stochastic analysis of a modified first fit decreasing packing we make a stochastic analysis of a modified version mffd of first fit decreasing in which each bin is closed after it receives its first fallback item consider a probability measure on 0 1 and independent random variables x1 xn distributed according to let rn rx1 xn be the number of unit size bins that mffd needs to pack items of size x1 xn we prove that c limn ern n exists and that the random variable rn nc n converges in distribution the main tools are deterministic inequalities concerning mffd that might be of independent interest

synthesis of binary machines the problem of constructing a binary machine with the minimum number of stages generating a given binary sequence is addressed binary machines are a generalization of nonlinear feedback shift registers nlfsrs in which both connections feedback and feedforward are allowed and no chain connection between the register stages is required an algorithm for constructing a shortest binary machine generating a given periodic binary sequence is presented

extremes of volterra series expansions with heavy tailed innovations abstract in this paper we look at the extremal behavior of volterra series expansions generated by heavy tailed innovations via a point process formulation volterra series expansions are known to be the most general nonlinear representation for any stationary sequence the so called complete convergence theorem on point processes we prove enable us to give in detail the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics in particular we investigate the limiting distribution of the sample maxima and the corresponding extremal index the study of the extremal properties of finite order volterra series expansions would be highly valuable in understanding the extremal behavior of nonlinear processes as well as understanding of order identification and adequacy of volterra series when used as models in signal processing in fact such extremal properties may suggest a way of finding the order of a finite volterra expansions which is consistent with the nonlinearities of the observed process

periodic moving averages of random variables with regularly varying tails in this paper we establish the basic asymptotic theory for periodic moving averages of i i d random variables with regularly varying tails the moving average coefficients are allowed to vary according to the season a simple reformulation yields the corresponding results for moving averages of random vectors our main result is that when the underlying random variables have finite variance but infinite fourth moment the sample autocorrelations are asymptotically stable it is well known in this case that sample autocorrelations in the classical stationary moving average model are asymptotically normal

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

local explosion modelling by non causal process the noncausal autoregressive process with heavy tailed errors possesses a nonlinear causal dynamics which allows for unit root local explosion or asymmetric cycles often observed in economic and financial time series it provides a new model for multiple local explosions in a strictly stationary framework the causal predictive distribution displays surprising features such as the existence of higher moments than for the marginal distribution or the presence of a unit root in the cauchy case aggregating such models can yield complex dynamics with local and global explosion as well as variation in the rate of explosion the asymptotic behavior of a vector of sample autocorrelations is studied in a semi parametric noncausal ar 1 framework with pareto like tails and diagnostic tests are proposed empirical results based on the nasdaq composite price index are provided

optimal bin packing with items of random sizes consider a probability on 0 1 and i i d random variables x1 x2 xn distributed like let qn denote the optimal minimum number of unit size bins needed to pack items of size x1 x2 xn we characterize the class of which have the property that limn qn n e x1 a s or equivalently that the expected level of occupancy of bins converges to one

on the linear complexity of nonlinearly filtered pn sequences binary sequences of period 2 sup n 1 generated by a linear feedback shift register lfsr whose stages are filtered by a nonlinear function f are studied new iterative formulas are derived for the calculation of the linear complexity of the output sequences it is shown that these tools provide an efficient mechanism for controlling the linear complexity of the nonlinearly filtered maximal length sequences

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

multidimensional optimal bin packing with items of random size consider a probability measure on 0 1 d and independent random variables x 1 x n distributed according to let vn vnx 1 x n resp rn rnx 1 x n be the minimal number of unit size d dimensional bins needed to pack items of size x 1 x n in the vector packing resp rectangular packing problem for a large class of distributions we characterize v limn vn n and r limn rn n

tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling we study the partitioning problem consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized this is equivalent to minimizing the completion time of n jobs on m parallel identical processors we establish upper bounds on the deviation from optimality of two heuristics the well known lpt heuristic and the on line rlp heuristic these bounds serve to establish a probabilistic analysis of these heuristics for both of them the absolute deviation from optimality remains finite when the size of the list of numbers becomes infinite this is a stronger result than previous convergence theorems and it is valid whenever the processing times are iid random variables with finite mean and arbitrary distributions

on the extremes of a class of non linear processes with heavy tailed innovations the aim of this paper is to look at the limiting form of certain empirical point processes induced by a particular class of non linear processes generated by heavy tailed innovations such asymptotic results will be highly useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics in particular we investigate the maximum limiting distribution and its corresponding extremal index the results are applied to the study of the extremal properties of bilinear processes

