the berry esseen bound for student s statistic we prove the berry esseen bound for the student t statistic under the assumption of a third moment this bound coincides up to an absolute constant with the classical berry esseen bound for the mean in general the distribution of the student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean and sometimes faster for example rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law

saddlepoint approximation for student s t statistic with no moment conditions a saddlepoint approximation of the student s t statistic was derived by daniels and young biometrika 78 1991 169 179 under the very stringent exponential moment condition that requires that the underlying density function go down at least as fast as a normal density in the tails this is a severe restriction on the approximation s applicability in this paper we show that this strong exponential moment restriction can be completely dispensed with that is saddlepoint approximation of the student s t statistic remains valid without any moment condition this confirms the folklore that the student s t statistic is robust against outliers the saddlepoint approximation neat only provides a very accurate approximation for the student s t statistic but it also can be applied much more widely in statistical inference as a result saddlepoint approximations should always be used whenever possible some numerical work will be given to illustrate these points

symmetrization approach to concentration inequalities for empirical processes we introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional as an application we give a new easy proof of talagrand s concentration inequality for empirical processes where besides symmetrization we use only talagrand s concentration inequality on the discrete cube 0 1 n as another application of this technique we prove new vapnik chervonenkis type inequalities for example for vc classes of functions we prove a classical inequality of vapnik and chervonenkis only with normalization by the sum of variance and sample variance

bootstrapping the student t statistic let x 1 x n n 1 be independent identically distributed random variables and consider the student t statistic t n based upon these random variables gine gotze and mason 1997 proved that t n converges in distribution to a standard normal random variable if and only if x is in the domain of attraction of a normal random variable and ex 0 we shall show that roughly the same holds true for the bootstrapped student t statistic t n in the process we shall disclose all the possible subsequential limiting laws of t n the proofs introduce a number of amusing tricks that may be of independent interest

precise asymptotics in the self normalized law of the iterated logarithm abstract let x x 1 x 2 be i i d nondegenerate random variables with zero means s n j 1 n x j and v n 2 j 1 n x j 2 we investigate the precise asymptotics in the law of the iterated logarithm for self normalized sums s n v n also for the maximum of self normalized sums max 1 k n s k v n when x belongs to the domain of attraction of the normal law

towards a universal self normalized moderate deviation this paper is an attempt to establish a universal moderate deviation for self normalized sums of independent and identically distributed random variables without any moment condition the exponent term in the moderate deviation is specified when the distribution is in the centered feller class an application to the law of the iterated logarithm is given

a list type reduced constraint generalization of the viterbi algorithm the viterbi algorithm va an optimum decoding rule for a q ary trellis code of constraint length k operates by taking the best survivor from each of q k 1 lists of candidates at each decoding step a generalized va gva is proposed that makes comparisons on the basis of a label of length l l leq k it selects incorporating the notion of list decoding the s best survivors from each of q l 1 lists of candidates at each decoding step coding theorems for a discrete memoryless channel are proved for gva decoding and shown to be natural generalizations of those for va decoding an example of intersymbol interference removal is given to illustrate the practical benefits that the gva can provide

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

darling erdős theorem for self normalized sums let x x1 x2 be i i d nondegenerate random variables sn nj 1xj and v2n nj 1x2j we investigate the asymptotic vspace 1pt behavior in distribution of the maximum of self normalized sums max1 k nsk vk and the law of the iterated logarithm for self normalized sums sn vn when x belongs to the domain of attraction of the normal law in this context we establish a darling erd s type theorem as well as an erd s feller kolmogorov petrovski type test for self normalized sums

an upper bound on the distribution of computation of a sequential decoder for multiple access channels the computational distribution of sequential decoding for discrete memoryless multiple access channels is examined it is shown that all possible collections of incorrect paths in the code tree may be described by bipartite graphs using this fact and a new decoding metric it is proved that the number of computations in the first incorrect subtree is a paretean random variable and that the parameter of the pareto distribution is estimated similarly to the parameter for systems of information transmission with one source

