Moderate deviations for particle filtering

Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^n\delta_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean \frac{1}{n}\sum_1^n\psi(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.Comment: Published at http://dx.doi.org/10.1214/105051604000000657 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a continuous function called a variance profile. Consider now a deterministic $N\times n$ matrix $\Lambda_n=(\Lambda_{ij}^{n})$ whose non diagonal elements are zero. Denote by $\Sigma_n$ the non-centered matrix $Y_n + \Lambda_n$. Then under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$\frac{1}{N} \sum_{i=1}^{N} \delta_{(\frac{i}{N}, (\Lambda_{ii}^n)^2)} \xrightarrow[n\to \infty]{} H(dx,d\lambda),$$ where $H$ is a probability measure, it is proven that the empirical distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication.Comment: 25 pages, revised version. Assumption (A2) has been relaxe

The empirical eigenvalue distribution of a Gram matrix: From independence to stationarity

Consider a $N\times n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary gaussian random field. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $n\to \infty$ and $\frac Nn \to c \in (0,\infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al., Girko, etc.).Comment: 15 page

Cooperative Spectrum Sensing Using Random Matrix Theory

In this paper, using tools from asymptotic random matrix theory, a new cooperative scheme for frequency band sensing is introduced for both AWGN and fading channels. Unlike previous works in the field, the new scheme does not require the knowledge of the noise statistics or its variance and is related to the behavior of the largest and smallest eigenvalue of random matrices. Remarkably, simulations show that the asymptotic claims hold even for a small number of observations (which makes it convenient for time-varying topologies), outperforming classical energy detection techniques.Comment: Submitted to International Symposium on Wireless Pervasive Computing 200

Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures$({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$, where${\mathcal N}_n(\Delta)$represents the number of eigenvalues within$\Delta$, are asymptotically independent as the size$n$goes to infinity,$p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.Comment: 15 page

Energy and Exergy Analysis of Dual Channel Solar Air Collector with Perforating “V” Corrugated Absorber Plate

In This paper, an experimental study was carried out on a dual channel with perforating “V” corrugated absorber plate of solar air collector which the air flows both in upper channel and lower channel of the absorber plate for increasing heat transfer coefficient and improving thermal performance. The results of experimental procedures for dual channel with perforating “V” corrugated absorber plate of solar collector were compared with the flat plate dual channel of solar collector. Experimental calculations had been performed under Baghdad (33.34° North latitude, 44.4° East longitude) climatic conditions at different values of mass flow rates 0.021 kg/s, 0.027 kg/s and 0.032 kg/s. The results showed that the dual channel with perforating “V” corrugated absorber plate of solar collector is found to perform more efficiently than the flat plate dual channel of solar collector with increased 39% in thermal efficiency. Then, it showed that the efficiency increases with increasing mass flow rates. Also, it showed that the heat removal factor of the dual channel with perforating “V” corrugated absorber plate of solar collector is 36% more than the flat plate dual channel of solar collector. At the last part of the study, the exergy relations were derived for both collectors. The results of this part showed that the flat plate dual channel of solar collector is having largest irreversibility (exergy loss) and the dual channel with perforating “V” corrugated absorber plate of solar collector is having a greatest exergetic efficiency. Keywords:Solar air collector; Dual channel; Double flow; Exergy analysis; perforating “V” corrugated; Heat removal facto

Determination of Cadmium and Lead in Sewage Sludge from the Middle Region (Misrata, Msallata and Tarhünah Cities) of Libya

The concentrations of cadmium and lead in sewage sludge samples were determined by Atomic Absorption Spectrometric Method. Samples of sewage sludge were obtained from three sewage treatment plants localised in Middle Region of Libya (Misrata, Msallata and Tarhünah cities). The results shows that, the mean levels of Cadmium for all regions are ranges from 81 to 123.4 ppm and these values are higher than the limitations for the international standard which are not registered more than 50 ppm (dry weight) in USA, Egypt and the EU countries. While, the lead concentrations are ranged from 8.0 to 189.2 ppm and all values are within the standard limits which graduated between (275–613) ppm

Thermophysical optimization of specialized concrete pavement materials for collection of surface heat energy and applications for shallow heat storage

There is great potential to use pavement structures to collect and/or store solar energy for the heating and cooling of adjacent buildings, e.g. airport terminals, shopping malls, etc. Therefore, pavement materials comprising both conventional and unconventional concrete mixtures with a wide range of densities, thermal conductivities, specific heat capacities, and thermal diffusivities were investigated. Their thermo-physical properties were then used as inputs to a one dimensional transient heat transport model in order to evaluate the temperature changes at the various depths at which heat might be abstracted or stored. The results indicated that a high diffusivity pavement, e.g. incorporating high conductive aggregates and/or metallic fibres, can significantly enhance heat transfer as well as reduction of thermal stresses across the concrete slab. On the other hand a low diffusivity concrete can induce a more stable temperature at shallower depth enabling easier heat storage in the pavement as well as helping to reduce the risk of damage due to freeze-thaw cycling in cold regions