Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMoments, cumulants and diagram formulae for non-linear functionals of random measures
286
0
0.0
(
0
)
Authors
Giovanni Peccati
Ask ChatGPT about the research
No Arabic abstract
This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Mobius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.
rate research
Read More
We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtained energy bounds are sharp for several infinite families of codes.
Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.
Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Steins method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.
The convex hull $P_{n}$ of a Gaussian sample $X_{1},...,X_{n}$ in $R^{d}$ is a Gaussian polytope. We prove that the expected number of facets $E f_{d-1} (P_n)$ is monotonically increasing in $n$. Furthermore we prove this for random polytopes generated by uniformly distributed points in a $d$-dimensional ball.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