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

diffusive corrections to asymptotics of a strong field quantum transport equation abstract the asymptotic analysis of a linear high field wigner bgk equation is developed by a modified chapman enskog procedure by an expansion of the unknown wigner function in powers of the knudsen number evolution equations are derived for the terms of zeroth and first order in in particular a quantum drift diffusion equation for the position density of electrons with an order correction on the field terms is obtained well posedness and regularity of the approximate problems are established and a rigorous proof that the difference between exact and asymptotic solutions is of order 2 uniformly in time and for arbitrary initial data is given

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

an almost sure central limit theorem for self normalized partial sums let x i i 1 be a sequence of i i d random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance denote s email protected i 1 nx i v n email protected i 1 nx i 2 then an almost sure central limit theorem for self normalized partial sums s n v n is studied under a mild condition in this paper

an almost sure central limit theorem for self normalized products of sums of i i d random variables abstract let x x 1 x 2 be a sequence of independent and identically distributed positive random variables with ex 0 in this paper we show that the almost sure central limit theorem for self normalized products of sums holds only under the assumptions that x belongs to the domain of attraction of the normal law

non central limit theorems for random selections selection from finite sets is a basic procedure of statistics and the partial sum behavior of selected elements is completely known under the uniform asymptotic negligibility condition of central limit theory the purpose of the present paper is to determine the asymptotic behavior of partial sums when the central limit theorem fails as an application we describe the limiting properties of permutation and bootstrap statistics in case of infinite variance

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

an exponential nonuniform berry esseen bound for self normalized sums in this paper we shall derive exponential nonuniformberry esseen bounds in the central limit theorem for self normalized sums we show that the size of the error can be reduced considerably by replacing the usual standardization by self normalization in particular we establish the exponential bounds for the probability of the self normalized sums under the condition that the third moment is finite whereas an exponential moment assumption is required for the standardized sums applications to t statistics and the probabilities of moderate deviations of self normalized sums are also discussed

an analysis of the wigner poisson problem with inflow boundary conditions abstract we present a study of the wigner poisson problem in a bounded spatial domain with non homogeneous and time dependent inflow boundary conditions this system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts we prove well posedness of the linearized n dimensional problem as well as existence and uniqueness of a global in time regular solution of the one dimensional nonlinear problem

limit distributions of studentized means let x xj j n be independent identically distributed random variables with probability distribution f it is shown that student s statistic of the sample xj nj 1 has a limit distribution g such that g 1 1 1 if and only if 1 x is in the domain of attraction of a stable law with some exponent 0 0 such that nx 2 1 x x y2f dy 1 if g 1 1 1 then student s statistic of the sample xj nj 1 has a limit distribution if and only if p x x x 0 is a slowly varying function at

random embedding of ell_p n into ell_r n for any 0 n we give an explicit definition of a random operator s ell p n to mathbb r n such that for every 0 0 is a real number depending only on p and r one of the main tools that we develop is a new type of multidimensional esseen inequality for studying small ball probabilities

when is the student t statistic asymptotically standard normal let x x i i in mathbb n be independent identically distributed random variables it is shown that the student t statistic based upon the sample x i i 1 n is asymptotically n 0 1 if and only if x is in the domain of attraction of the normal law it is also shown that for any x if the self normalized sums u n sum i 1 n x i sum i 1 n x i 2 1 2 n in mathbb n are stochastically bounded then they are uniformly subgaussian that is sup n mathbb e exp lambda u n 2 infty for some lambda 0

bayesian motivated tests of function fit and their asymptotic frequentist properties we propose and analyze nonparametric tests of the null hypothesis that a function belongs to a specified parametric family the tests are based on bic approximations bic to the posterior probability of the null model and may be carried out in either bayesian or frequentist fashion we obtain results on the asymptotic distribution of bic under both the null hypothesis and local alternatives one version of bic call it bic uses a class of models that are orthogonal to each other and growing in number without bound as sample size n tends to infinity we show that n 1 bic converges in distribution to a stable law under the null hypothesis we also show that bic can detect local alternatives converging to the null at the rate log n n a particularly interesting finding is that the power of the bic based test is asymptotically equal to that of a test based on the maximum of alternative log likelihoods simulation results and an example involving variable star data illustrate desirable features of the proposed tests

necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix let xk i i 1 k 1 be a double array of nondegenerate i i d random variables and let pn n 1 be a sequence of positive integers such that n pn is bounded away from 0 and in this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry l n rm max 1 leq i 0 we show that the following six statements are equivalent bf i quad lim n to infty n 2 int limits n log n 1 4 infty left f n 1 x f n 1 left frac sqrt n log n x right right df x 0 bf ii quad n mathbb p left max 1 leq i j leq n x 1 i x 1 j geq sqrt n log n right to 0 quad rm as n to infty bf iii quad frac w mu n sqrt n log n stackrel mathbb p rightarrow 2 sigma 2 bf iv quad left frac n log n right 1 2 l n stackrel mathbb p rightarrow 2 bf v quad lim n rightarrow infty mathbb p left frac w mu n 2 n sigma 4 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty bf vi quad lim n rightarrow infty mathbb p left n l n 2 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty where mu mathbb e x 1 1 sigma 2 mathbb e x 1 1 mu 2 and an 4 log pn log log pn the equivalences between i ii iii and v assume that only mathbb e x 1 1 2 1 2 and n 1 alpha l n stackrel mathbb p rightarrow 0 1 2 alpha leq 1 respectively the current work thus provides weak limit analogues of the strong limit theorems of li and rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of ln obtained by jiang some open problems are also posed

non finite fisher information and homogeneity an em approach summary even simple examples of finite mixture models can fail to fulfil the regularity conditions that are routinely assumed in standard parametric inference problems many methods have been investigated for testing for homogeneity in finite mixture models for example but all rely on regularity conditions including the finiteness of the fisher information and the space of the mixing parameter being a compact subset of some euclidean space very simple examples where such assumptions fail include mixtures of two geometric distributions and two exponential distributions and more generally mixture models in scale distribution families to overcome these difficulties we propose and study an em test statistic which has a simple limiting distribution for examples in this paper simulations show that the em test has accurate type i errors and is more efficient than existing methods when they are applicable a real example is also included

precise asymptotics in complete moment convergence for self normalized sums let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and v n 2 x 1 2 x n 2 ex 2i x x is a slowly varying function at we prove that for any b 2 d 2 b 1 email protected email protected b d 1 email protected n 2 logn d 2 bne s n v n 2i s n email protected n logn 1 b email protected n b d 1 b d 1 2 where n is a standard normal random variable

precise asymptotics in the deviation probability series of self normalized sums abstract let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and w n 2 x 1 2 x n 2 in the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self normalized sums s n w n for positive functions g x x x and x we obtain the precise asymptotics for the following deviation probabilities of self normalized sums n 1 g n n e s n w n 2 i s n w n n n the results given can be considered as the generalization of that in the complete moment convergence law of iterated logarithm and large deviation for self normalized sums

large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation let x n 1 leq n infty be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index 0 alpha 2 the limit of x n 1 log p s n max x i geq x n is found when x n to infty and x n n to 0 the large deviation result is used to prove the law of the iterated logarithm for the self normalized partial sums

characterization of almost surely continuous 1 stable random fourier series and strongly stationary processes we complete the results of m marcus and g pisier by showing that a strongly stationary 1 stable process x t t g defined on a locally compact group has a version with sample continuous paths if and only if the entropy integral 0 log log n k d x e de is finite where k is a given neighborhood of the unit and d x is the distance induced by the process

sums of independent triangular arrays and extreme order statistics let x n i denote an infinitesimal array of independent random variables with convergent partial sums z n n i 1 x n i a n d throughout we find conditions for the convergence of the portion k n of lower extremes l n k n kn i 1 x i n b n given by order statistics x i n similarly w n r n denotes the sum of the r n upper extremes and m n z n l n w n stands for the middle part of the sum it is shown that l n m n w n d 1 2 3 jointly converges for various sequences k n r n where the components of the limit law are independent such that 1 2 3 d the limit of the middle part 2 is asymptotically normal and 1 3 gives the negative positive spectral poisson part of in the case of a compound poisson limit distribution we obtain rates of convergence that can be used for applications to insurance mathematics

