ambiguities in estimates of critical exponents for long range dependent processes abstract in this article we show that the hurst parameter and a well known wavelet estimator fail to distinguish on off models with substantially different distributions of on times we identify the problem as arising when standard power spectrum techniques which are well defined for commonly considered processes are applied to processes with extremely heavy tailed on times we provide an elementary mechanism whereby such extremely heavy tailed on time processes arise in a simple queuing model

large deviation analysis of subexponential waiting times in a processor sharing queue we investigate the distribution of the waiting timev in a stable m g 1 processor sharing queue with traffic intensityp x p b 1 p x 1 o 1 as x 8 furthermore we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail thane vx

tail asymptotics for the busy period in the gi g 1 queue we characterise the tail behaviour of the busy period distribution in thegi g 1 queue under the assumption that the tail of the service time distribution is of intermediate regular variation this extends a result of de meyer and teugels de meyer and teugels 1980 who treated them g 1 queue with a regularly varying service time distribution our method of proof is opposed to the one in de meyer and teugels 1980 probabilistic and reveals an insightful relationship between the busy period and the cycle maximum

tail approximations of integrals of gaussian random fields this paper develops asymptotic approximations of p tef t dt b as b for a homogeneous smooth gaussian random field f living on a compact d dimensional jordan measurable set t the integral of an exponent of a gaussian random field is an important random variable for many generic models in spatial point processes portfolio risk analysis asset pricing and so forth the analysis technique consists of two steps 1 evaluate the tail probability p ef t dt b over a small domain depending on b where mes 0 as b and mes is the lebesgue measure 2 with appropriately chosen we show that p tef t dt b 1 o 1 mes t mes 1 p ef t dt b

on the existence of moments for the busy period in a single server queue we obtain sufficient conditions for the finiteness of the moments of the single server queue busy period in discrete and continuous time in the cases of power and exponential moments the conditions found coincide with the known necessary ones

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

one sided local large deviation and renewal theorems in the case of infinite mean if s n n 0 is an integer valued random walk such that s n a n converges in distribution to a stable law of index 0 1 as n then gnedenko s local limit theorem provides a useful estimate for p s n r for values of r such that r a n is bounded the main point of this paper is to show that under certain circumstances there is another estimate which is valid when r a n in other words to establish a large deviation local limit theorem we also give an asymptotic bound for p s n r which is valid under weaker assumptions this last result is then used in establishing some local versions of generalized renewal theorems

direct meshless kernel techniques for time dependent equations we provide a class of positive definite kernels that allow to solve certain evolution equations of parabolic type for scattered initial data by kernel based interpolation or approximation avoiding time intergation completely some numerical illustrations are given

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

large deviations for solutions to stochastic recurrence equations under kesten s condition in this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation we assume kesten s condition acta math 131 1973 207 248 under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails while the noise sequence of the equations can have light tails the results of the paper are analogs to those obtained by a v nagaev theory probab appl 14 1969 51 64 193 208 and s v nagaev ann probab 7 1979 745 789 in the case of partial sums of i i d random variables in the latter case the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum for the solution to a stochastic recurrence equation the magnitude of the large deviation probabilities is again given by the tail of the maximum summand but the exact asymptotic tail behavior is also influenced by clusters of extreme values due to dependencies in the sequence we apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model

large deviations of square root insensitive random sums we provide a large deviation result for a random sum s nxn 0 x n wheren xis a renewal counting process and x n n 0are i i d random variables independent ofn x with a common distribution that belongs to a class of square root insensitive distributions asymptotically the tails of these distributions are heavier thane vxand have zero relative decrease in intervals of length v x hence square root insensitive using this result we derive the asymptotic characterization of the busy period distribution in the stable gi g 1 queue with square root insensitive service times this characterization further implies that the tail behavior of the busy period exhibits a functional change for distributions that are lighter thane vx

liouville metric of star scale invariant fields tails and weyl scaling we study the liouville metric associated to an approximation of a log correlated gaussian field with short range correlation we show that below a parameter gamma c 0 the left right length of rectangles for the riemannian metric e gamma phi 0 n ds 2 with various aspect ratio is concentrated with quasi lognormal tails that the renormalized metric is tight when gamma min gamma c 0 4 and that subsequential limits are consistent with the weyl scaling

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

response time distribution for a processor sharing system we consider the response time for jobs in a processor sharing system with a poisson arrival process and exponentially distributed required service time i e an m m 1 ps queue the response time tilde w is the sum of the delay and the required service time we derive an integral representation for the equilibrium response time distribution pr tilde w t and evaluate this integral numerically for several values of the traffic intensity rho t in the heavy traffic case when rho is close to 1 for rho 1 alpha 0 t for 0 75 leqq rho leqq 0 99 the relative error of this approximation is less than 0 62 alpha 2 in the range for which 1 geqq pr tilde w t geqq 0 01

