On Random Operator-Valued Matrices: Operator-Valued Semicircular Mixtures and Central Limit Theorem

Motivated by a random matrix theory model from wireless communications, we define random operator-valued matrices as the elements of $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes M_d({\mathcal A})$ where $(\Omega,{\mathcal F},{\mathbb P})$ is a classical probability space and $({\mathcal A},\varphi)$ is a non-commutative probability space. A central limit theorem for the mean $M_d(\mathbb{C})$-valued moments of these random operator-valued matrices is derived. Also a numerical algorithm to compute the mean $M_d({\mathbb C})$-valued Cauchy transform of operator-valued semicircular mixtures is analyzed.Comment: 17 pages, 1 figur

Privacy-Aware Guessing Efficiency

We investigate the problem of guessing a discrete random variable $Y$ under a privacy constraint dictated by another correlated discrete random variable $X$, where both guessing efficiency and privacy are assessed in terms of the probability of correct guessing. We define $h(P_{XY}, \epsilon)$ as the maximum probability of correctly guessing $Y$ given an auxiliary random variable $Z$, where the maximization is taken over all $P_{Z|Y}$ ensuring that the probability of correctly guessing $X$ given $Z$ does not exceed $\epsilon$. We show that the map $\epsilon\mapsto h(P_{XY}, \epsilon)$ is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for $h(P_{XY}, \epsilon)$ when $X$ and $Y$ are connected via a binary-input binary-output channel. For $(X^n, Y^n)$ being pairs of independent and identically distributed binary random vectors, we similarly define $\underline{h}_n(P_{X^nY^n}, \epsilon)$ under the assumption that $Z^n$ is also a binary vector. Then we obtain a closed form expression for $\underline{h}_n(P_{X^nY^n}, \epsilon)$ for sufficiently large, but nontrivial values of $\epsilon$.Comment: ISIT 201

Information Extraction Under Privacy Constraints

A privacy-constrained information extraction problem is considered where for a pair of correlated discrete random variables $(X,Y)$ governed by a given joint distribution, an agent observes $Y$ and wants to convey to a potentially public user as much information about $Y$ as possible without compromising the amount of information revealed about $X$. To this end, the so-called {\em rate-privacy function} is introduced to quantify the maximal amount of information (measured in terms of mutual information) that can be extracted from $Y$ under a privacy constraint between $X$ and the extracted information, where privacy is measured using either mutual information or maximal correlation. Properties of the rate-privacy function are analyzed and information-theoretic and estimation-theoretic interpretations of it are presented for both the mutual information and maximal correlation privacy measures. It is also shown that the rate-privacy function admits a closed-form expression for a large family of joint distributions of $(X,Y)$. Finally, the rate-privacy function under the mutual information privacy measure is considered for the case where $(X,Y)$ has a joint probability density function by studying the problem where the extracted information is a uniform quantization of $Y$ corrupted by additive Gaussian noise. The asymptotic behavior of the rate-privacy function is studied as the quantization resolution grows without bound and it is observed that not all of the properties of the rate-privacy function carry over from the discrete to the continuous case.Comment: 55 pages, 6 figures. Improved the organization and added detailed literature revie

How Do Dispositional Mindfulness and Statistical Anxiety Affect Student Performance on an Exam?

The current study investigated if a specific disposition, mindfulness, offers an advantage for undergraduate students’ success on statistical exams by decreasing the impact of an academic-related strain, anxiety. Prior research utilizing state mindfulness has found evidence of this performance advantage, but it has yet to be investigated if dispositional mindfulness offers similar results. The current study found that dispositional mindfulness does impact one’s anxiety, but that relation does not offer a performance advantage for those with higher levels of mindfulness in comparison to those with lower levels of the disposition. To improve student success, engaging in mindfulness practices appears more beneficial instead of a student organically holding high levels of mindfulness prior to an exam

Graviton-photon oscillation in alternative theories of gravity

In this paper we investigate graviton-photon oscillation in the presence of an external magnetic field in alternative theories of gravity. Whereas the effect of an effective refractive index for the electromagnetic radiation was already considered in the literature, we develop the first approach to take into account the effect of the modification of the predictions for gravitational waves in alternative theories of gravity in the phenomenon of graviton-photon mixing.Comment: V1: 5 pages. V2: 9 pages (new style); clarifications in the discussion included, no physics changes; 3 references added. V3: 10 pages, 4 references added, discussion extended. Version accepted for publication in Classical and Quantum Gravit