sequential decoding for multiple access channels the use of sequential decoding in multiple access channels is considered the fano metric which achieves all achievable rates in the one user case fails to do so in the multiuser case a new metric is introduced and an inner bound is given to it achievable rate region this inner bound region is large enough to encourage the use of sequential decoding in practice the new metric is optimal in the sense of achieving all achievable rates in the case of one user and pairwise reversible channels whether the metric is optimal for all multiple access channels remains an open problem it is noted that even in the one user case the new metric differs from the fano metric in a nontrivial way showing that the fano metric is not uniquely optimal for such channels a new and stricter criterion of achievability in sequential decoding is also introduced and examined

characterizations of almost surely continuous p stable random fourier series and strongly stationary processes on etend le probleme de dudley fernique aux processus p stables fortement stationnaires 1 p 2

how do bootstrap and permutation tests work resampling methods are frequently used in practice to adjust critical values of nonparametric tests in the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points from this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality the results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses they are conditionally correct iff a central limit theorem holds for the original test statistic we prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs special cases are the m n out of k n bootstrap the weighted bootstrap the wild bootstrap and all kinds of permutation statistics the program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail these results are used to compare power functions of conditional resampling tests and their unconditional counterparts the proof uses the method of random scores for permutation type statistics

donsker s theorem for self normalized partial sums processes let x x1 x2 be a sequence of nondegenerate i i d random variables with zero means in this paper we show that a self normalized version of donsker s theorem holds only under the assumption that x belongs to the domain of attraction of the normal law a thus resulting extension of the arc sine law is also discussed we also establish that a weak invariance principle holds true for self normalized self randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero if and only if max1 j n xj vn p0 as n where v2n nj 1x2j

asymptotics for the random coefficient first order autoregressive model with possibly heavy tailed innovations consider a random coefficient ar 1 model x t n n x t 1 u t where n n 1 is a sequence of real numbers n n 1 is a sequence of random variables and the innovations of the model form a sequence of i i d random variables belonging to the domain of attraction of the normal law by imposing some weaker conditions the conditional least squares estimator of the autoregressive coefficient n is achieved and shown to be asymptotically normal by allowing the second moment of the innovation to be possibly infinite

exact convergence rate and leading term in central limit theorem for student s t statistic the leading term in the normal approximation to the distribution of student s t statistic is derived in a general setting with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law the form of the leading term is shown to have its origin in the way in which extreme data influence properties of the studentized sum the leading term approximation is used to give the exact rate of convergence in the central limit theorem up to order n 1 2 where n denotes sample size it is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points moreover the exact rate is identical to that for the non studentized sum when the latter is normalized for scale using a truncated form of variance but when the corresponding truncated centering constant is omitted examples of characterizations of convergence rates are also given it is shown that in some instances their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points

robustness and accuracy of methods for high dimensional data analysis based on student s t statistic summary student s t statistic is finding applications today that were never envisaged when it was introduced more than a century ago many of these applications rely on properties e g robustness against heavy tailed sampling distributions that were not explicitly considered until relatively recently we explore these features of the t statistic in the context of its application to very high dimensional problems including feature selection and ranking the simultaneous testing of many different hypotheses and sparse high dimensional signal detection robustness properties of the t ratio are highlighted and it is established that those properties are preserved under applications of the bootstrap in particular bootstrap methods correct for skewness and therefore lead to second order accuracy even in the extreme tails indeed it is shown that the bootstrap and also the more popular but less accurate t distribution and normal approximations are more effective in the tails than towards the middle of the distribution these properties motivate new methods e g bootstrap based techniques for signal detection that confine attention to the significant tail of a statistic

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

invariance principles for some farima and nonstationary linear processes in the domain of a stable distribution we prove some invariance principles for processes which generalize farima processes when the innovations are in the domain of attraction of a nongaussian stable distribution the limiting processes are extensions of the fractional levy processes the technique used is interesting in itself it extends an older idea of splitting a sample into a central part and an extreme one analyzing each part with different techniques and then combining the results this technique seems to have the potential to be useful in other problems in the domain of nongaussian stable distributions

