moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

existence and regularity for an energy maximization problem in two dimensions we consider the variational problem of maximizing the weighted equilibrium green s energy of a distribution of charges free to move in a subset of the upper half plane under a particular external field we show that this problem admits a solution and that under some conditions this solution is an s curve in the sense of gonchar rakhmanov the above problem appears in the theory of the semiclassical limit of the integrable focusing nonlinear schrodinger equation in particular its solution provides a justification of a crucial step in the asymptotic theory of nonlinear steepest descent for the inverse scattering problem of the associated linear non self adjoint zakharov shabat operator and the equivalent riemann hilbert factorization problem

partition functions and the continuum limit in penner matrix models we present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of penner random matrix models to obtain their explicit forms in the exactly solvable cases and to determine the coefficients of their perturbative expansions in the continuum limit the method relies on identities satisfied by the resolvent of the jacobi matrix in the three term recursion relation of the associated families of orthogonal polynomials these identities lead to a convenient formulation of the string equations as an application we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double penner models can be written as a sum of gaussian contributions plus linear terms to illustrate the one cut case we discuss the linear double and cubic penner models and for the two cut case we discuss theoretically and numerically the existence of a double branch structure of the free energy for the gaussian penner model

random matrix model with external source and a constrained vector equilibrium problem we consider the random matrix model with external source in the case where the potential v x is an even polynomial and the external source has two eigenvalues a of equal multiplicity we show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures 1 2 that solve a constrained vector equilibrium problem the proof is based on the steepest descent analysis of the associated riemann hilbert problem for multiple orthogonal polynomials we illustrate our results in detail for the case of a quartic double well potential v x 1 4x 4 t 2x 2 we are able to determine the precise location of the phase transitions in the ta plane where either the constraint becomes active or the two intervals in the support come together or both 2010 wiley periodicals inc

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

the supercritical regime in the normal matrix model with cubic potential abstract the normal matrix model with a cubic potential is ill defined and it develops a critical behavior in finite time we follow the approach of bleher and kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a hermitian form this reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime namely that the zeros of the polynomials tend to a number of segments the motherbody inside a domain the droplet that attracts the eigenvalues in the normal matrix model in the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the boutroux condition the boutroux condition determines a system of contours 1 consisting of the motherbody and whiskers sticking out of the domain we find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain in the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 3 matrix valued riemann hilbert problem it follows that the zeros of the orthogonal polynomials tend to 1 with the exception of at most three spurious zeros

strong asymptotics for cauchy biorthogonal polynomials with application to the cauchy two matrix model we apply the nonlinear steepest descent method to a class of 3 3 riemann hilbert problems introduced in connection with the cauchy two matrix random model the general case of two equilibrium measures supported on an arbitrary number of intervals is considered in this case we solve the riemann hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three sheeted riemann surface of arbitrary genus and establish strong asymptotic results for the cauchy biorthogonal polynomials

critical measures quadratic differentials and weak limits of zeros of stieltjes polynomials we investigate the asymptotic zero distribution of heine stieltjes polynomials polynomial solutions of second order differential equations with complex polynomial coefficients in the case when all zeros of the leading coefficients are all real zeros of the heine stieltjes polynomials were interpreted by stieltjes as discrete distributions minimizing an energy functional in a general complex situation one deals instead with a critical point of the energy we introduce the notion of discrete and continuous critical measures saddle points of the weighted logarithmic energy on the plane and prove that a weak limit of a sequence of discrete critical measures is a continuous critical measure thus the limit zero distributions of the heine stieltjes polynomials are given by continuous critical measures we give a detailed description of such measures showing their connections with quadratic differentials in doing that we obtain some results on the global structure of rational quadratic differentials on the riemann sphere that have an independent interest