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

probabilistic analysis of bin packing heuristics in this paper we present both a probabilistic and statistical analysis of several bin packing heuristics with mild conditions on the distribution of problem data we show that the ratio of the expected number of bins packed for these heuristics to the number of elements converges to some constant this constant can either be derived analytically or estimated through simulation we analyze the variance in performance of these heuristics and we further show that as the number of elements n increases the probability that the number of bins required for these heuristics exceeds any fixed percentage of their expected performance goes to zero

statistical estimation of multivariate ornstein uhlenbeck processes and applications to co integration abstract ornstein uhlenbeck models are continuous time processes which have broad applications in finance as e g volatility processes in stochastic volatility models or spread models in spread options and pairs trading the paper presents a least squares estimator for the model parameter in a multivariate ornstein uhlenbeck model driven by a multivariate regularly varying levy process with infinite variance we show that the estimator is consistent moreover we derive its asymptotic behavior and test statistics the results are compared to the finite variance case for the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems furthermore we embed this model in the setup of a co integrated model in continuous time

results on the nonlinear span of binary sequences the problem of finding the length of a shortest feedback shift register that generates a given finite length sequence is considered an efficient algorithm for the determination of the span is proposed that takes advantage of the special block structure of the associated system of linear equations the span distribution of finite length binary sequences is also studied

heavy tailed distributions correlations kurtosis and taylor s law of fluctuation scaling pillai meng pillai meng 2016 ann stat 44 20892097 p 2091 speculated that the dependence among random variables rvs can be overwhelmed by the heaviness of their marginal tails we giv

lower bounds and heuristic algorithms for the ki partitioning problem we consider the problem of partitioning a set of positive integers values into a given number of subsets each having an associated cardinality limit so that the maximum sum of values in a subset is minimized and the number of values in each subset does not exceed the corresponding limit the problem is related to scheduling and bin packing problems we give combinatorial lower bounds reduction criteria constructive heuristics a scatter search approach and a lower bound based on column generation the outcome of extensive computational experiments is presented

an analysis of the lpt algorithm for the max min and the min ratio partition problems given a set of positive numbers the max min partition problem asks for a k partition such that the minimum part is maximized the min ratio partition problem has the similar definition but the objective is to minimize the ratio of the maximum to the minimum parts in this paper we analyze the performances of the longest processing time lpt algorithm for the two problems we show that the tight bounds of the lpt are respectively 4k 2 3k 1 and 7 5

robust score and portmanteau tests of volatility spillover abstract this paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or garch type feedback the tests are couched in a general conditional heteroskedasticity framework with idiosyncratic shocks that are only required to have a finite variance if they are independent we negligibly trim test equations or components of the equations and construct heavy tail robust score and portmanteau statistics trimming is either simple based on an indicator function or smoothed in particular we develop the tail trimmed sample correlation coefficient for robust inference and prove that its gaussian limit under the null hypothesis of no spillover has the same standardization irrespective of tail thickness further if spillover occurs within a specified horizon our test statistics obtain power of one asymptotically we discuss the choice of trimming portion including a smoothed p value over a window of extreme observations a monte carlo study shows our tests provide significant improvements over extant garch based tests of spillover and we apply the tests to financial returns data finally based on ideas in patton 2011 we construct a heavy tail robust forecast improvement statistic which allows us to demonstrate that our spillover test can be used as a model specification pre test to improve volatility forecasting

limit theory for the sample covariance and correlation functions of moving averages abstract document describes a moving average process which have regularly varying tail probabilities with index alpha 0 the limit distribution of the sample covariance function is derived in the case that the process has a finite variance but an infinite variance but an infinite fourth moment furthermore in the infinite variance case 0 alpha 2 the sample correlation function is shown to converge in distribution to the ratio of two independent stable random variables with indices alpha and alpha 2 respectively this result immediately gives the limit distribution for the least squares estimates of the parameters in an autoregressive process author

