Some statistical properties of random noise

1949 ◽  
Vol 45 (3) ◽  
pp. 368-372 ◽  
Author(s):  
D. K. C. MacDonald
Keyword(s):  
Experimental Investigation ◽  
Correlation Function ◽  
Power Spectrum ◽  
Narrow Range ◽  
Random Noise ◽  
Practical Importance ◽  
Statistical Properties ◽  
Regular Oscillation ◽  
Image Position ◽  
Random Fluctuations

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein(7), Fürth(1) and Rice (4), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form(where R(t) and θ(t) are variables changing slowly in comparison with sinω0t), it is clear that the magnitude of the envelope R(t) and the phase θ(t) are now the significant quantities. Rice (4), among others, has made a study of the statistical properties of R, deriving in particular the correlation function R(t) R(t + τ) in terms of the characteristics of the (power) spectrum ω(f). Fürth and the writer (2) have extended this work and carried out a collateral experimental investigation.

scholarly journals Statistical Properties of the LEDA Redshift Database

Fractals ◽  
1997 ◽  
Vol 05 (04) ◽  
pp. 635-660 ◽  
Author(s):  
Luca Amendola ◽  
Helene di Nella ◽  
Marco Montuori ◽  
Francesco Sylos Labini
Keyword(s):  
Correlation Function ◽  
Power Spectrum ◽  
Large Scale ◽  
Density Fluctuations ◽  
Statistical Properties ◽  
Redshift Surveys ◽  
Galaxy Distribution ◽  
Fractal Properties

We present the statistical properties of large scale galaxy distribution in the LEDA redshift database. This catalog contains more than 40,000 redshifts over all the sky. We find that LEDA, although seriously affected by incompleteness, shows quite stable statistical properties. In particular, we have considered the behavior of the two points correlation function and of the power spectrum of the density fluctuations, and we have done several tests to check whether the incompleteness of the catalog affect these statistical quantities. Our conclusion is that galaxy distribution in this catalog has fractal properties up to the distance ~ 200h-1 Mpc with D = 2.1 ± 0.2, with no sign towards homogeneity: this result is statistically stable. Finally, we compare these results with those obtained in other redshift surveys, finding that this sample well reproduces the properties of galaxy distribution found in the different catalogs.

scholarly journals Statistical modelling of the cosmological dispersion measure

2021 ◽  
Vol 502 (2) ◽  
pp. 2615-2629
Author(s):  
Ryuichi Takahashi ◽  
Kunihito Ioka ◽  
Asuka Mori ◽  
Koki Funahashi
Keyword(s):  
Dark Matter ◽  
Correlation Function ◽  
Probability Distribution ◽  
Power Spectrum ◽  
Free Electron ◽  
Statistical Modelling ◽  
Dispersion Measure ◽  
Host Galaxies ◽  
Angular Power Spectrum ◽  
Bias Factor

ABSTRACT We have investigated the basic statistics of the cosmological dispersion measure (DM)—such as its mean, variance, probability distribution, angular power spectrum, and correlation function—using the state-of-the-art hydrodynamic simulations, IllustrisTNG300, for the fast radio burst cosmology. To model the DM statistics, we first measured the free-electron abundance and the power spectrum of its spatial fluctuations. The free-electron power spectrum turns out to be consistent with the dark matter power spectrum at large scales, but it is strongly damped at small scales (≲  Mpc) owing to the stellar and active galactic nucleus feedback. The free-electron power spectrum is well modelled using a scale-dependent bias factor (the ratio of its fluctuation amplitude to that of the dark matter). We provide analytical fitting functions for the free-electron abundance and its bias factor. We next constructed mock sky maps of the DM by performing standard ray-tracing simulations with the TNG300 data. The DM statistics are calculated analytically from the fitting functions of the free-electron distribution, which agree well with the simulation results measured from the mock maps. We have also obtained the probability distribution of source redshift for a given DM, which helps in identifying the host galaxies of FRBs from the measured DMs. The angular two-point correlation function of the DM is described by a simple power law, $\xi (\theta) \approx 2400 (\theta /{\rm deg})^{-1} \, {\rm pc}^2 \, {\rm cm}^{-6}$, which we anticipate will be confirmed by future observations when thousands of FRBs are available.

scholarly journals Sufficiency of a Gaussian power spectrum likelihood for accurate cosmology from upcoming weak lensing surveys

2021 ◽  
Author(s):  
Robin E Upham ◽  
Michael L Brown ◽  
Lee Whittaker
Keyword(s):  
Power Spectrum ◽  
Random Noise ◽  
Power Spectra ◽  
Stage Iv ◽  
Wishart Distribution ◽  
Weak Lensing ◽  
Dependence Structure ◽  
Gaussian Fields ◽  
Non Gaussian ◽  
Parameter Constraints

