G, (f1(x))x∈G, we prove the existence of a curve (ft(k)) t∈[0,1],k∈Z satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem

A natural derivative on [0,n] and a binomial Poincar\'e inequality

We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator $\nabla_n$ plays a key role in a new Poincar\'e (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures

An exponentially averaged Vasyunin formula

We prove a Vasyunin-type formula for an autocorrelation function arising from a Nyman-Beurling criterion generalized to a probabilistic framework. This formula can also be seen as a reciprocity formula for cotangent sums, related to the ones proven in [BC13], [ABB17].Comment: This paper has been written from results already stated in a previous version of another paper in 2018, but has been now submitted separately. arXiv admin note: text overlap with arXiv:1805.0673

Polynomial Moments with a weighted Zeta Square measure on the critical line

We prove closed-form identities for the sequence of moments $\int t^{2n}|\Gamma(s)\zeta(s)|^2dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $\pi$, especially featuring the numbers $\zeta(n)B_n/n$ unveiled by Bettin and Conrey in 2013. Their main power series identity allows for a short proof of an auxiliary result: the computation of the $k$-th derivatives at $1$ of the "exponential auto-correlation" function studied in a recent paper by the authors. We also provide an elementary and self-contained proof of this secondary result. The starting point of our work is a remarkable identity proven by Ramanujan in 1915. The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $|\zeta|$ on the critical line. They arise in some generalizations of the Nyman-Beurling criterion, but might be of independent interest for the numerous connections concerning the above mentioned numbers

On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $\theta_k$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong B\'aez-Duarte criterion. %We start here the study of these criteria, with a special focus on exponential and gamma distributions. The main goal of the present paper is the study of the interplay between these probabilistic Nyman-Beurling criteria and the Riemann hypothesis. By means of our probabilistic point of view, we answer a question raised by B\'aez-Duarte in 2005. These new structures open the door to calculable determinants.Comment: 14 page