the asymptotic null distribution of the box pierce q statistic for random variables with infinite variance an application to german stock returns abstract the asymptotic null distribution of the box pierce q statistic is considered for random variables with probability distribution in the domain of attraction of a stable law with characteristic exponent 1

the limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes we investigate the maximum increment of a random walk with heavy tailed jump size distribution here heavy tailedness is understood as regular variation of the finite dimensional distributions the jump sizes constitute a strictly stationary sequence using a continuous mapping argument acting on the point processes of the normalized jump sizes we prove that the maximum increment of the random walk converges in distribution to a frechet distributed random variable

limit theory for moving averages of random variables with regularly varying tail probabilities let z k infty k infty be iid where the z k s have regularly varying tail probabilities under mild conditions on a real sequence c j j geq 0 the stationary process x n sum infty j 0 c jz n j n geq 1 exists a point process based on x n converges weakly and from this a host of weak limit results for functionals of x n ensue we study sums extremes excedences and first passages as well as behavior of sample covariance functions

constructing periodic binary sequences with maximum nonlinear span the nonlinear span of a sequence x is defined as the length of the shortest feedback shift register that generates x in this correspondence we solve the problem of designing periodic binary sequences with given linear complexity and maximal possible nonlinear span we concentrate on the case of sequences with period n 2n 1 and we introduce two construction methods

some distributions that allow perfect packing a probability distribution on 0 1 allows perfect packing if n items of size x 1 x n independent and identically distributed according to can be packed in unit size bins in such a way that the expected wasted space is o n a large class of distributions that allow perfect packing is exhibited as a corollary the intervals a b for which the uniform distribution on a b allows perfect packing are determined

a note on expected makespans for largest first sequences of independent tasks on two processors we consider the scheduling of sets of n simultaneously available tasks on two identical processors the task execution times are assumed to be independent samples from the uniform distribution on 0 1 we analyze the expected makespan schedule length performance of two well known largest task first nonpreemptive approximation rules with the largest first lf rule tasks are assigned in nonincreasing order of execution time to the processors as they become available with the restricted largest first rlf rule tasks are assigned in pairs one to a processor the larger task of a pair is the first to be scheduled if n is odd the last task is simply assigned to the earner finishing processor in the schedule for the first n 1 tasks we prove that the expected makespan for lf is bounded by frac n 4 frac e 2n 1 and for rlf it is precisely frac n 4 frac 1 2n 1 from these results simple bounds are derived on the ratio of the expected performance of lf and rlf to the expected performance of an optimization rule finally a lower bound on expected lf makespans is shown to be frac n 4 frac 1 4n 1

convolution tails product tails and domains of attraction a distribution function is said to have an exponential tail f t f t if e uf t u is asymptotically equivalent to f t t t for all u in this case f lnt is regularly varying for two such distributions f and g the convolution h f g also has an exponential tail we investigate the relationship between h and its components f and g providing conditions for lim h f to exist in addition we are able to describe the asymptotic nature of h when the limit is infinite for many cases this corresponds to determining both the domain of attraction and the norming constants for the product of independent variables whose distributions have regularly varying tails

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

the structure of the class of subexponential distributions letx 1 x 2 x n be a sequence of positive independent identically distributed random variables with the same distribution function d f f and denote byx 1 n x 2 n x n n the order statistics of the sample we characterize the class of d f f for which p x 1 n x 2 n ldots x n i n x sim p x n i n x as x to infty for fixedn andi i n 1 and we show that it is independent ofn this leads to the genesis of a new class of d f l i we show that the sequence l i 0 is strictly decreasing and we illustrate how the classesl i determine the probabilistic structure of the classl of subexponential distributions

sign tests for long memory time series abstract this paper proposes sign based tests for simple and composite hypotheses on the long memory parameter of a time series process the tests allow for nonstationary hypothesis such as unit root as well as for stationary hypotheses such as weak dependence or no integration the proposed generalized lagrange multiplier sign tests for simple hypotheses on the long memory parameter are exact and locally optimal among those in their class we also propose tests for composite hypotheses on the parameters of arfima p d q processes the resulting tests statistics have a standard normal limiting distribution under the null hypothesis

