Let K be a convex set in R d and let K $λ$ be the convex hull of a homogeneous Poisson point process P $λ$ of intensity $λ$ on K. When K is a simple polytope, we establish scaling limits as $λ$ $\rightarrow$ $\infty$ for the boundary of K $λ$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $λ$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $λ$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity $\sqrt$ de dh dhdv.
Let $K \subset \R^d$ be a smooth convex set and let $¶_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $¶_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.
Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, co…
The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this missing piece of the jigsaw in the theory of Gaussian polytopes are discussed.
Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional and the intrinsic volumes of this random polytope. Additionally we derive a central limit theorem for the oracle estimator, that is an unbiased an minimal variance estimator for the volume of a convex set. Finally we obtain a multivariate limit theorem for the intrinsic volumes and the components of the $\mathbf{f}$-vector of the random polytope.
Stationary and isotropic iteration stable random tessellations are considered, which can be constructed by a random process of cell division. The collection of maximal polytopes at a fixed time $t$ within a convex window $W\subset{\Bbb R}^d$ is regarded and formulas for mean values, variances, as well as a characterization of certain covariance measures are proved. The focus is on the case $d\geq 3$, which is different from the planar one, treated separately in \cite{ST2}. Moreover, a multivariate limit theorem for the vector of suitably rescaled intrinsic volumes is established, leading in ea…
The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of $K_n$ are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.
Lower bounds for variances are often needed to derive central limit theorems. In this paper, we establish a lower bound for the variance of Poisson functionals that uses the difference operator of Malliavin calculus. Poisson functionals, i.e. random variables that depend on a Poisson process, are frequently studied in stochastic geometry. We apply our lower variance bound to statistics of spatial random graphs, the $L^p$ surface area of random polytopes and the volume of excursion sets of Poisson shot noise processes. Thereby we do not only bound variances from below but also show positive def…
Fix a space dimension $d\ge 2$, parameters $α> -1$ and $β\ge 1$, and let $γ_{d,α, β}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*} ||x||^α\, \exp\left(-\frac{\|x\|^β}β\right), \qquad x\in \mathbb{R}^d. \end{align*} By $K_λ$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $\mathbb{R}^d$ with intensity measure $λγ_{d,α,β}$, $λ>0$. We establish that the scaling limit of the boundary of $K_λ$, as $λ\rightarrow \infty$, is given by a universal `festoon' of piecewise pa…
Schreiber and Yukich [Ann. Probab. 36 (2008) 363-396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^d, d\geq2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high…
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rollin…
Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Bárány and La…
Let $Γ$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $Γ^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $Γ$). In particular, we show that $$ ΓB_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right). $$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results…
We study proportions of consecutive occurrences of permutations of a given size. Specifically, the limit of such proportions on large permutations forms a region, called \emph{feasible region}. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called \emph{overlap graph}. This allows us to compute the dimension, vertices and faces of the polytope, and to determine the equations that define it. Finally we prove that the limit of classical occurrences and consecutive occurrences are in some sense independent. As a consequence, the scaling limi…
In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offsprin…
Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution…
We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope. Finally, we prove that the limits of classical occurrences and consecutive occurrences are independent, in some sense made precise in the extended abstract. As a consequence, the scaling limit of a sequ…
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a sub-exponential tail inequality possess this property RIP with overwhelming probability. We show that such "sensing" matrices are valid for the exact reconstruction process of $…
In the paper we present simple examples of linear random fields defined on $\ZZ^2$ and $\ZZ^3$ which exhibit the scaling transition phenomenon. These examples lead to more general definition of the scaling transition and allow to understand the mechanism of appearance of this phenomenon better. In previous papers devoted to the scaling transition it was proved mainly for random fields with finite variance and long-range dependence. We consider random fields with finite and infinite variance. Our examples show that the scaling transition phenomenon can be observed for linear random fields with …
There are two ways of speeding up MCMC algorithms: (1) construct more complex samplers that use gradient and higher order information about the target and (2) design a control variate to reduce the asymptotic variance. While the efficiency of (1) as a function of dimension has been studied extensively, this paper provides first rigorous results linking the growth of the asymptotic variance in (2) with dimension. Specifically, we construct a control variate for a $d$-dimensional Random walk Metropolis chain with an IID target using the solution of the Poisson equation for the scaling limit in t…
For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the c…
We consider a particle system with weights and the scaling limits derived from its occupation time. We let the particles perform independent recurrent Lévy motions and we assume that their initial positions and weights are given by a Poisson point process. In the limit we obtain a number of recently discovered stationary stable self-similar processes recently studied in [SAM1] and [SAM2]. We also observe very different limit processes depending on whether the function, whose occupation time is considered, integrates to zero or not.
With respect to a class of long-range exclusion processes on $\mathbb{Z}^d$, with single particle transition rates of order $|\cdot|^{-(d+α)}$, starting under Bernoulli invariant measure $ν_ρ$ with density $ρ$, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on $α$, $d$ and $ρ$ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian …
We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables with its usual $SU(m + 1)$-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope $P$. We then determine the asymptotics of the conditional expectation $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ p…
The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article, we show that $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian measure and then consider the conditional measures $γ_{|NP}$ induced on the subspace of polynomials whose Newton polytop…
The prevalence of many urban phenomena changes systematically with population size. We propose a theory that unifies models of economic complexity and cultural evolution to derive urban scaling. The theory accounts for the difference in scaling exponents and average prevalence across phenomena, as well as the difference in the variance within phenomena across cities of similar size. The central ideas are that a number of necessary complementary factors must be simultaneously present for a phenomenon to occur, and that the diversity of factors is logarithmically related to population size. The …
Let $η_t$ be a Poisson point process of intensity $t\geq 1$ on some state space $\Y$ and $f$ be a non-negative symmetric function on $\Y^k$ for some $k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $η_t$ generates a point process $ξ_t$ on the positive real-half axis. The scaling limit of $ξ_t$ as $t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the $m$-th smallest point of $ξ_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-Itô…
Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Lévy process with non-zero Lévy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the Lévy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under gener…
We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit of finite variance Galton-Watson trees condition on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to this is a generalization of the classical Otter-Dwass formula.
We study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP pa…
The Random Walk Metropolis (RWM) algorithm is a Metropolis- Hastings MCMC algorithm designed to sample from a given target distribution πwith Lebesgue density on R^N. RWM constructs a Markov chain by randomly proposing a new position (the "proposal move"), which is then accepted or rejected according to a rule which makes the chain reversible with respect to π. When the dimension N is large a key question is to determine the optimal scaling with N of the proposal variance: if the proposal variance is too large, the algorithm will reject the proposed moves too often; if it is too small, the alg…
The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on-off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the tra…
We provide Monte Carlo estimates of the scaling of the length $L_{n}$ of the longest increasing subsequences of $n$-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, $L_{n} \sim n^θ$ with the leading scaling exponent $0.60 \lesssim θ\lesssim 0.69$ for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and $θ\simeq 0.57$ for distributions of finite variance, irrespective of the particular distribution. …
In this paper we characterize the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. The extremal sums behave completely different by compared to the i.i.d case. In particular, though we still have asymptotic normality, the scaling factor is relatively bigger than in the i.i.d case, meaning that the maximal terms have relatively smaller contribution to the whole sum. Also, the scaling need not depend on the tail index of the underlying marginal distribution, as it is well-known to be so in the i.i.d. si…
For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (Random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as $n^{-1}$ and as $n^{-\frac{1}{3}}$ with dimension $n$, and that the acceptance rates should be tuned to $0.234$ and $0.574$. We establish counterexamples to demonstrate that smoothness assumptions of the order of $\mathcal{C}^1(\mathbb{R})$ for RWM and $\mathcal{C}^3(\mathbb{R})$ for MALA are indeed required if these scaling rates are to hold. The counterexamples identify …
We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs-Seppalainen for asymmetric simple exclusion. However, some modifications we…
