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Register a new userCentral Limit Theorem for product of dependent random variables
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Given ${X_k}$ is a martingale difference sequence. And given another ${Y_k}$ which has dependency within the sequence. Assume ${X_k}$ is independent with ${Y_k}$, we study the properties of the sums of product of two sequences $sum_{k=1}^{n} X_k Y_k$. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.
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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.
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on R d , with d $ge$ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials , namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Ces`aro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gartner-Ellis theorem and exponential equivalent method.
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.
In this article, we are interested in the normal approximation of the self-normalized random vector $Big(frac{sum_{i=1}^{n}X_{i1}}{sqrt{sum_{i=1}^{n}X_{i1}^2}},dots,frac{sum_{i=1}^{n}X_{ip}}{sqrt{sum_{i=1}^{n}X_{ip}^2}}Big)$ in $mathcal{R}^p$ uniformly over the class of hyper-rectangles $mathcal{A}^{re}={prod_{j=1}^{p}[a_j,b_j]capmathcal{R}:-inftyleq a_jleq b_jleq infty, j=1,ldots,p}$, where $X_1,dots,X_n$ are non-degenerate independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of $log p$ in the uniform central limit theorem (UCLT) under variety of moment conditions. When $X_{ij}$s have $(2+delta)$th absolute moment for some $0< deltaleq 1$, the optimal rate of $log p$ is $obig(n^{delta/(2+delta)}big)$. When $X_{ij}$s are independent and identically distributed (iid) across $(i,j)$, even $(2+delta)$th absolute moment of $X_{11}$ is not needed. Only under the condition that $X_{11}$ is in the domain of attraction of the normal distribution, the growth rate of $log p$ can be made to be $o(eta_n)$ for some $eta_nrightarrow 0$ as $nrightarrow infty$. We also establish that the rate of $log p$ can be pushed to $log p =o(n^{1/2})$ if we assume the existence of fourth moment of $X_{ij}$s. By an example, it is shown however that the rate of growth of $log p$ can not further be improved from $n^{1/2}$ as a power of $n$. As an application, we found respecti
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