Showing papers on "Central limit theorem published in 2017"
Book•
17 Apr 2017
TL;DR: In this article, a preliminary survey of the central limit theorem for heads and tails with independent increments is presented. But the authors do not consider the case of independence and dependence exchangeability.
Abstract: Part 7 A preliminary survey: heads and tails - preliminary considerations heads and tails - the random process laws of "large numbers" the "central limit theorem". Part 8 Random processes with independent increments: the case of asymptotic normality the Wiener-Levy process behaviour and asymptotic behaviour ruin problems ballot problems. Part 9 An introduction to other types of stochastic process: Markov processes stationary processes. Part 10 Problems in higher dimensions: second-order characteristics and the normal distribution the discrete case the continuous case the case of spherical symmetry. Part 11 Inductive reasoning, statistical inference: the basic formulation and preliminary clarifications the case of independence and the case of dependence exchangeability. Part 12 Mathematical statistics: the scope and limits of the treatment the likelihood principle and sufficient statistics a Bayesian approach to "estimation" and "hypothesis testing" the connections with decision theory.
614 citations
TL;DR: This review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations.
Abstract: According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding.
349 citations
TL;DR: In this article, a new general version of Stein's method for univariate distributions is proposed, which is based on a linear difference or differential-type operator, and the resulting Stein identity highlights the unifying theme behind the literature on Stein's methods both for continuous and discrete distributions.
Abstract: We propose a new general version of Stein’s method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein’s method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions: normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Frechet and Gumbel.
125 citations
TL;DR: In this article, it was shown that certain quantities associated to the energy density of solutions are essentially additive and that the first-order correctors converge to a Gaussian free field in the large-scale limit.
Abstract: One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in Armstrong et al. (Commun. Math. Phys. 347(2):315–361, 2016): using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.
113 citations
TL;DR: The authors used central limit theorems for randomization-based causal analyses of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment.
Abstract: Frequentists’ inference often delivers point estimators associated with confidence intervals or sets for parameters of interest. Constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analyses of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under th...
112 citations
TL;DR: In this article, the authors derived optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$ under the assumption that a spectral gap estimate holds in probability, and proved that there exists a stationary corrector field in dimensions$d>2$ and the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages.
Abstract: We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.
94 citations
TL;DR: In this article, the authors consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits, showing that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems.
84 citations
TL;DR: In this paper, the Betti numbers of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices were studied, obtaining limit theorems for means, strong laws, concentration inequalities and central limit.
Abstract: We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.
72 citations
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TL;DR: In this article, the authors provided several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants.
Abstract: This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure convergence of the estimators and the corresponding central limit theorems are obtained for all the Hurst parameter range $H\in (0, 1)$. The least squares estimator is used for the drift parameter. A central limit theorem is proved when the Hurst parameter $H \in (0, 1/2)$ and a noncentral limit theorem is proved for $H\in[3/4, 1)$. Thus, the open problem left in the paper by Hu and Nualart (2010) is completely solved, where a central limit theorem for least squares estimator is proved for $H\in [1/2, 3/4)$.
71 citations
TL;DR: The elephant random walk (ERW) as mentioned in this paper is a non-Markovian discrete-time random walk on Ω with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior.
Abstract: We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on ℤ with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime, the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.
69 citations
TL;DR: In this paper, the authors considered the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian.
Abstract: We consider the KPZ equation in one space dimension driven by a stationary centred space–time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf–Cole solution to the KPZ equation driven by space–time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.
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TL;DR: A class of adaptive tests that are computationally tractable and completely data-driven that achieve nontrivial powers in the contiguous regime and consistency in the singular regime whenever $n p_{n,av} \to\infty$ is the average connection probability.
Abstract: The present paper considers testing an Erdos--Renyi random graph model against a stochastic block model in the asymptotic regime where the average degree of the graph grows with the graph size n. Our primary interest lies in those cases in which the signal-to-noise ratio is at a constant level. Focusing on symmetric two block alternatives, we first derive joint central limit theorems for linear spectral statistics of power functions for properly rescaled graph adjacency matrices under both the null and local alternative hypotheses. The powers in the linear spectral statistics are allowed to grow to infinity together with the graph size. In addition, we show that linear spectral statistics of Chebyshev polynomials are closely connected to signed cycles of growing lengths that determine the asymptotic likelihood ratio test for the hypothesis testing problem of interest. This enables us to construct a sequence of test statistics that achieves the exact optimal asymptotic power within $O(n^3 \log n)$ time complexity in the contiguous regime when $n^2 p_{n,av}^3 \to\infty$ where $p_{n,av}$ is the average connection probability. We further propose a class of adaptive tests that are computationally tractable and completely data-driven. They achieve nontrivial powers in the contiguous regime and consistency in the singular regime whenever $n p_{n,av} \to\infty$. These tests remain powerful when the alternative becomes a more general stochastic block model with more than two blocks.