Abstract We investigate whether a Gaussian likelihood is sufficient to obtain accurate parameter constraints from a Euclid-like combined tomographic power spectrum analysis of weak lensing, galaxy clustering and their cross-correlation. Testing its performance on the full sky against the Wishart distribution, which is the exact likelihood under the assumption of Gaussian fields, we find that the Gaussian likelihood returns accurate parameter constraints. This accuracy is robust to the choices made in the likelihood analysis, including the choice of fiducial cosmology, the range of scales included, and the random noise level. We extend our results to the cut sky by evaluating the additional non-Gaussianity of the joint cut-sky likelihood in both its marginal distributions and dependence structure. We find that the cut-sky likelihood is more non-Gaussian than the full-sky likelihood, but at a level insufficient to introduce significant inaccuracy into parameter constraints obtained using the Gaussian likelihood. Our results should not be affected by the assumption of Gaussian fields, as this approximation only becomes inaccurate on small scales, which in turn corresponds to the limit in which any non-Gaussianity of the likelihood becomes negligible. We nevertheless compare against N-body weak lensing simulations and find no evidence of significant additional non-Gaussianity in the likelihood. Our results indicate that a Gaussian likelihood will be sufficient for robust parameter constraints with power spectra from Stage IV weak lensing surveys.

Bohm's Mysterious 'Quantum Force' and 'Active Information': Alternative Interpretation and Statistical Properties

2001 ◽  
Vol 56 (1-2) ◽  
pp. 205-207 ◽  
Author(s):  
Boon Leong Lan
Keyword(s):  
Time Series ◽  
Power Spectrum ◽  
Important Implication ◽  
Alternative Interpretation ◽  
Quantum States ◽  
Statistical Properties ◽  
Numerical Evidence ◽  
Stationary Quantum ◽  
Non Gaussian ◽  
Quantum Force

AbstractAn alternative interpretation to Bohm’s ‘quantum force’ and ‘active information’ is proposed. Numerical evidence is presented, which suggests that the time series of Bohm’s ‘quantum force’ evaluated at the Bohmian position for non-stationary quantum states are typically non-Gaussian stable distributed with a flat power spectrum in classically chaotic Hamitonian systems. An important implication of these statistical properties is briefly mentioned

An experimental investigation of the Rossby two-slit problem

2020 ◽  
Vol 893 ◽  
Author(s):  
A. K. Kaminski ◽  
K. R. Helfrich ◽  
J. Pedlosky
Keyword(s):  
Experimental Investigation ◽  
Image Position

scholarly journals Cosmological Spectrum of Two-Point Correlation Function from Vacuum Fluctuation of Stringy Axion Field in De Sitter Space: A Study of the Role of Quantum Entanglement

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 79
Author(s):  
Sayantan Choudhury ◽  
Sudhakar Panda
Keyword(s):  
Correlation Function ◽  
Power Spectrum ◽  
Quantum Entanglement ◽  
De Sitter Space ◽  
Entangled State ◽  
De Sitter ◽  
Vacuum Fluctuation ◽  
Axion Field ◽  
Point Correlation ◽  
Point Correlation Function

In this work, we study the impact of quantum entanglement on the two-point correlation function and the associated primordial power spectrum of mean square vacuum fluctuation in a bipartite quantum field theoretic system. The field theory that we consider is the effective theory of axion field arising from Type IIB string theory compacted to four dimensions. We compute the expression for the power spectrum of vacuum fluctuation in three different approaches, namely (1) field operator expansion (FOE) technique with the quantum entangled state, (2) reduced density matrix (RDM) formalism with mixed quantum state and (3) the method of non-entangled state (NES). For a massless axion field, in all three formalisms, we reproduce, at the leading order, the exact scale invariant power spectrum which is well known in the literature. We observe that due to quantum entanglement, the sub-leading terms for these thee formalisms are different. Thus, such correction terms break the degeneracy among the analysis of the FOE, RDM and NES formalisms in the super-horizon limit. On the other hand, for massive axion field we get a slight deviation from scale invariance and exactly quantify the spectral tilt of the power spectrum in small scales. Apart from that, for massless and massive axion field, we find distinguishable features of the power spectrum for the FOE, RDM, and NES on the large scales, which is the result of quantum entanglement. We also find that such large-scale effects are comparable to or greater than the curvature radius of the de Sitter space. Most importantly, in near future if experiments probe for early universe phenomena, one can detect such small quantum effects. In such a scenario, it is possible to test the implications of quantum entanglement in primordial cosmology.

scholarly journals Statistical Properties of the Lyman α Forest

1989 ◽  
Vol 134 ◽  
pp. 549-554
Author(s):  
Jill Bechtold ◽  
Stephen A. Shectman
Keyword(s):  
Correlation Function ◽  
High Redshift ◽  
Statistical Properties ◽  
Echelle Spectrograph ◽  
Metal Line ◽  
Photon Counter ◽  
Point Correlation ◽  
Point Correlation Function

We have observed two high-redshift quasars with the echelle spectrograph and 2D-Frutti Photon Counter at Las Campanas, in order to investigate the statistical properties of the Lyman-α forest. The two-point correlation function for the Lyman-α forest lines at z ≈ 3 is consistent with zero, for all velocity splittings Δv > 50 km/sec. When Lyman-α lines and other metal lines from known metal-line systems are included, the correlation function shows a weak non-zero signal at small Δv. We suggest that the weak clustering of the Lyman-α forest detected by other workers may be the result of contamination by a small number of metal-line systems and their associated Lyman-α lines.

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