Central limit theorem for a class of globally correlated random variables

The standard central limit theorem with a Gaussian attractor for the sum of independent random variables may lose its validity in presence of strong correlations between the added random contributions. Here, we study this problem for similar interchangeable globally correlated random variables. Under these conditions, a hierarchical set of equations is derived for the conditional transition probabilities. This result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables. Depending on the correlation mechanisms and single statistics, the corresponding sums are characterized by distinct statistical probability densities. For a class of urn models it is also possible to characterize their domain of attraction which, as in the standard case, is parametrized by the probability density of each random variable. Symmetric and asymmetric $q$-Gaussian attractors $(q<1)$ are a particular case of these models.Comment: 11 pages, 7 figures, Appendixes 5 page

Weak ergodicity breaking induced by global memory effects

We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, set the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the mean residence time in each state may result divergent, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, global memory effects are an alternative mechanism able to induce this property. Analytical and numerical calculations support these results.Comment: 11 pages, 3 figure

Non-Markovian quantum jumps from measurements in bipartite Markovian dynamics

The quantum jump approach allows to characterize the stochastic dynamics associated to an open quantum system submitted to a continuous measurement action. In this paper we show that this formalism can consistently be extended to non-Markovian system dynamics. The results rely in studying a measurement process performed on a bipartite arrangement characterized by a Markovian Lindblad evolution. Both a renewal and non-renewal extensions are found. The general structure of non-local master equations that admit an unravelling in terms of the corresponding non-Markovian trajectories are also found. Studying a two-level system dynamics, it is demonstrated that non-Markovian effects such as an environment-to-system flow of information may be present in the ensemble dynamics.Comment: 13 pages, 3 figure

Extended q-Gaussian and q-exponential distributions from Gamma random variables

The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar non-equilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained by maximizing Tsallis "non-extensive" entropy under appropriate constraints, as well as from superstatistical models. In this paper we provide an alternative and complementary scheme for deriving these objects. We show that q-Gaussian and q-exponential random variables can always be expressed as function of two statistically independent Gamma random variables with the same scale parameter. Their shape index determine the complexity q-parameter. This result also allows to define an extended family of asymmetric q-Gaussian and modified $q$-exponential densities, which reduce to the previous ones when the shape parameters are the same. Furthermore, we demonstrate that simple change of variables always allow to relate any of these distributions with a Beta stochastic variable. The extended distributions are applied in the statistical description of different complex dynamics such as log-return signals in financial markets and motion of point defects in fluid flows.Comment: 11 pages, 6 figure

Fluctuation relations with intermittent non-Gaussian variables

Non-equilibrium stationary fluctuations may exhibit a special symmetry called fluctuation relations (FR). Here, we show that this property is always satisfied by the subtraction of two random and independent variables related by a thermodynamic-like change of measure. Taking one of them as a modulated Poisson process, it is demonstrated that intermittence and FR are compatibles properties that may coexist naturally. Strong non-Gaussian features characterize the probability distribution and its generating function. Their associated large deviation functions (LDF) develop a kink at the origin and a plateau regime respectively. Application of this model in different stationary nonequilibrium situations is discussed.Comment: 7 pages, 3 figure

Development and characterisation of an easily configurable range imaging system

Range imaging is becoming a popular tool for many applications, with several commercial variants now available. These systems find numerous real world applications such as interactive gaming and the automotive industry. This paper describes the development of a range imaging system employing the PMD-19 k sensor from PMD technologies. One specific advantage of our system is that it is extremely customisable in terms of modulation patterns to act as a platform for further research into time-of-flight range imaging. Experimental results are presented giving an indication of the precision and accuracy of the system, and how modifying certain operating parameters can improve system performance

Heterodyne range imaging in real-time

A versatile full-field range imaging system has previously been constructed. This system is configurable in software to produce either high precision or fast acquisition range images. Indicatively a 10 second exposure has been shown to produce a range image of sub-millimeter precision, whilst video frame rate (30 fps) acquisition provides for centimetre precision. Currently the acquisition time of the system is to a large degree constrained by the off-line processing of the frames by an external computer. This paper presents an alternative to the off-line PC image processing utilising an Altera Stratix II FPGA. Processing rates up to 30 frames per second have been achieved with the added advantage that many of the previous systempsilas existing digital electronics can also be accommodated, providing for an even more compact and flexible system

The distribution of red and blue galaxies in groups: an empirical test of the halo model

The popular halo model predicts that the power spectrum of the galaxy fluctuations is simply the sum of the large scale linear halo-halo power spectrum and the weighted power spectrum of the halo profile. Previous studies have derived halo parameters from the observed galaxy correlation function. Here we test the halo model directly for self-consistency with a minimal set of theoretical assumptions by utilising the 2dF Galaxy Redshift Survey (2dFGRS). We derive empirically the halo occupation and galaxy radial distributions in the haloes of the 2dF Percolation-Inferred Galaxy Group (2PIGG) catalogue. The mean halo occupation number is found to be well-fitted by a power-law, ~ M^b, at high masses, with b = 1.05, 0.88, 0.99 for red, blue and all galaxies respectively (with 1-sigma errors of 15-19%). We find that the truncated NFW profile provides a good fit to the galaxy radial distributions, with concentration parameters c=3.9, 1.3, 2.4 for red, blue and all galaxies respectively (with 1-sigma errors of 8-15%). Adding the observed linear power spectrum to these results, we compare these empirical predictions of the halo model with the observed correlation functions for these same 2dF galaxy populations. We conclude that subject to some fine tuning it is an acceptable model for the two-point correlations. Our analysis also explains why the correlation function slope of the red galaxies is steeper than that of the blue galaxies. It is mainly due to the number of red and blue galaxies per halo, rather than the radial distribution within the haloes of the two galaxy species.Comment: 15 pages, 15 figures. MNRAS accepted version. Adds appx. on profile fitting; now use truncated NF

A simple microcontroller based digital lock-in amplifier for the detection of low level optical signals

Traditionally digital lock-in amplifiers sample the input signal at a rate much higher than the lock-in reference frequency and perform the lock-in algorithm with high-speed processors. We present a small and simple digital lock-in amplifier that uses a 20 bit current integrating analogue-to-digital converter interfaced to a microcontroller. The sample rate is set to twice the reference frequency placing the sampled lock-in signal at the Niquest frequency allowing the lock-in procedure to be performed with one simple algorithm. This algorithm consists of a spectral inversion technique integrated into a highly optimised low-pass filter. We demonstrate a system with a dynamic range of 103dB recovering signals up to 85dB below the interference