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

on a closure property of convolution equivalent class of distributions abstract in this note we reconsider the closure problem in the convolution equivalent class of distributions s 0 we show that if f g l then inclusions of f g fg and p f 1 p g for all some p 0 1 into the class s are equivalent also we provide an example which shows that p f 1 p g can be in s even if f and g are not in s

average case analysis of cutting and packing in two dimensions abstract stock cutting and packing in two dimensions is a source of problems of great practical significance much has been written on the subject including several surveys but the focus of the mathematically oriented research has been largely on combinatorial optimization and worst case analysis recently substantial progress has been made in the average case analysis of algorithms for two dimensional packing this paper reviews selected illustrative results in this area of research and discusses desirable extensions and open problems several cutting and packing algorithms are defined and the results of their probabilistic analysis reported from the presentation of this material it is clear that the field is still in early stages of development the algorithms and probability models tend to be simplistic and stimates of performance are far more common that exact measures in spite of these limitations valuable and in some cases unexpected insights have emerged

stable limits for sums of dependent infinite variance random variables the aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution conditions for this convergence to hold are known in the literature however most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process in this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence we will apply our results to some standard time series models including the garch 1 1 process and its squares the stochastic volatility models and solutions to stochastic recurrence equations

the differencing algorithm ldm for partitioning a proof of a conjecture of karmarkar and karp the algorithm ldm largest differencing method divides a list of n random items into two blocks the parameter of interest is the expected difference between the two block sums it is shown that if the items are i i d and uniform then the rate of convergence of this parameter to zero is n log n an algorithm for balanced partitioning is constructed with the same rate of convergence to zero

martingale inequalities and np complete problems we use martingale inequalities in the probabilistic analysis of stochastic models of np complete problems we give examples of applications to the bin packing and traveling salesman problems in both cases we obtain sharp bounds on pr un eun t where un is the objective function value of the problem with n random data

on the lagarias odlyzko algorithm for the subset sum problem we give a simple analysis of an algorithm for solving subset sum problems proposed by lagarias and odlyzko 2

the durbin watson ratio under infinite variance errors abstract this paper studies the properties of the von neumann ratio for time series with infinite variance our asymptotics cover the null of iid variates and general moving average ma alternatives regression residuals are also considered in the static regression model the durbin watson statistic has the same limit distribution as the von neumann ratio in dynamic models the results are more complex in finite variance models our results specialize to those of the durbin h statistic and equivalent lm test asymptotics some monte carlo results are reported illustrating the effects of infinite variance errors and regressors in finite samples

stable limits for partial sums of dependent random variables let x n be a stationary sequence of random variables whose marginal distribution f belongs to a stable domain of attraction with index alpha 0 alpha 2 under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences nonnormal stable limits are established for the normalized partial sums the method of proof relies heavily on a recent paper by lepage woodroofe and zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear also an example of a process which is an instantaneous function of a stationary gaussian process with covariance function r n behaving like r n log n rightarrow 0 as n rightarrow infty is shown to satisfy these conditions

on minimum period of nonlinear feedback shift registers in grain like structure grain is one of three hardware oriented finalists of the estream project a nonlinear feedback shift register nfsr in grain like structure is a cascade connection of a linear feedback shift register lfsr into an nfsr in which the characteristic polynomial of the lfsr is primitive and the feedback function of the nfsr is nonsingular in 2011 hu and gong pointed out that the period of the sequence generated by an nfsr in grain like structure is a multiple of the period of the sequence generated by its lfsr if the initial state of the lfsr is nonzero meanwhile they proposed an open problem for fixed feedback functions of an nfsr and an lfsr determine whether the sequences generated by the nfsr in grain like structure can achieve the minimum period i e the period of the lfsr when the initial state of the lfsr is nonzero and if they can achieve provide at least one pair of the initial states of the nfsr and lfsr clearly from a security point of view it is not preferable if the sequences generated by an nfsr in grain like structure achieve the minimum period this paper converts the open problem into a problem of solving an integer equation with respect to two unknown integers that uniquely correspond to the initial states of the nfsr and lfsr by viewing the nfsr as a boolean control network based on the integer equation this paper shows that for any given initial state of an n stage nfsr and any given nonzero initial state of an m stage lfsr the probability that the sequence generated by the nfsr in grain like structure achieves the minimum period 2 m 1 is at most 2 n this implies that the probability of the cascade connection used in grain achieving the minimum period is very small