nonparametric policy analysis abstract this article considers the problem of predicting the mean effect of a change in the distribution of certain policy related variables on a dependent variable y this is conventionally done using a parametric model if however the conditional expectation of y given policy and nonpolicy variables x is unaltered by the policy intervention and if the support of x after the policy intervention lies within the support of x before the intervention then this analysis can be performed nonparametrically the proposed nonparametric estimator is developed for the model yj g xj a dj uj where g is a continuous unknown function of the continuous variables x dj is an m vector of dummy variables a is an m vector of unknown parameters representing fixed cell specific effects and uj is an error term with e uj xj dj 0 the estimand is b eg xj eg xj where x and x respectively denote the values of x before and after the policy intervention a nonparametric estimator bn is propo

cross validation in density estimation considerable interest has been shown recently in the use of cross validatory procedures much of this attention is due to the efforts of stone 1974a b 1977a b who has provided a general discussion of the method of cross validatory choice dawid 1974 suggested that in certain circumstances cross validation might lead to inconsistency and some of these fears were confirmed by stone 1977b in the present paper we examine a popular cross validatory argument for selecting the smoothing parameter or window size in the nonparametric estimation of a density we derive results describing the order of magnitude of the cross validatory window and show that the resulting estimator will perform suboptimally our results are confirmed by monte carlo trials alternative techniques such as that proposed by silverman 1978a seem preferable habbema hermans van den broek 1974 suggested a method for discriminant analysis based on nonparametric density estimation they considered kernel type estimates and in the univariate case these usually have the form

consistency of cross validation for comparing regression procedures theoretical developments on cross validation cv have mainly focused on selecting one among a list of finite dimensional models e g subset or order selection in linear regression or selecting a smoothing parameter e g bandwidth for kernel smoothing however little is known about consistency of cross validation when applied to compare between parametric and nonparametric methods or within nonparametric methods we show that under some conditions with an appropriate choice of data splitting ratio cross validation is consistent in the sense of selecting the better procedure with probability approaching 1 our results reveal interesting behavior of cross validation when comparing two models procedures converging at the same nonparametric rate in contrast to the parametric case it turns out that the proportion of data used for evaluation in cv does not need to be dominating in size furthermore it can even be of a smaller order than the proportion for estimation while not affecting the consistency property

orthogonal polynomials in the normal matrix model with a cubic potential abstract we consider the normal matrix model with a cubic potential the model is ill defined and in order to regularize it elbau and felder introduced a model with a cut off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut off region on the complex plane in the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane without any cut off which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of elbau and felder the main goal of this paper is to develop the riemann hilbert rh approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity as the first step in the rh approach we introduce an auxiliary vector equilibrium problem for a pair of measures 1 2 on the complex plane we then formulate a 3 3 matrix valued rh problem for the orthogonal polynomials in hand and we apply the nonlinear steepest descent method of deift zhou to the asymptotic analysis of the rh problem the central steps in our study are a sequence of transformations of the rh problem based on the equilibrium vector measure 1 2 and the construction of a global parametrix the main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane we prove that the distribution of zeros of the orthogonal polynomials converges to the measure 1 the first component of the equilibrium measure we also obtain analytical results for the measure 1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve

critical measures for vector energy global structure of trajectories of quadratic differentials abstract saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector valued critical measures a notion that finds a natural motivation in several branches of analysis we study in depth the case of measures 1 2 3 when the mutual interaction comprises both attracting and repelling forces for arbitrary vector polynomial external fields we establish general structural results about critical measures such as their characterization in terms of an algebraic equation solved by an appropriate combination of their cauchy transforms and the symmetry properties or the s properties exhibited by such measures in consequence we conclude that vector valued critical measures are supported on a finite number of analytic arcs that are trajectories of a quadratic differential globally defined on a three sheeted riemann surface the complete description of the so called critical graph for such a differential is the key to the construction of the critical measures we illustrate these connections studying in depth a one parameter family of critical measures under the action of a cubic external field this choice is motivated by the asymptotic analysis of a family of non hermitian multiple orthogonal polynomials that is subject of a forthcoming paper here we compute explicitly the riemann surface and the corresponding quadratic differential and analyze the dynamics of its critical graph as a function of the parameter giving a detailed description of the occurring phase transitions when projected back to the complex plane this construction gives us the complete family of vector valued critical measures that in this context turn out to be vector valued equilibrium measures

