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Register a new userA central limit like theorem for Fourier sums
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We consider the probability distributions of values in the complex plane attained by Fourier sums of the form sum_{j=1}^n a_j exp(-2pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficients a_j are i.i.d. drawn with finite third moment, the distance of these distributions to an isotropic two-dimensional Gaussian on C converges in probability to zero for any pseudometric on the set of distributions for which the distance between empirical distributions and the underlying distribution converges to zero in probability.
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The discounted central limit theorem concerns the convergence of an infinite discounted sum of i.i.d. random variables to normality as the discount factor approaches $1$. We show that, using the Fourier metric on probability distributions, one can obtain the discounted central limit theorem, as well as a quantitative version of it, in a simple and natural way, and under weak assumptions.
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
We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $sqrt{n}{mathcal{T}_c(P_n,Q)-mathcal{W}_c(P,Q)}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $pgeq 1$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials.
We consider a class of interacting particle systems with values in $[0,8)^{zd}$, of which the binary contact path process is an example. For $d ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of nodes with threshold zero which consists of initially activated nodes, whereas every other node is inactive. Subsequently, in each round, if an inactive node with threshold $theta$ has at least $theta$ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more nodes become activated. The main result of this paper provides a central limit theorem for the final size of activated nodes. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated nodes has asymptotically Gaussian fluctuations.
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