TL;DR: In this article, the authors established a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable.
Abstract: Summary
Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable. This result is of independent interest in large dimensional random-matrix theory. We also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed. Meanwhile, we apply the linear spectral statistic to an independence test for p random variables, and then an equivalence test for p factor loadings and n factors in a factor model. The finite sample performance of the test proposed shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from various cities in China is also conducted.
TL;DR: In this article, the bias and variance of the root-n consistent entropy estimator proposed by Bickel and Breiman (1983) for dimension 1 and 2 were studied.
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TL;DR: In this article, the authors considered Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0, and proved limit theorems for the nodal statistics associated with a smooth compact domain, in the high energy limit.
Abstract: We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).
TL;DR: For a general class of Bernoulli percolation models with long-range correlations, this paper showed that mesoscopic subballs have regular volume growth and satisfy a weak Poincare inequality and deduced quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with polynomial growth.
Abstract: For a general class of percolation models with long-range correlations on $\mathbb{Z}^{d}$, $d\geq2$, introduced in [J. Math. Phys. 55 (2014) 083307], we establish regularity conditions of Barlow [Ann. Probab. 32 (2004) 3024–3084] that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincare inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of [Probab. Theory Related Fields 166 (2016) 619–657] allow to extend various other known results about Bernoulli percolation by mimicking their proofs, for instance, the local central limit theorem of [Electron. J. Probab. 14 (209) 1–27] or the result of [Ann. Probab. 43 (2015) 2332–2373] that the dimension of at most linear harmonic functions on the infinite cluster is $d+1$. In terms of specific models, all these results are new for random interlacements at every level in any dimension $d\geq3$, as well as for the vacant set of random interlacements [Ann. of Math. (2) 171 (2010) 2039–2087; Comm. Pure Appl. Math. 62 (2009) 831–858] and the level sets of the Gaussian free field [Comm. Math. Phys. 320 (2013) 571–601] in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime for these models).
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TL;DR: This work derives the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss) and proposes a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions.
Abstract: The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss). We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. Our work is inspired by the results of Sommerfeld and Munk (2016) on the asymptotic distribution of empirical Wasserstein distance on finite space using unregularized transportation costs. Incidentally we also analyze the asymptotic distribution of entropy-regularized Wasserstein distances when the regularization parameter tends to zero. Simulated and real datasets are used to illustrate our approach.
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TL;DR: In this paper, the authors study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space.
Abstract: We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.
TL;DR: In this article, the authors consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density, and analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices.
Abstract: We consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satises a uniform X 2 logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Centered by a multiple of logn, where the constant is the inverse of the Malthusian rate of growth of the associated branching process, the minimal weight converges in distribution. The limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices, and a Gumbel random variable, and thus shows a remarkably universal behavior. The proofs rely on a rened coupling between the shortest path problems on these graphs and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination. The results extend to a host of related random graph models, ranging from random rregular graphs, inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence.
TL;DR: In this paper, a robust estimator of the autocorrelation parameter, which is consistent and satisfies a central limit theorem in the Gaussian case, is proposed to follow the classical inference approach, by plugging this estimator in the criteria used for change-points estimation.
Abstract: We consider the problem of multiple change-point estimation in the mean of an $\operatorname{AR}(1)$ process. Taking into account the dependence structure does not allow us to use the dynamic programming algorithm, which is the only algorithm giving the optimal solution in the independent case. We propose a robust estimator of the autocorrelation parameter, which is consistent and satisfies a central limit theorem in the Gaussian case. Then, we propose to follow the classical inference approach, by plugging this estimator in the criteria used for change-points estimation. We show that the asymptotic properties of these estimators are the same as those of the classical estimators in the independent framework. The same plug-in approach is then used to approximate the modified BIC and choose the number of segments. This method is implemented in the R package AR1seg and is available from the Comprehensive R Archive Network (CRAN). This package is used in the simulation section in which we show that for finite sample sizes taking into account the dependence structure improves the statistical performance of the change-point estimators and of the selection criterion.
TL;DR: In this paper, a non-Markovian discrete-time random walk with unbounded memory called the elephant random walk (ERW) is considered and a strong invariance principle for the ERW is shown.
Abstract: We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.
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TL;DR: In this article, the convergence results for discrete approximations of a large class of rough models were obtained for a large range of rough volatility models, including American or Bermudan.
Abstract: We extend Donsker's approximation of Brownian motion to fractional Brownian motion with Hurst exponent $H \in (0,1)$ and to Volterra-like processes. Some of the most relevant consequences of our `rough Donsker (rDonsker) Theorem' are convergence results for discrete approximations of a large class of rough models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement (for a large range of values of~$H$). This rDonsker Theorem further provides a weak convergence proof for the Hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan.
TL;DR: In this paper, a new Berry-Esseen bound for nonlinear functionals of nonsymmetric and nonhomogeneous infinite Rademacher sequences is established based on a discrete version of the Malliavin-Stein method and an analysis of the discrete Ornstein-Uhlenbeck semigroup.
Abstract: A new Berry–Esseen bound for nonlinear functionals of nonsymmetric and nonhomogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin–Stein method and an analysis of the discrete Ornstein–Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős–Renyi random graph. A further application deals with a percolation problem on trees.
TL;DR: In this article, the authors show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power p>0p>0, the Blumenthal-Getoor index β∈[ 0,2)β∈ [0,2] of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα.
Abstract: In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.
TL;DR: In this article, a new quantitative approach to a multidimensional version of the well-known de Jong central limit theorem under optimal conditions was developed, which allows one to deduce explicit (and presumably optimal) Wasserstein bounds in the case of general $U$-statistics of arbitrary order.
Abstract: We develop a new quantitative approach to a multidimensional version of the well-known de Jong’s central limit theorem under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Wasserstein bounds in the case of general $U$-statistics of arbitrary order $d\geq 1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong’ s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.
TL;DR: Sabot and Zeng as mentioned in this paper introduced a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the inverse Gaussian distribution, considered as the potential of a random Schrodinger operator.
Abstract: We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the inverse Gaussian distribution. Considered as the potential of a random Schrodinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot and Zeng (2015)], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.
TL;DR: In this article, a multivariate central limit theorem for lq-norms of high-dimensional random vectors that are chosen uniformly at random in an lpn-ball is presented.
Abstract: In this paper, we prove a multivariate central limit theorem for lq-norms of high-dimensional random vectors that are chosen uniformly at random in an lpn-ball. As a consequence, we provide several applications on the intersections of lpn-balls in the flavor of Schechtman and Schmuckenschlager and obtain a central limit theorem for the length of a projection of an lpn-ball onto a line spanned by a random direction 𝜃 ∈ 𝕊n−1. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 ≤ p < q this displays in speed and rate function deviations of the q-norm on an lpn-ball obtained by Schechtman and Zinn, but we obtain explicit constants.
TL;DR: In this article, the capacity of the range of a simple random walk on Z^d was studied and the main result is a central limit theorem for the capacity for d ≥ 6.
Abstract: We study the capacity of the range of a transient simple random walk on Z^d. Our main result is a central limit theorem for the capacity of the range for d ≥ 6. We present a few open questions in lower dimensions.
TL;DR: In this paper, the authors considered a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction and showed a two-dimensional phase transition with respect to the temperature and the coupling constant.
Abstract: We consider a spherical spin system with pure 2-spin spherical Sherrington–Kirkpatrick Hamiltonian with ferromagnetic Curie–Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices.
TL;DR: In this article, the authors provided a general quantitative central limit theorem for sequences of (correctly normalised) discrete probability measures of the type ρ(V{L}(V_{C_{n}}) = ρ ∈ {0,1,\ldots,d\}, which implies that most conic volumes encountered in applications can be approximated by a suitable Gaussian distribution.
Abstract: Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset\mathbb{R}^{d}$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,\ldots,d\}$, customarily denoted by $\mathcal{L}(V_{C})$. The aim of the present paper is to provide a Berry–Esseen bound for the normal approximation of $\mathcal{L}(V_{C})$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_{n}})$, $n\geq1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein’s method and second-order Poincare inequality, (3) concentration estimates and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp [Inf. Inference 3 (2014) 224–294] and McCoy and Tropp [Discrete Comput. Geom. 51 (2014) 926–963] about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.