Convex hulls of stable random walks

Features related to the perimeter of the convex hull ${C_n}$ of a random walk in ${\mathbb {R}^2}$ are studied, with particular attention given to its length ${L_n}$. Bounds on the variance of ${L_n}$ are obtained to show that, for walks with drift, ${L_n}$ obeys a strong law. Exponential bounds on the tail probabilities of ${L_n}$ under special conditions are also obtained. We then develop simple expressions for the expected values of other features of ${C_n}$, including the number of faces, the sum of the lengths and squared lengths of the faces, and the number of faces of length $t$ or less.

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

We consider in this paper two types of queueing systems which operate under a specific and fixed scheduling policy. The first system consists of a single server and several buffers in which arriving jobs are stored. We assume that arriving parts may require several stages of processing in which case each stage corresponds to a different buffer. The second system is a communication type queueing network given by a graph. The arriving jobs (packets) request a simple path along which they need to be processed. In both models the jobs arrive in a completely deterministic fashion: the interarrival times are fixed and known. All the processing times are also deterministic. A scheduling policy specifies a rule using which arriving parts are processed in the queueing system. A scheduling policy is defined to be stable if there is a finite uniform upper bound on the total number of parts in the system at all times. A necessary condition for stability of any work conserving policy is that the traffic intensity of the station (of each link in the graph in the communication model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([24],[20],[5],[6],[10]), deterministic fluid networks ([6],[9],[7],[8],[2]), deterministic adversarial networks ([4],[1],[14],[12]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks. Bramson [5] and Seidman [25] showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et. al. [1]. Dai [6] established that stability of a fluid deterministic queueing network implies stability of a stochastic queueing network. A similar result was established by Gamarnik [12], which connects stability of fluid and adversarial queueing networks. A complete characterization of two-station fluid networks which are stable under any work conserving policy was established by Bertsimas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik [13]. Motivated by a queueing network model stability of homogeneous random walks in a nonnegative orthant was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. Such random walks Q ( t ), t = 0, 1, 2,¡­ have Z + d as a state space ( Z + is the set of nonnegative integers). The transition vectors ¦¤ have deterministically bounded length in max norm and the transition probability p (¦«, ¦¤) along the direction ¦¤ depends only on the face ¦« that the random walk is currently on (the transition probabilities depend only on which coordinates of the current state are positive and which are zero). Such a random walk is defined to be stable if it is positive recurrent. We will also consider deterministic walks, for which p (¦«, ¦¤) is always zero or one (the transition deterministically depends on the face that the walk is currently on). A complete characterization of stable homogeneous random walks in Z + d for d ¡Ü 4 was obtained in Malyshev [21], Menshikov [23] and Ignatyuk and Malyshev [18], but no extension of this classification to higher dimensions has been obtained. Malyshev in [22] establishes a connection between homogeneous random walks and general dynamical systems on compact manifolds and shows that the difficulty of classifying stable random walks is of the same nature as the difficulty of understanding the dynamics of these dynamical systems. Specifically, the complicated dynamics precludes obtaining classification of stable random walks for d = 5. Thus despite many efforts the classification of stable random walks for general dimensions is not known. Likewise classification of stable policies in queueing systems is an open problem.

We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.

Stable distributions are special types of probability distributions whose origin is a particular limit regime of other types of distributions. They are closely related to the simple convolution process, which is introduced first for continuous and then for discrete random variables. This leads to the central limit theorem as one of the most important results of probability theory, as well as to its generalized version which is useful in the analysis of random walks. Extreme-value distributions are also presented, as they possess a limit theorem of their own (Fisher–Tippett–Gnedenko). The last part is devoted to the discussion of discrete-time and continuous-time random walks, together with their characteristic diffusion properties.

The objective of this research was to explore the effect of the treatment regimen of Spiriva combined with Symbicort on the immune function of non-small-cell lung cancer (NSCLC) based on computed tomography (CT) imaging features. An automatic CT image segmentation algorithm (RW-CT) was constructed based on random walk (RW) and image segmentation technology. The image segmentation algorithm based on the Toboggan method (C-CT) was introduced to compare with the traditional RW algorithm. 60 subjects were divided into four groups: a Chinese combined with Western medicine group (treated with Spiriva combined with Symbicort, group C+W), a Chinese medicine group (treated with Spiriva, group C), a Western medicine group (treated with Symbicort, group W), and a model group for control (group M). The results show that the Dice coefficient of the RW-CT algorithm was obviously larger than that of the C-CT algorithm and the RW algorithm, while the Hausdorff distance (HD) of the RW-CT algorithm was much smaller than that of the other two algorithms (P < 0.05). The proportion of positive cells of hypoxia-inducible factor-1α (HIF-1α) in group C+W was the least (15%-23%), followed by the group W (21%-29%) and the group C (28%-37%), and that in the group M was the highest (39%-49%). There was a remarkable difference in the immunohistochemical scores (HIS) of vascular endothelial growth factor (VEGF) in the tumor tissues between group C+W and the group M (H = 14.200, P = 0.001), but there was no great difference from the group C and the group W (P > 0.05). There was a notable difference in the IHS of vascular endothelial factor recepto-2 (VEGFR-2) between the group C+W medication group and the group M (H = 12.800, P = 0.002), and there was no statistical difference between the group C and W (P > 0.05). In short, the RW-CT constructed based on RW was better than the traditional algorithms for CT image segmentation. The Spiriva combined with Symbicort could effectively inhibit the expression of VEGF, VEGFR-2, and HIF-1α in NSCLC and promote the immunologic function of the body.

Abstract-In this paper, a novel feature set in images for registration is identified. Unique, geometrically invariant and easily extractable features in images called convex diagonal, convex quadrilateral are used for accurate image registration. Convex diagonals, convex quadrilaterals have attractive properties like easy extraction, geometric invariance and frequent occurrence. Coordinates, length and orientation information of corresponding convex diagonals in different images is used for initial transformation estimate. Corresponding convex hulls of scene objects are matched using Hausdorff distance as similarity measure operator. Coarse level estimate facilitates efficient, real time computation for final registration process. Initial transformation estimate based on convex diagonals, extracted from convex hull of scene objects, is refined using fine level image details to minimize errors originating from quantization and same convex hull information for different object shapes. The behavior of reference quadrilateral is robust against noise, outliers and broken edges.

The isoperimetric problem for convex hulls and large deviations rate functionals of random walks

Let R be a normed linear space, K be an arbitrary convex subset of an n-dimensional subspace Φn⊂R. This paper first gives a general charactaerization for a best approximation from K in form of “zero in the convex hull”. Applying it to the uniform approximation by generalized polynomials with restricted ranges, we get further an alternation characterization. Our results ocntains the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction.

The purpose of this thesis is to give a detail study of the well known Brouwer's fixed point theorem and Nash's equilibrium theorem. Section 1 is an introduction. Notions about triangulations of simplexes are recalled in section 2, the definitions of the affine independence, affne combination, affine hull, convex hull, barycentric coordinate, simplex, face, and triangulation are given and some basic properties of them are listed. In section 3, the definitions of barycenters of simplexes and sequences of ascending simplexes are introduced. After some discussions, the existence of a triangulation, the k-th barycentric subdivision, of a simplex is established, in case k is large enough, the maximum diameter of its members can be arbitrarily small. In section 4, we study some topological properties of convex sets and convex bodies, we see that some special homeomorphisms map convex bodies onto simplexes of suitable dimensions. In section 5, we study the celebrate theorems of Sperner, Knaster-Kuratowski-Mazurkiewicz, and Brouwer in a typical way. In section 6, the definition of noncooperative finite games in normal form is given, notions of the player, pure or mixed strategy, situation, payoff and equilibrium point are introduced with some examples. We see that the fixed points of Nash mappings are exactly the equilibrium points in mixed strategies of the corresponding games.

We study an extreme scenario of the Marstrand projection theorem for which a fractal has the property that its orthogonal projection is the same as the orthogonal projection of its convex hull. We extend results in current literature and establish checkable criteria for self-affine sets to have this property. Using this, we show that every convex polytope on \R^{d} contains a totally disconnected compact set, which is a union of self-affine sets, of dimension as close to 1 as possible, as well as a rectifiable 1-set, such that the fractal projects to an interval in every 1-dimensional subspace and its convex hull is the given polytope. Other convex sets and projections onto higher dimensional subspaces are also discussed.

Abstract. Urban planning increasingly relies on structured 3D city models for analysis and decision-making, but while point clouds capture rich detail, their unstructured nature complicates direct use in CityJSON-based workflows. We address the challenge of translating raw point clouds into lightweight, interoperable models by confronting the lack of point connectivity, the difficulty of extracting stable planes, edges, and boundaries at scale, and the need for CityJSON-compliant outputs. We introduce a four-phase pipeline based on basic primitives (cuboid, pyramid, sphere, torus) that comprises specification and setup, data capture with EyesCloud3D, preprocessing through downsampling, noise removal, and normal estimation, and processing with RANSAC plane detection, DBSCAN clustering, Convex Hull and Alpha Shapes for boundary extraction, Triangle Normal Analysis for sharp-edge detection, volume-based RMSE evaluation, CityJSON conversion, and visualization in CityJSON Ninja. The approach yielded volume RMSEs of 309.14 cm³ for the cuboid, 70.2541 cm³ for the pyramid, 352.0292 cm³ for the sphere, and 58.1517 cm³ for the torus, while downsampling and denoising reduced point counts substantially, for example from 1,554,781 to 5,151 and then to 5,124 for the cuboid, and wireframes rendered efficiently and validated in CityJSON for the cuboid, pyramid, and sphere although the torus inner hole was not represented correctly. These results indicate that a plane and edge driven pipeline works best for sharp-edged, convex objects, with thresholds around 35 degrees for the cuboid and 30 degrees for the pyramid balancing sensitivity and noise, whereas performance degrades on smooth or non-convex shapes because Triangle Normal Analysis over detects tessellation artifacts on spheres and Convex Hull or Alpha Shapes cannot recover the torus topology. We conclude that the method is effective for sharp-edged urban forms and CityJSON-ready, and we suggest extending it with curvature-aware or implicit-surface reconstruction for smooth or non-convex geometries, adding semantic attributes, auto-tuning thresholds, and evaluating on compound building assemblies representative of real urban scenes.

To exploit convexity, a non-convex object can be represented by a boolean combination of convex components. A convex decomposition method of polyhedral objects uses convex hulls and set difference operations. This decomposition, however, may not converge. In this article, we formalize this decomposition method and find local cause of non-convergence.

An image recovery algorithm based on alternating projections onto convex sets (POCS) is proposed for block-based coding. In the proposed algorithm, vectors are extracted from the surrounding neighborhood of a missing block to make a convex hull for the reference of recovery. The recovery vectors including both known and lost pixels are used for the projections to find optimal spectrum coefficients for the missing pixels. The convex hull and two other convex sets are formulated for application of POCS. Simulation results show that the proposed algorithm recovers spatial structure both at low and high frequencies.

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk in a sparse random environment is a nearest neighbor random walk on that jumps to the left or to the right with probability 1/2 from every point of and jumps to the right (left) with the random probability λ k+1 (1 - λ k+1) from the point S k , . Assuming that are independent copies of a random vector and the mean is finite (moderate sparsity) we obtain stable limit laws for X n , properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case (strong sparsity) will be analyzed in a forthcoming paper.