consistent order determination for processes with infinite variance summary order determination for autoregressive processes when the residuals have a distribution belonging to the domain of attraction of an infinite variance stable law is examined the use of the fpe criterion of bhansali and downham 1977 is shown to provide a consistent estimator of the order with any a 0 order determination for moving average processes is considered a procedure based on autoregressive estimates of the inverse correlations see bhansali 1983 is shown to provide a consistent estimator of the order consistency of the estimated autoregressive and moving average parameters of an arbitrary order is established the question of consistent discrimination between autoregressive and moving average models is also considered the finite sample behaviour of the discrimination procedure is investigated by means of a simulation study

average performance of greedy heuristics for the integer knapsack problem abstract this paper derives a lower bound on the average performance of a total value greedy heuristic for the integer knapsack problem this heuristic selects items in order of their maximum possible contribution to the solution value at each stage we show that as for the worst case bound the average performance bound for the total value heuristic dominates the corresponding bound for the density ordered greedy heuristic

average case performance analysis of an approximation algorithm for maximum subset sum using recurrence relations given a positive integer m and a set s x1 x2 xn of positive integers the maximum subset sum problem is to find a subset s of s such that x s x is maximized under the constraint that the summation is no larger than m in addition the cardinality of s is also a maximum among all subsets of s which achieve the maximum subset sum this problem is known to be np hard we analyze the average case performance of a simple on line approximation algorithm assuming that all integers in s are independent and have the same probability distribution we develop a general methodology i e using recurrence relations to evaluate the expected values of the content and the cardinality of s for any distribution the maximum subset sum problem has important applications especially in static job scheduling in multiprogrammed parallel systems the algorithm studied can also be easily adapted for dynamic job scheduling in such systems

asymptotic methods in the probabilistic analysis of sequencing and packing heuristics recently there has been considerable interest in the average case performance of heuristics this paper pursues that interest where it concerns sequencing and packing problems in particular we survey the methods that have been used to obtain formal probabilistic analyses of heuristics for makespan scheduling and one dimensional bin packing in so doing we present many of the key results in these research areas

moment condition tests for heavy tailed time series abstract we develop an asymptotically chi squared statistic for testing moment conditions e m t 0 0 where m t 0 may be weakly dependent scalar components of m t 0 may have an infinite variance and e m t need not exist for any under the alternative score tests are a natural application and in general a variety of tests can be heavy tail robustified by our method including white noise garch affects omitted variables distribution functional form causation volatility spillover and over identification the test statistic is derived from a tail trimmed sample version of the moments evaluated at a consistent plug in t for 0 depending on the test in question and heaviness of tails t may be any consistent estimator including sub t 1 2 convergent and or asymptotically non gaussian ones since t can be assured not to affect the test statistic asymptotically we adapt bootstrap p value occupation time and covariance determinant methods for selecting the trimming fractile in any sample and apply our statistic to tests of white noise omitted variables and volatility spillover we find it obtains sharp empirical size and strong power while conventional tests exhibit size distortions

optimal bin packing with items of random sizes iii consider a probability measure mu on 0 1 and independent random variables x 1 cdots x n distributed according to mu let q n q n x 1 cdots x n be the minimum number of unit size bins needed to pack items of size x 1 cdots x n let c mu lim n to infty e q n n in this paper it is proved that the random variable q n nc mu sqrt n converges in distribution the limit is identified as a distribution of the supremum of a certain gaussian process canonically attached to mu

a large deviations approach to limit theory for heavy tailed time series in this paper we propagate a large deviations approach for proving limit theory for generally multivariate time series with heavy tails we make this notion precise by introducing regularly varying time series we provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin we study a variety of such functionals including large deviations of random walks their suprema the ruin functional and further derive weak limit theory for maxima point processes cluster functionals and the tail empirical process one of the main results of this paper concerns bounds for the ruin probability in various heavy tailed models including garch stochastic volatility models and solutions to stochastic recurrence equations

random walk and the area below its path a simple random walk with reflected origin is considered the walk starts at the origin and it must return to the origin at time 2n we show that the expected area below the path of this walk is n22n 1 cn2n if however the walk is required to return to the origin for the first time at time 2n then the expected area below the path of this bernoulli excursion is 2n 1 22n 1 cn2n we also show that if v1 vn is the order statistics based on a sample of size n from a uniform distribution over 0 1 and that if u1 un is another independent set of order statistics from the same distribution then e i 1n vi ui n22n 1 2n 1 cn2n we use this result to find an average case performance of the earliest due date edd heuristic for one machine scheduling problem with earliness and tardiness penalties we also apply some of the results to larson s queue inference engine 1990