hyperelliptic uniformization of algebraic curves of the third order an algebraic function of the third order plays an important role in the problem of asymptotics of hermite pade approximants for two analytic functions with branch points this algebraic function appears as the cauchy transform of the limiting measure of the asymptotic distribution of the poles of the approximants in many cases this algebraic function can be determined by using the given position of the branch points of the functions which are approximated and by the condition that its abelian integral has purely imaginary periods in the present paper we obtain a hyperelliptic uniformization of this algebraic function in the case when each approximated function has only two branch points the genus of this function can be equal to 0 1 elliptic case or 2 ultra elliptic case we use this uniformization to parametrize the elliptic case this parametrization allows us to obtain a numerical procedure for finding this elliptic curve and as a result we can describe the limiting measure of the distribution of the poles of the approximants

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

power laws in cities population financial markets and internet sites scaling in systems with a variable number of components we study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules we compare them with respect to the emergence of power laws in the size distribution of their components we show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components in particular a growing system with a xed number of components and a xed smallest component size does not converge to a power law we present a new model with variable number of components that converges to a power law for a very wide range of parameters in a very large subset of this range one obtains for the exponent the special value 1 specic for the city populations distribution we discuss the conditions in which can take dierent values in the case of the stock market the distribution of the investors wealth is related to the ratio between the new capital invested in stock and the rate of increase of the stock index c 2000 elsevier science b v all rights reserved

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

modulated branching processes origins of power laws and queueing duality power law distributions have been repeatedly observed in a wide variety of socioeconomic biological and technological areas in many of the observations e g city populations and sizes of living organisms the objects of interest evolve because of the replication of their many independent components e g births and deaths of individuals and replications of cells furthermore the rates of replications are often controlled by exogenous parameters causing periods of expansion and contraction e g baby booms and busts economic booms and recessions etc in addition the sizes of these objects often have reflective lower boundaries e g cities do not fall below a certain size low income individuals are subsidized by the government companies are protected by bankruptcy laws etc hence it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations indeed our main results show that the proposed mathematical models result in power law distributions under quite general polynomial gartner ellis conditions the generality of which could explain the ubiquitous nature of power law distributions in addition on a logarithmic scale we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones the latter as recognized by goldie goldie c m 1991 implicit renewal theory and tails of solutions of random equations ann appl probab 1 1 126 166 is known to be dual to queueing additive processes we emphasize this duality further in the generality of stationary and ergodic processes

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

quadratic differentials and asymptotics of laguerre polynomials with varying complex parameters abstract in this paper we study the asymptotics as n of the sequences of laguerre polynomials with varying complex parameters depending on the degree n more precisely we assume that n n a n and lim n a n a c this study has been carried out previously only for n r but complex values of a introduce an asymmetry that makes the problem more difficult the main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane which may have an independent interest while the weak asymptotics is obtained by reduction to the theorem of gonchar rakhmanov stahl the strong asymptotic results are derived via the non commutative steepest descent analysis based on the riemann hilbert characterization of the laguerre polynomials

