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"Distribution (Probability theory)"
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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.
eBook
The Mother Body Phase Transition in the Normal Matrix Model
The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to
several other topics as quadrature domains, inverse potential problems and the Laplacian growth.
In this present paper we
consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and
introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain
We also study in detail the mother body problem associated to
To construct the mother body measure, we define a quadratic differential
Following previous works of Bleher & Kuijlaars
and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou
nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials.
Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of
the associated
eBook
Mixtures : estimation and applications
This book uses the EM (expectation maximization) algorithm to simultaneously estimate the missing data and unknown parameter(s) associated with a data set.The parameters describe the component distributions of the mixture; the distributions may be continuous or discrete.
eBook
Spatial and spatio-temporal bayesian models with R-INLA
Spatial and Spatio-Temporal Bayesian Models with R-INLA provides a much needed, practically oriented & innovative presentation of the combination of Bayesian methodology and spatial statistics. The authors combine an introduction to Bayesian theory and methodology with a focus on the spatial and spatio-temporal models used within the Bayesian framework and a series of practical examples which allow the reader to link the statistical theory presented to real data problems. The numerous examples from the fields of epidemiology, biostatistics and social science all are coded in the R package R-INLA, which has proven to be a valid alternative to the commonly used Markov Chain Monte Carlo simulations
eBook
Ubiquity of human-induced changes in climate variability
While climate change mitigation targets necessarily concern maximum mean state changes, understanding impacts and developing adaptation strategies will be largely contingent on how climate variability responds to increasing anthropogenic perturbations. Thus far Earth system modeling efforts have primarily focused on projected mean state changes and the sensitivity of specific modes of climate variability, such as the El Niño–Southern Oscillation. However, our knowledge of forced changes in
the overall spectrum of climate variability and higher-order statistics is
relatively limited. Here we present a new 100-member large ensemble of
climate change projections conducted with the Community Earth System Model
version 2 over 1850–2100 to examine the sensitivity of internal climate
fluctuations to greenhouse warming. Our unprecedented simulations reveal
that changes in variability, considered broadly in terms of probability distribution, amplitude, frequency, phasing, and patterns, are ubiquitous
and span a wide range of physical and ecosystem variables across many
spatial and temporal scales. Greenhouse warming in the model alters
variance spectra of Earth system variables that are characterized by
non-Gaussian probability distributions, such as rainfall, primary
production, or fire occurrence. Our modeling results have important
implications for climate adaptation efforts, resource management, seasonal
predictions, and assessing potential stressors for terrestrial and
marine ecosystems.
Journal Article
On the rate of convergence in Wasserstein distance of the empirical measure
Let
μ
N
be the empirical measure associated to a
N
-sample of a given probability distribution
μ
on
R
d
. We are interested in the rate of convergence of
μ
N
to
μ
, when measured in the Wasserstein distance of order
p
>
0
. We provide some satisfying non-asymptotic
L
p
-bounds and concentration inequalities, for any values of
p
>
0
and
d
≥
1
. We extend also the non asymptotic
L
p
-bounds to stationary
ρ
-mixing sequences, Markov chains, and to some interacting particle systems.
Journal Article
Approximate NFormula omittedLO parton distribution functions with theoretical uncertainties: MSHT20aNFormula omittedLO PDFs
We present the first global analysis of parton distribution functions (PDFs) at approximate N [Formula omitted]LO in the strong coupling constant [Formula omitted], extending beyond the current highest NNLO achieved in PDF fits. To achieve this, we present a general formalism for the inclusion of theoretical uncertainties associated with the perturbative expansion in the strong coupling. We demonstrate how using the currently available knowledge surrounding the next highest order (N [Formula omitted]LO) in [Formula omitted] can provide consistent, justifiable and explainable approximate N [Formula omitted]LO (aN [Formula omitted]LO) PDFs. This includes estimates for uncertainties due the currently unknown N [Formula omitted]LO ingredients, but also implicitly some missing higher order uncertainties (MHOUs) beyond these. Specifically, we approximate the splitting functions, transition matrix elements, coefficient functions and K-factors for multiple processes to N [Formula omitted]LO. Crucially, these are constrained to be consistent with the wide range of already available information about N [Formula omitted]LO to match the complete result at this order as accurately as possible. Using this approach we perform a fully consistent approximate N [Formula omitted]LO global fit within the MSHT framework. This relies on an expansion of the Hessian procedure used in previous MSHT fits to allow for sources of theoretical uncertainties. These are included as nuisance parameters in a global fit, controlled by knowledge and intuition based prior distributions. We analyse the differences between our aN [Formula omitted]LO PDFs and the standard NNLO PDF set, and study the impact of using aN [Formula omitted]LO PDFs on the LHC production of a Higgs boson at this order. Finally, we provide guidelines on how these PDFs should be used in phenomenological investigations.
Journal Article
Time-like Graphical Models
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure-- so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
eBook