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Register a new userA Central Limit Theorem, and related results, for a two-color randomly reinforced urn
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We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.
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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 describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.
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.
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the cost function is continuous, non-decreasing and depends on the distance. As an application, the CLT in the transportation distance is proved for independent and some dependent stationary sequences.
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.
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