Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwo limit theorems for the high-dimensional two-stage contact process
80
0
0.0
(
0
)
Authors
Xiaofeng Xue
Ask ChatGPT about the research
No Arabic abstract
In this paper we are concerned with the two-stage contact process introduced in cite{Krone1999} on a high-dimensional lattice. By comparing this process with an auxiliary model which is a linear system, we obtain two limit theorems for this process as the dimension of the lattice grows to infinity. The first theorem is about the upper invariant measure of the process. The second theorem is about asymptotic behavior of the critical value of the process. These two theorems can be considered as extensions of their counterparts for the basic contact processes proved in cite{Grif1983} and cite{Schonmann1986}.
rate research
Read More
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse $alpha$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombes theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Gotze (1991) in Steins method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $log n$ factor. We also discuss an application to multiple Wiener-It^{o} integrals.
The Pitman-Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson-Dirichlet distribution with parameters $0<alpha<1, theta>-alpha$. The parameters $alpha$ and $theta$ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability $ u$. In this article we consider the limit theorems for the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. These include law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either $alpha$ tends to zero or one. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to generalized random energy model for disordered system.
We consider eigenvalues of generalized Wishart processes as well as particle systems, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, central limit theorems for the Dysons Brownian motion and the eigenvalues of the Wishart process are recovered under slightly more general initial conditions, and a central limit theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process is obtained.
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