large deviations for random walks under subexponentiality the big jump domain for a given one dimensional random walk sn with a subexponential step size distribution we present a unifying theory to study the sequences xn for which as n uniformly for x xn we also investigate the stronger local analogue our theory is self contained and fits well within classical results on domains of partial attraction and local limit theory when specialized to the most important subclasses of subexponential distributions that have been studied in the literature we reproduce known theorems and we supplement them with new results

local behaviour of first passage probabilities suppose that s is an asymptotically stable random walk with norming sequence cn and that tx is the time that s first enters x where x 0 the asymptotic behaviour of p t0 n has been described in a recent paper of vatutin and wachtel probab theory relat fields 143 177 217 2009 and here we build on that result to give three estimates for p tx n which hold uniformly as n in the regions x o cn x o cn and x cn respectively

notes on random walks in the cauchy domain of attraction the goal of these notes is to fill some gaps in the literature about random walks in the cauchy domain of attraction which has been in many cases left aside because of its additional technical difficulties we prove here several results in that case a fuk nagaev inequality and a local version of it a large deviation theorem two types of local large deviation theorems we also derive two important applications of these results a sharp estimate of the tail of the first ladder epochs and renewal theorems extending standard renewal theorems to the case of random walks most of our techniques carry through to the case of random walks in the domain of attraction of an alpha stable law with alpha in 0 2 so we also present results in that case since many of them seem to be missing in the literature

large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy tailed innovations in this paper we consider the stochastic recurrence equation y t a t y t 1 b t for an i i d sequence of pairs a t b t of nonnegative random variables where we assume that b t is regularly varying with index k 0 and ea k t x c 1 np y 1 x as x holds for some constant c 1 0 for k 1 we also study the large deviation probabilities p s n es n x x x n for some sequence x n whose growth depends on the heaviness of the tail of the distribution of y 1 we show that the relation p s n es n x c 2 np y 1 x holds uniformly for x x n and some constant c 2 0 then we apply the large deviation results to derive bounds for the ruin probability u p sup n 1 s n es n n u for any 0 we show that u c 3 up y 1 u 1 1 1 for some constant c 3 0 in contrast to the case of i i d regularly varying y t s when the above results hold with c 1 c 2 c 3 1 the constants c 1 c 2 and c 3 are different from 1

random walks with non convolution equivalent increments and their applications abstract this paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in the intermediate case where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class some of the obtained results can include the classical results for this the paper first introduces some new distribution classes using the transform of distributions and investigates their properties and relations with some other existing distribution classes based on the above results some equivalent conditions for the global and local asymptotics of the transform of the distribution of the supremum of the above random walk are given applying these results to risk theory and infinitely divisible laws the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non convolution equivalent claims and the global and local asymptotics of an infinitely divisible law with a non convolution equivalent levy measure

delay analysis of the max weight policy under heavy tailed traffic via fluid approximations we consider switched queueing networks with a mix of heavy tailed i e arrival processes with infinite variance and exponential type traffic and study the delay performance of the max weight policy known for its throughput optimality and asymptotic delay optimality properties our focus is on the impact of heavy tailed traffic on exponential type queues flows which may manifest itself in the form of subtle rate dependent phenomena we introduce a novel class of lyapunov functions piecewise linear and nonincreasing in the length of heavy tailed queues whose drift analysis provides exponentially decaying upper bounds to queue length tail asymptotics despite the presence of heavy tails to facilitate a drift analysis we employ fluid approximations proving that if a continuous and piecewise linear function is also a lyapunov function for the fluid model then the same function is a lyapunov function for the original stochastic system furthermore we use fluid approximations and renewal theory i

gaussian multiplicative chaos revisited in this article we extend the theory of multiplicative chaos for positive definite functions in rd of the form f x 2 ln t x g x where g is a continuous and bounded function the construction is simpler and more general than the one defined by kahane in 1985 as main application we give a rigorous mathematical meaning to the kolmogorov obukhov model of energy dissipation in a turbulent flow

subexponential distributions and characterizations of related classes letl 0 0 denote the class of distributionsf satisfying mathop lim limits x to infty bar f 2 x bar f x 2 mathop smallint limits 0 infty e gamma y df y infty i mathop lim limits x to infty bar f 2 x y bar f x e gamma y forall y in mathbb r ii the classesl for 0 are characterized by means of subexponential densities as an application we derive a result on the asymptotic behaviour of densities of random sums in particular for anm g 1 queue we relate the tail behaviour of the stationary waiting time density to that of the service time distribution

the topological structure of scaling limits of large planar maps we discuss scaling limits of large bipartite planar maps if p 2 is a fixed integer we consider for every integer n 2 a random planar map mn which is uniformly distributed over the set of all rooted 2p angulations with n faces then at least along a suitable subsequence the metric space consisting of the set of vertices of mn equipped with the graph distance rescaled by the factor n 1 4 converges in distribution as n towards a limiting random compact metric space in the sense of the gromov hausdorff distance we prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence and that this space can be obtained as the quotient of the continuum random tree for an equivalence relation which is defined from brownian labels attached to the vertices we also verify that the hausdorff dimension of the limit is almost surely equal to 4