critical measures for vector energy asymptotics of non diagonal multiple orthogonal polynomials for a cubic weight abstract we consider the type i multiple orthogonal polynomials mops a n m b n m deg a n m n 1 deg b n m m 1 and type ii mops p n m deg p n m n m satisfying non hermitian orthogonality with respect to the weight e z 3 on two unbounded contours 1 and 2 on c with in the case of type ii mops n conditions on 1 and m on 2 under the assumption that n m n n m 0 1 we find the detailed rescaled asymptotics of a n m b n m and p n m on c and describe the phase transitions of this limit behavior as a function of this description is given in terms of the vector critical measure 1 2 3 the saddle point of an energy functional comprising both attracting and repelling forces this critical measure is characterized by a cubic equation spectral curve and its components j live on trajectories of a canonical quadratic differential on the riemann surface of this equation the structure of these trajectories and their deformations as functions of was object of study in our previous paper 42 and some of the present results strongly rely on the analysis carried out there we conclude that the asymptotic zero distribution of the polynomials a n m and p n m are given by appropriate combinations of the components j of the vector critical measure however in the case of the zeros of b n m the behavior is totally different and can be described in terms of the balayage of the signed measure 2 3 onto certain curves on the plane these curves are constructed with the aid of the canonical quadratic differential and their topology has three very distinct characters depending on the value of and are obtained from the critical graph of once the trajectories and vector critical measures are studied the main asymptotic technical tool is the deift zhou nonlinear steepest descent analysis of a 3 3 matrix valued riemann hilbert problem characterizing both a n m b n m and p n m which allows us to obtain the limit behavior of both types of mops simultaneously we illustrate our findings with results of several numerical experiments explain the computational methodology and formulate some conjectures and empirical observations based on these experiments

general large deviations of longest gaps in homogeneous poisson processes abstract let n t t 0 be a homogeneous poisson process with unit intensity the longest gap l t of this process is the maximal time subinterval between epochs of arrival times up to time t and it finds applications in the theory of reliability it has been known since the 1980s that the longest gap obeys the laws of large numbers l t ln t 1 as t almost surely however large deviation probabilities of the longest gap have not been well developed yet and in this paper we derive such large deviations with a very general speed the main tool is to establish a global estimation for the distribution function of the longest gap which is of independent interest

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

from gene families and genera to incomes and internet file sizes why power laws are so common in nature we present a simple explanation for the occurrence of power law tails in statistical distributions by showing that if stochastic processes with exponential growth in expectation are killed or observed randomly the distribution of the killed or observed state exhibits power law behavior in one or both tails this simple mechanism can explain power law tails in the distributions of the sizes of incomes cities internet files biological taxa and in gene family and protein family frequencies

the pareto law of incomes an explanation and an extension abstract a stochastic model for the generation of observed income distributions is used to provide an explanation for the pareto law of incomes analysis of the model also yields a prediction of paretian power law behaviour in the lower tail of the distribution and this is shown to occur in a number of empirical distributions a tractable four parameter distribution is derived and shown to fit extremely well to a number of different empirical income distributions

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

hammerstein system identification with the nearest neighbor algorithm the nonlinear characteristic in a hammerstein system i e a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade is recovered with the nonparametric nearest neighbor regression estimate the a priori information is nonparametric both the nonlinear characteristic and the impulse response are completely unknown and can be of any form local and global properties of the estimate are examined whatever the probability density of the input signal the estimate converges at every continuity point of the characteristic as well as in the global sense we derive the asymptotic bias and variance of the proposed estimate as a result the optimal rate of convergence is established that additionally is independent of the shape of the input density results of numerical simulations are also presented

asymptotic behavior of total times for jobs that must start over if a failure occurs many processes must complete in the presence of failures different systems respond to task failure in different ways the system may resume a failed task from the failure point or a saved checkpoint shortly before the failure point it may give up on the task and select a replacement task from the ready queue or it may restart the task the behavior of systems under the first two scenarios is well documented but the third restart has resisted detailed analysis in this paper we derive tight asymptotic relations between the distribution of task times without failures and the total time when including failures for any failure distribution in particular we show that if the task time distribution has an unbounded support then the total time distribution h is always heavy tailed asymptotic expressions are given for the tail of h in various scenarios the key ingredients of the analysis are the cramer lundberg asymptotics for geometric sums and integral asymptotics which in some cases are obtained

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved