Normality of a non-linear transformation of AR parameters: application to reflection and cepstrum coefficients

Two sets of random vectors cannot both be Gaussian if they are nonlinearly related. Thus, Autoregressive (AR)parameters and reflection coefficient (resp. cepstrum coefficient) estimators cannot both be Gaussian for a finite number of samples. However, most estimators of AR parameters and reflection coefficients (resp. cepstrum coefficients) are Gaussian asymptotically. Thus, the distribution of AR parameter and reflection coefficient (resp. cepstrum coefficient) estimates are close to Gaussian for large samples. This paper studies the ``closeness'' between the Gaussian distribution and the ``non-linear transformation of Gaussian AR parameters'' distribution. A new distance is defined which is based on the Taylor expansion of the non-linear transformation. This ``Taylor'' distance called $M$-distance is used to measure the deviations from the Gaussian distribution of reflection coefficient and cepstrum coefficient statistics. A comparison is presented between this distance and Kullback's divergence. The main advantage of the M-distance with respect to other distances is a very simple closed form expression of the deviations from normality. This closed form expression shows that the convergence of the reflection and cepstrum coefficient distribution to its asymptotic Gaussian distribution (when the number of samples tends to infinity) depends on the position of AR model poles in the unit circle

Gaussian distribution of LMOV numbers

Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects.Comment: 23 page

Gaussian Entanglement Distribution via Satellite

In this work we analyse three quantum communication schemes for the generation of Gaussian entanglement between two ground stations. Communication occurs via a satellite over two independent atmospheric fading channels dominated by turbulence-induced beam wander. In our first scheme the engineering complexity remains largely on the ground transceivers, with the satellite acting simply as a reflector. Although the channel state information of the two atmospheric channels remains unknown in this scheme, the Gaussian entanglement generation between the ground stations can still be determined. On the ground, distillation and Gaussification procedures can be applied, leading to a refined Gaussian entanglement generation rate between the ground stations. We compare the rates produced by this first scheme with two competing schemes in which quantum complexity is added to the satellite, thereby illustrating the trade-off between space-based engineering complexity and the rate of ground-station entanglement generation.Comment: Closer to published version (to appear in Phys. Rev. A) 13 pages, 6 figure