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Register a new userLimit theory for point processes in manifolds
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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 ${Y_i}_{i=1}^n$.
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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 windows $W_n:= [-{1 over 2}n^{1/d},{1 over 2}n^{1/d}]^d subset mathbb{R}^d$. We consider the statistic $H_n^xi:= sum_{x in P_n}xi(x,P_n)$ where $xi(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^{xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $xi$-weighted point measures $mu_n^{xi} := sum_{x in P_n} xi(x,P_n) delta_{n^{-1/d}x}$, as $W_n uparrow mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $alpha$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $alpha$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $mu_n^xi$ via an extension of the cumulant method.
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 point processes on union of two copies of $mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
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 P(Z_n=v_n) as v_n earrow infty, and use this to study conditional large deviations of {Y_{Z_n}:nge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:nge1} conditioned on Z_nge v_n.
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 supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
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