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

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

On electromagnetics of an isotropic chiral medium moving at constant velocity

A medium which is an isotropic chiral medium from the perspective of a co-moving observer is a Faraday chiral medium (FCM) from the perspective of a non-co-moving observer. The Tellegen constitutive relations for this FCM are established. By an extension of the Beltrami field concept, these constitutive relations are exploited to show that planewave propagation is characterized by four generally independent wavenumbers. This FCM can support negative phase velocity at certain translational velocities and with certain wavevectors, even though the corresponding isotropic chiral medium does not. The constitutive relations and Beltrami--like fields are also used to develop a convenient spectral representation of the dyadic Green functions for the FCM

Country, Sector or Style: What Matters Most When Constructing Global Equity Portfolios? An Empirical Investigation from 1990-2001

Equity returns are believed to be strongly influenced by country, sector and style effects. A key issue is to be able to disentangle those various effects from one another. In particular, differences between country returns may simply reflect differences in the sector composition of country markets, which makes it clearly difficult to disassociate both effects. Similarly, from 1999-2001 the relative perfor-mance of Growth versus Value might be solely due to the striking performance of the Technology and Telecommunication sectors. For global equity portfolio man-agers, it is crucial to identify which factors offer the highest diversification benefits and return potential. We apply a multi-factor approach to estimate ”pure” coun-try, sector and style factor returns. Using data going back to 1990, we identify the major changes that have occurred in developed markets until 2001. Our various indicators clearly point out the growing influence of sector factors. However, coun-try effects remain important and there is no clear-cut evidence that sector factors dominate country factors. Style factors such as Growth, Value and Size also remain significant, even once sector and country effects are deduced. Finally, we show that momentum strategies based on sector returns offer substantial gains, while momen-tum strategies based on country returns do not. These findings suggest that, while diversification and return benefits from sector strategies have become substantial, managers should continue to monitor carefully country as well as style rewards and risks.