Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCentral Limit Theorems for Multicolor Urns with Dominated Colors
171
0
0.0
(
0
)
Authors
Patrizia Berti
Ask ChatGPT about the research
No Arabic abstract
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.
rate research
Read More
The approximation of integral type functionals is studied for discrete observations of a continuous It^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color given the past is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.
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.
We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.
An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0leq L<Uleq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_noverset{a.s.}longrightarrow Z$ for some random variable $Z$, and begin{gather*} D_n:=sqrt{n},(Z_n-Z)longrightarrowmathcal{N}(0,sigma^2)quadtext{conditionally a.s.} end{gather*} where $sigma^2$ is a certain random variance. Almost sure conditional convergence means that begin{gather*} Pbigl(D_nincdotmidmathcal{G}_nbigr)overset{weakly}longrightarrowmathcal{N}(0,,sigma^2)quadtext{a.s.} end{gather*} where $Pbigl(D_nincdotmidmathcal{G}_nbigr)$ is a regular version of the conditional distribution of $D_n$ given the past $mathcal{G}_n$. Thus, in particular, one obtains $D_nlongrightarrowmathcal{N}(0,sigma^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
