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Let $Y_i,igeq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $mathcal {M}subset mathbb{R}^d$ and consider sums $sum_{i=1}^nxi(n^{1/m}Y_i,{n^{1/m}Y_j}_{j=1}^n)$, where $xi$ is a real valued function defined on pairs $(y,mathcal {Y})$, with $yin mathbb{R}^d$ and $mathcal {Y}subset mathbb{R}^d$ locally finite. Subject to $xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $mathcal {M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, R{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on ${Y_i}_{i=1}^n$.
Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P cap W_n$ be its restriction to w
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal poin
In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:nge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of
Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes [math.PR/0407059]
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the super