Cram\'er transform and t-entropy

t-entropy is the convex conjugate of the logarithm of the spectral radius of a weighted composition operator (WCO). Let $X$ be a nonnegative random variable. We show how the Cram\'er transform with respect to the spectral radius of WCO is expressed by the t-entropy and the Cram\'er transform of the given random variable X.Comment: 12 pages; Positivity(2013

Asymptotic Distributions of the Overshoot and Undershoots for the L\'evy Insurance Risk Process in the Cram\'er and Convolution Equivalent Cases

Recent models of the insurance risk process use a L\'evy process to generalise the traditional Cram\'er-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a L\'{e}vy process which drifts to $-\infty$ and satisfies a Cram\'er or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cram\'er case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cram\'er model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the L\'evy process belongs to the "GTSC" class

Cram\'{e}r-type large deviations for samples from a finite population

Cram\'{e}r-type large deviations for means of samples from a finite population are established under weak conditions. The results are comparable to results for the so-called self-normalized large deviation for independent random variables. Cram\'{e}r-type large deviations for the finite population Student $t$-statistic are also investigated.Comment: Published at http://dx.doi.org/10.1214/009053606000001343 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ziv-Zakai Error Bounds for Quantum Parameter Estimation

I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cram\'er-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a "Heisenberg" error limit that scales with the average energy and a limit similar to the quantum Cram\'er-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cram\'er-Rao bound for states with highly non-Gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.Comment: v1: preliminary result, 3 pages; v2: major update, 4 pages + supplementary calculations, v3: another major update, added proof of "Heisenberg" limit, v4: accepted by PR

Channel Estimation for Diffusive MIMO Molecular Communications

In diffusion-based communication, as for molecular systems, the achievable data rate is very low due to the slow nature of diffusion and the existence of severe inter-symbol interference (ISI). Multiple-input multiple-output (MIMO) technique can be used to improve the data rate. Knowledge of channel impulse response (CIR) is essential for equalization and detection in MIMO systems. This paper presents a training-based CIR estimation for diffusive MIMO (D-MIMO) channels. Maximum likelihood and least-squares estimators are derived, and the training sequences are designed to minimize the corresponding Cram\'er-Rao bound. Sub-optimal estimators are compared to Cram\'er-Rao bound to validate their performance.Comment: 5 pages, 5 figures, EuCNC 201

Cram\'{e}r type large deviations for trimmed L-statistics

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed $L$-statistics and we apply it to the Cram\'{e}r type large deviation problem. Our results can be compared with ones in Callaert et al.(1982) -- the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed $L$-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed $L$-statistic by a non-trimmed $L$-statistic (with smooth weight function) based on Winsorized random variables. Using this method, we establish the Cram\'{e}r type large deviation results for the trimmed $L$-statistics under quite mild and natural conditions.Comment: 17 page