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Register a new userNormal approximation of Gibbsian sums in geometric probability
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This paper concerns the asymptotic behavior of a random variable $W_lambda$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_lambda uparrow R^d$. We establish conditions ensuring that $W_lambda$ has volume order fluctuations, that is they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Steins method to deduce rates of normal approximation for $W_lambda$, as $lambdatoinfty$. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.
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We employ stabilization methods and second order Poincare inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s geq 1$, of statistics of marked Poisson processes on $mathbb{R}^d$, $d geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $iin{1,...,m}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,jin{1,...,m}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.
Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdos-Renyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramer-type moderate deviations via Steins method.
The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy [5]) with one degree of freedom. Its generalization to all integer degrees of freedom and sufficiently large non-integer degrees of freedom was recently proved in [10]. Here, this inequality is partly extended to smaller non-integer degrees of freedom and in particular - in a weaker form - to all infinitely divisible multivariate gamma distributions. A further monotonicity property - sometimes called more PLOD (positively lower orthant dependent) - for increasing correlations is proved for multivariate gamma distributions with integer or sufficiently large degrees of freedom.
We view the classical Lindeberg principle in a Markov process setting to establish a universal probability approximation framework by It^{o}s formula and Markov semigroup. As applications, we consider approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by additive Brownian motion, and obtain an approximation error with explicit dependence on the dimension which makes it possible to analyse high dimensional models. We also apply our framework to study stable approximation and normal approximation and obtain their optimal convergence rates (up to a logarithmic correction for normal approximation).
We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analog of this geometric inequality is derived, giving large and small deviation inequalities from a median, in terms of a modulus of regularity. Our methods are of independent interest, we find that log-affine sequences are the extreme points of the set of log-concave sequences belonging to a half-space slice of the simplex. This amounts to a discrete analog of the localization lemma of Lovasz and Simonovits. Further applications of this lemma are used to produce a discrete version of the Prekopa-Leindler inequality, large deviation inequalities for log-concave measures about their mean, and provide insight on the stability of generalized log-concavity under convolution.
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