the wiener condition and the conjectures of embrechts and goldie we show that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots it gives a negative answer to the classical conjectures of embrechts and goldie moreover we establish two sufficient conditions in order that the class of convolution equivalent distributions is closed under convolution roots

a robust goodness of fit test for generalized autoregressive conditional heteroscedastic models summary the estimation of time series models with heavy tailed innovations has been widely discussed but corresponding goodness of fit tests have attracted less attention primarily because the autocorrelation function commonly used in constructing goodness of fit tests necessarily imposes certain moment conditions on the innovations as a bounded random variable has finite moments of all orders we address the problem by first transforming the residuals with a bounded function more specifically we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model with the corresponding residual empirical distribution function naturally employed as the transformation a robust goodness of fit test is then constructed the asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived and monte carlo experiments are conducted to examine finite sample properties the proposed test is shown to be more powerful than existing tests when the innovations are heavy tailed

the full solution of the convolution closure problem for convolution equivalent distributions assume f and g are distribution functions on 0 co having unbounded support and let h f g be their convolution for n e n denote by f the n fold convolution of f furthermore we shall denote distribution tails byf i f r l f g andp l f forallnen the moment generating function of f is defined by if f has a finite expectation it will be denoted by p f if f and g are absolutely continuous with densities f and g then f g is absolutely continuous with density f g for n e n denote by f the density of f the aim of this paper is to prove that none of the following classes is closed under convolution

the sample autocorrelations of heavy tailed processes with applications to arch we study the sample acvf and acf of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying this includes linear and bilinear processes with regularly varying noise and arch processes their squares and absolute values we show that the distributional limits of the sample acf can be random provided that the variance of the marginal distribution is infinite and the process is nonlinear this is in contrast to infinite variance linear processes if the process has a finite second but infinite fourth moment then the sample acp is consistent with scaling rates that grow at a slower rate than the standard root n consequently asymptotic confidence bands are wider than those constructed in the classical theory we demonstrate the theory in full detail far an arch 1 process

on the nonlinear complexity and lempel ziv complexity of finite length sequences the nonlinear complexity of binary sequences and its connections with lempel ziv complexity is studied in this paper a new recursive algorithm is presented which produces the minimal nonlinear feedback shift register of a given binary sequence moreover it is shown that the eigenvalue profile of a sequence uniquely determines its nonlinear complexity profile thus establishing a connection between lempel ziv complexity and nonlinear complexity furthermore a lower bound for the lempel ziv compression ratio of a given sequence is proved that depends on its nonlinear complexity

self weighted least absolute deviation estimation for infinite variance autoregressive models summary how to undertake statistical inference for infinite variance autoregressive models has been a long standing open problem to solve this problem we propose a self weighted least absolute deviation estimator and show that this estimator is asymptotically normal if the density of errors and its derivative are uniformly bounded furthermore a wald test statistic is developed for the linear restriction on the parameters and it is shown to have non trivial local power simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given the results are entirely different from other published results and should provide new insights for future research on heavy tailed time series

estimating the stable index alpha in order to measure tail thickness a critique stable laws are often fit to outlier prone data and if the index alpha is estimated to be much less than two then the normal law is rejected in favor of an infinite variance stable law this paper derives the theoretical properties of such a procedure when the true distribution is stable the distribution of the m l e of alpha is non regular if alpha 2 when the true distribution is not stable the estimate of alpha is not a robust measure of the rate of decrease of the tail probabilities a more robust procedure is developed and a statistic for describing and comparing the tail shapes of arbitrary samples is proposed

safer parameters for the chor rivest cryptosystem vaudenay s cryptanalysis against chor rivest cryptosystem is applicable when the parameters p and h originally proposed by the authors are used nevertheless if p and h are both prime integers then vaudenay s attack is not applicable in this work a choice of these parameters resistant to the existing cryptanalytic attacks is presented the parameters are determined in a suitable range guaranteeing its security and the computational feasibility of implementation regrettably the obtained parameters are scarce in practice