Discrete events: Perspectives from system theory

Systems Theory;differentiaal/ integraal-vergelijkingen

Parameter estimation and the statistics of nonlinear cosmic fields

The large scale distribution of matter in the universe contains valuable information about fundamental cosmological parameters, the properties of dark matter and the formation processes of galaxies. The best hope of retrieving this information lies in providing a statistical description of the matter distribution that may be used for comparing models with observation. Unfortunately much of the important information lies on scales below which nonlinear gravitational effects have taken hold, complicating both models and statistics considerably. This thesis deals with the distribution of matter - mass and galaxies - on such scales. The aim is to develop new statistical tools that make use of the nonlinear evolution for the purposes of constraining cosmological models.A new derivation for the 1 -point probability distribution function (PDF) for density inhomogeneities is presented first. The calculation is based upon an exact statistical treatment, using the Chapman -Kolmogorov equation and second order Eulerian perturbation theory to propagate the initial density field into the nonlinear regime. The analysis yields the generating function for moments, allowing for a straightforward derivation of the skewness. A new dependance upon the perturbation spectrum is found for the skewness at second order. The results of the analysis are compared against other methods for deriving the 1 -point PDF and against data from numerical N -body simulations. Good agreement is found in both cases.The 1 -point PDF for galaxies is derived next, taking into account nonlinear biasing of the density field and the distorting effects associated with working in redshift space. Once again perturbation theory is used to evolve the density field into the nonlinear regime and the Chapman -Kolmogorov equation to propagate the initial probabilities. Transformation of the dark matter density to a biased galaxy distribution is done through an Eulerian biasing prescription, expanding the nonlinear bias function to second order. An advantage of the Chapman- Kolmogorov approach is the natural way that different initial conditions and biasing models may be incorporated. It is shown that the method is general enough to allow a non -deterministic (hidden variable) bias. The dependance on cosmological parameters of the evolution of the galaxy 1 -point PDF is demonstrated and a method for differentiating between degenerate models in linear theory is presented. A new derivation of the skewness for a biased density field in red - shift space is also given and shown to depend significantly on the density and bias parameters. The results are compared favourably with those of numerical simulations.Finally a new, general formalism for analysing parameter information from non - Gaussian cosmic fields is developed. The method is general enough for application to a range of problems including the measurement of parameters from galaxy redshift surveys, weak lensing surveys and velocity field surveys. It may also be used to test for non -Gaussianity in the Cosmic Microwave Background. Generalising maximum likelihood analysis to second order, the non -Gaussian Fisher information matrix is derived and the detailed shapes of likelihood surfaces in parameter space are explored via a parameter entropy function. Concentrating on non -Gaussianity due to nonlinear evolution under gravity, the generalised Fisher analysis is applied to a model of a Galaxy redshift survey, including the effects of biasing, redshift space distortions and shot noise. Incorporating second order moments into the parameter estimation is found to have a large effect, breaking all of the degeneracies between parameters. The results indicate that using nonlinear likelihood analysis may yield parameter uncertainties around the few percent level from forthcoming large galaxy redshift surveys

Stability Problems for Stochastic Models: Theory and Applications II

Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 21­25 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia