scholarly journals The Asymptotic Behavior of the Average $L^p-$Discrepancies and a Randomized Discrepancy

10.37236/378 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Stefan Steinerberger
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Lower Bounds ◽  
Probabilistic Approach ◽  
Upper Bounds ◽  
Upper And Lower Bounds

This paper gives the limit of the average $L^p-$star and the average $L^p-$extreme discrepancy for $[0,1]^d$ and $0 < p < \infty$. This complements earlier results by Heinrich, Novak, Wasilkowski & Woźnia-kowski, Hinrichs & Novak and Gnewuch and proves that the hitherto best known upper bounds are optimal up to constants.We furthermore introduce a new discrepancy $D_{N}^{\mathbb{P}}$ by taking a probabilistic approach towards the extreme discrepancy $D_{N}$. We show that it can be interpreted as a centralized $L^1-$discrepancy $D_{N}^{(1)}$, provide upper and lower bounds and prove a limit theorem.

scholarly journals Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Huyuan Chen ◽  
Laurent Véron
Keyword(s):  
Asymptotic Behavior ◽  
Fourier Transform ◽  
Dirichlet Problem ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Laplacian Operator ◽  
Lower And Upper Bounds

Abstract We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢  in  ⁢ Ω , u = 0 ⁢  in  ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

On a Voronoi aggregative process related to a bivariate Poisson process

1996 ◽  
Vol 28 (04) ◽  
pp. 965-981 ◽  
Author(s):  
S. G. Foss ◽  
S. A. Zuyev
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Distribution Functions ◽  
Poisson Processes ◽  
Upper And Lower Bounds ◽  
Additive Functionals ◽  
Second Moments ◽  
Explicit Formulae ◽  
Bivariate Poisson Process ◽  
Sum Of Distances

We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.

Asymptotic behavior of a multiplexer fed by a long-range dependent process

1999 ◽  
Vol 36 (01) ◽  
pp. 105-118 ◽  
Author(s):  
Zhen Liu ◽  
Philippe Nain ◽  
Don Towsley ◽  
Zhi-Li Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Lognormal Distribution ◽  
Arrival Rate ◽  
Upper Bounds ◽  
Single Server ◽  
Service Capacity ◽  
Single Server Queue ◽  
Input Process ◽  
Server Queue

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ &lt; 1, where ρ is the arrival rate at the buffer.

scholarly journals ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

2012 ◽  
Vol 29 (3) ◽  
pp. 642-658 ◽  
Author(s):  
Benedikt M. Pötscher
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Integrated Process ◽  
Order Of Magnitude ◽  
Image Position ◽  
Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

Composite Thermoelectrics - Exact Results and Calculational Methods

1991 ◽  
Vol 234 ◽  
Author(s):  
David J. Bergman ◽  
Ohad Levy
Keyword(s):  
Quality Factor ◽  
Computer Simulations ◽  
Lower Bounds ◽  
Theoretical Study ◽  
Transport Coefficients ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Thermoelectric Transport

ABSTRACTA theoretical study of composite thermoelectric media has resulted in the development of a number of simple approximations, as well as some exact results. The latter include exact upper and lower bounds on the bulk effective thermoelectric transport coefficients of the composite and upper bounds on the bulk effective thermoelectric quality factor Ze. In particular, as a result of some exact theorems and computer simulations we conclude that Ze can never be greater than the largest value of Z in the different components that make up the composite.

scholarly journals Computational Approaches for Stochastic Shortest Path on Succinct MDPs

2018 ◽  
Author(s):  
Krishnendu Chatterjee ◽  
Hongfei Fu ◽  
Amir Goharshady ◽  
Nastaran Okati
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Shortest Path ◽  
Upper Bounds ◽  
Decision Processes ◽  
Upper And Lower Bounds ◽  
Computational Approaches ◽  
Stochastic Shortest Path ◽  
Markov Decision ◽  
Infinite State

We consider the stochastic shortest path (SSP) problem for succinct Markov decision processes (MDPs), where the MDP consists of a set of variables, and a set of nondeterministic rules that update the variables. First, we show that several examples from the AI literature can be modeled as succinct MDPs. Then we present computational approaches for upper and lower bounds for the SSP problem: (a) for computing upper bounds, our method is polynomial-time in the implicit description of the MDP; (b) for lower bounds, we present a polynomial-time (in the size of the implicit description) reduction to quadratic programming. Our approach is applicable even to infinite-state MDPs. Finally, we present experimental results to demonstrate the effectiveness of our approach on several classical examples from the AI literature.

scholarly journals Lower Bounds for Maximal Matchings and Maximal Independent Sets

2021 ◽  
Vol 68 (5) ◽  
pp. 1-30
Author(s):  
Alkida Balliu ◽  
Sebastian Brandt ◽  
Juho Hirvonen ◽  
Dennis Olivetti ◽  
Mikaël Rabie ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Maximal Independent Set ◽  
Maximal Matching ◽  
Maximal Independent Sets ◽  
Open Question

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of  n .

On a Voronoi aggregative process related to a bivariate Poisson process

1996 ◽  
Vol 28 (4) ◽  
pp. 965-981 ◽  
Author(s):  
S. G. Foss ◽  
S. A. Zuyev
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Distribution Functions ◽  
Poisson Processes ◽  
Upper And Lower Bounds ◽  
Additive Functionals ◽  
Second Moments ◽  
Explicit Formulae ◽  
Bivariate Poisson Process ◽  
Sum Of Distances

We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.

Theoretical and Experimental Analysis of Failure for the Hemisphere Punch Hydroforming Processes

1996 ◽  
Vol 118 (3) ◽  
pp. 434-438 ◽  
Author(s):  
Tze-Chi Hsu ◽  
Shian-Jiann Hsieh
Keyword(s):  
Experimental Data ◽  
Limit Theorem ◽  
Sheet Metal ◽  
Lower Bounds ◽  
Yield Criterion ◽  
Fluid Pressure ◽  
Upper And Lower Bounds ◽  
Premature Failure ◽  
Hydroforming Process ◽  
Theoretical Results

A limit theorem of plasticity has been developed to investigate the hemisphere punch hydroforming process. The limit theorem of plasticity is used to predict the upper and lower bounds of the permissible fluid pressure. Loci representing the critical fluid pressures which result in the rupture and wrinkling are presented. The property of the sheet metal is governed by Hill’s quadratic yield criterion with a power-law hardening for anisotropic material. The premature failure is avoidable if the fluid pressure path is restricted to travel only within the suggested bounds. The theoretical results which include the failure prediction and wrinkling distribution are verified by conducting a serial of hydroforming experiments. The experimental data agrees well with the computed results and demonstrates the technological usefulness of the results.

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