weak lqg metrics and liouville first passage percolation for gamma in 0 2 we define a weak gamma liouville quantum gravity lqg metric to be a function h mapsto d h which takes in an instance of the planar gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms we show that these axioms are satisfied for any subsequential limits of liouville first passage percolation such subsequential limits were proven to exist by ding et al tightness of liouville first passage percolation for gamma in 0 2 2019 arxiv e prints arxiv 1904 08021 it is also known that these axioms are satisfied for the sqrt 8 3 lqg metric constructed by miller and sheffield 2013 2016 for any weak gamma lqg metric we obtain moment bounds for diameters of sets as well as point to point set to set and point to set distances we also show that any such metric is locally bi holder continuous with respect to the euclidean metric and compute the optimal holder exponents in both directions finally we show that lqg geodesics cannot spend a long time near a straight line or the boundary of a metric ball these results are used in subsequent work by gwynne and miller which proves that the weak gamma lqg metric is unique for each gamma in 0 2 which in turn gives the uniqueness of the subsequential limit of liouville first passage percolation however most of our results are new even in the special case when gamma sqrt 8 3

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

metastability in a condensing zero range process in the thermodynamic limit zero range processes with decreasing jump rates are known to exhibit condensation where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value we study such a process on a one dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed super critical particle density we show that the process exhibits metastability with respect to the condensate location i e the suitably accelerated process of the rescaled location converges to a limiting markov process on the unit torus this process has stationary independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice our result extends previous work for fixed lattices and diverging density in beltran and landim probab theory relat fields 152 3 4 781 807 2012 and we follow the martingale approach developed there and in subsequent publications besides additional technical difficulties in estimating error bounds for transition rates the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells corresponding to a typical set of configurations with a particular condensate location the total exit rates from individual wells turn out to diverge in the limit which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells which is of a general nature and can be adapted to other models

uniqueness and universality of the brownian map we consider a random planar map mn which is uniformly distributed over the class of all rooted q angulations with n faces we let mn be the vertex set of mn which is equipped with the graph distance dgr both when q 4 is an even integer and when q 3 there exists a positive constant cq such that the rescaled metric spaces mn cqn 1 4dgr converge in distribution in the gromov hausdorff sense toward a universal limit called the brownian map the particular case of triangulations solves a question of schramm

non universality for first passage percolation on the exponential of log correlated gaussian fields we consider first passage percolation fpp where the vertex weight is given by the exponential of two dimensional log correlated gaussian fields our work is motivated by understanding the discrete analog for the random metric associated with liouville quantum gravity lqg which roughly corresponds to the exponential of a two dimensional gaussian free field gff the particular focus of the present paper is an aspect of universality for such fpp among the family of log correlated gaussian fields more precisely we construct a family of log correlated gaussian fields and show that the fpp distance between two typically sampled vertices according to the lqg measure is n 1 o varepsilon where n is the side length of the box and varepsilon can be made arbitrarily small if we tune a certain parameter in our construction that is the exponents can be arbitrarily close to 1 combined with a recent work of the first author and goswami on an upper bound for this exponent when the underlying field is a gff our result implies that such exponent is not universal among the family of log correlated gaussian fields

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

large deviations of the front in a one dimensional model of x y 2x we investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction x y 2x on the integer lattice in which y particles do not move while x particles move as independent simple continuous time random walks of total jump rate 2 for a wide class of initial conditions we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval 0 where is the velocity of the front given by the law of large numbers we also give more precise estimates for the rate of decay of the slowdown probabilities our results indicate a gapless property of the generator of the process as seen from the front as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state

fast simulation of multi dimensional wave problems by the sparse scheme of the method of fundamental solutions in this work a meshless scheme is presented for the fast simulation of multi dimensional wave problems the present method is rather simple and straightforward the houbolt method is used to eliminate the time dependence of spatial variables then the original wave problem is converted into equivalent systems of modified helmholtz equations the sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables to demonstrate the effectiveness and simplicity of this new approach three numerical examples have been assessed with excellent performance

lognormal star scale invariant random measures in this article we consider the continuous analog of the celebrated mandelbrot star equation with lognormal weights mandelbrot introduced this equation to characterize the law of multiplicative cascades we show existence and uniqueness of measures satisfying the aforementioned continuous equation these measures fall under the scope of the gaussian multiplicative chaos theory developed by kahane in 1985 or possibly extensions of this theory as a by product we also obtain an explicit characterization of the covariance structure of these measures we also prove that qualitative properties such as long range independence or isotropy can be read off the equation

scaling limits of random planar maps with a unique large face we study random bipartite planar maps defined by assigning nonnegative weights to each face of a map we prove that for certain choices of weights a unique large face having degree proportional to the total number of edges in the maps appears when the maps are large it is furthermore shown that as the number of edges n of the planar maps goes to infinity the profile of distances to a marked vertex rescaled by n 1 2 is described by a brownian excursion the planar maps with the graph metric resealed by n 1 2 are then shown to converge in distribution toward aldous brownian tree in the gromov hausdorff topology in the proofs we rely on the bouttier di francesco guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large