Mod-Gaussian convergence and its applications for models of statistical mechanics

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of "breaking of symmetry" in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of $L^1$-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.Comment: 49 pages, 21 figure

The accuracy of merging approximation in generalized St. Petersburg games

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Cs\"org\H{o}, S., Merging asymptotic expansions in generalized St. Petersburg games, \textit{Acta Sci. Math. (Szeged)} \textbf{73} 297--331, 2007. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.Comment: 30 pages long version (to appear in Journal of Theoretical Probability); some corrected typo

An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and $$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}}$$ are proved for the uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution function $\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.Comment: 33 page

Lectures on Gaussian approximations with Malliavin calculus

In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the "Fourth Moment Theorem" in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre, giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper "Stein's method on Wiener chaos" (written in collaboration with Peccati) in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein's method and Malliavin calculus fit together admirably well. Their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of infinite-dimensional Gaussian fields. The current survey aims to introduce the main features of this recent theory. It originates from a series of lectures I delivered at the Coll\`ege de France between January and March 2012, within the framework of the annual prize of the Fondation des Sciences Math\'ematiques de Paris. It may be seen as a teaser for the book "Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality" (jointly written with Peccati), in which the interested reader will find much more than in this short survey.Comment: 72 pages. To be published in the S\'eminaire de Probabilit\'es. Mild update: typos, referee comment

On the distribution of linear combinations of independent Gumbel random variables

The distribution of linear combinations of independent Gumbel random variables is of great interest for modeling risk and extremes in the most different areas of application. In this paper we develop near-exact approximations for the distribution of linear combination of independent Gumbel random variables based on a shifted generalized near-integer gamma distribution and on the distribution of the difference of two independent generalized integer gamma distributions. These near-exact distributions are computationally appealing and numerical studies confirm their accuracy, as assessed by a proximity measure used in related studies. We illustrate the proposed approximations on applied problems in networks engineering, computational biology, and flood risk management

Large proportion of wood dependent lichens in boreal pine forest are confined to old hard wood

Intensive forest management has led to a population decline in many species, including those dependent on dead wood. Many lichens are known to depend on dead wood, but their habitat requirements have been little studied. In this study we investigated the habitat requirements of wood dependent lichens on coarse dead wood (diameter > 10 cm) of Scots pine Pinus sylvestris in managed boreal forests in central Sweden. Twenty-one wood dependent lichen species were recorded, of which eleven were confined to old (estimated to be > 120 years old) and hard dead wood. Almost all of this wood has emanated from kelo trees, i.e. decorticated and resin-impregnated standing pine trees that died long time ago. We found four red-listed species, of which two were exclusive and two highly associated with old and hard wood. Lichen species composition differed significantly among dead wood types (low stumps, snags, logs), wood hardness, wood age and occurrence of fire scars. Snags had higher number of species per dead wood area than logs and low stumps, and old snags had higher number of species per dead wood area than young ones. Since wood from kelo trees harbours a specialized lichen flora, conservation of wood dependent lichens requires management strategies ensuring the future presence of this wood type. Besides preserving available kelo wood, the formation of this substratum should be supported by setting aside P. sylvestris forests and subject these to prescribed burnings as well as to allow wild fires in some of these forests