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

spectrum of non hermitian heavy tailed random matrices let x jk j k 1 be i i d complex random variables such that x jk is in the domain of attraction of an alpha stable law with 0 alpha 2 our main result is a heavy tailed counterpart of girko s circular law namely under some additional smoothness assumptions on the law of x jk we prove that there exists a deterministic sequence a n n 1 alpha and a probability measure mu alpha on c depending only on alpha such that with probability one the empirical distribution of the eigenvalues of the rescaled matrix a n 1 x jk 1 j k n converges weakly to mu alpha as n tends to infinity our approach combines aldous steele s objective method with girko s hermitization using logarithmic potentials the underlying limiting object is defined on a bipartized version of aldous poisson weighted infinite tree recursive relations on the tree provide some properties of mu alpha in contrast with the hermitian case we find that mu alpha is not heavy tailed

asymptotics for self normalized random products of sums of i i d random variables abstract let x x i i 1 be a sequence of independent and identically distributed positive random variables which is in the domain of attraction of the normal law and t n be a positive integer random variable denote s n i 1 n x i v n 2 i 1 n x i x 2 where x denotes the sample mean then we show that the self normalized random product of the partial sums k 1 t n s k k v t n is still asymptotically lognormal under a suitable condition about t n

a kind of complete moment convergence for self normalized sums let x x n n 1 be a sequence of independent and identically distributed i i d random variables with x in the domain of attraction of the normal law and ex 0 in this note we obtain a new kind of complete moment convergence for self normalized sums

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 the three dimensional wigner poisson problem with inflow boundary conditions abstract we study the wigner poisson problem in a bounded spatial domain with non homogeneous and time dependent inflow boundary conditions this system is a quantum model of charge transport in a semiconductor device coupled with reservoirs in presence of a self consistent potential and of an external one we state a local in time well posedness result for the problem the main difficulty is proving in the three dimensional case that the non linear potential term is a lipschitz perturbation of the affine streaming operator in an appropriately weighted l 2 space

quantum moment hydrodynamics and the entropy principle this paper presents how a non commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models our starting point is the moment method which consists in integrating the quantum liouville equation with respect to momentum p against a given vector of monomials of p like in the classical case the so obtained moment system is not closed inspired from levermore s procedure in the classical case 26 we propose to close the moment system by a quantum wigner distribution function which minimizes the entropy subject to the constraint that its moments are given in contrast to the classical case the quantum entropy is defined globally and not locally as the trace of an operator therefore the relation between the moments and the lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators instead of purely local ones in the classical case in the present paper we discuss some practical aspects and consequences of this nonlocal feature

quantum dynamical semigroups for diffusion models with hartree interaction we consider a class of evolution equations in lindblad form which model the dynamics of dissipative quantum mechanical systems with mean field interaction particularly this class includes the so called quantum fokker planck poisson model the existence and uniqueness of global in time mass preserving solutions is proved thus establishing the existence of a nonlinear conservative quantum dynamical semigroup the mathematical difficulties stem from combining an unbounded lindblad generator with the hartree nonlinearity

small time compactness and convergence behavior of deterministically and self normalised levy processes the first author s research was partially supported by arc grant dp0664603 the second author s research was partially supported by nsf grant dms 0503908

nonasymptotic gaussian approximation for inference with stable noise the results of a series of theoretical studies are reported examining the convergence rate for different approximate representations of alpha stable distributions although they play a key role in modelling random processes with jumps and discontinuities the use of alpha stable distributions in inference often leads to analytically intractable problems the lepage series which is a probabilistic representation employed in this work is used to transform an intractable infinite dimensional inference problem into a finite dimensional conditionally gaussian parametric problem a major component of our approach is the approximation of the tail of this series by a gaussian random variable standard statistical techniques such as expectation maximization em markov chain monte carlo and particle filtering can then be readily applied in addition to the asymptotic normality of the tail of this series we establish explicit nonasymptotic bounds on the approximation error their proofs follow classical fourier analytic arguments using esseen s smoothing lemma specifically we consider the distance between the distributions of i the tail of the series and an appropriate gaussian ii the full series and the truncated series and iii the full series and the truncated series with an added gaussian term in all three cases sharp bounds are established and the theoretical results are compared with the actual distances computed numerically in specific examples of symmetric alpha stable distributions this analysis facilitates the selection of appropriate truncations in practice and offers theoretical guarantees for the accuracy of resulting estimates one of the main conclusions obtained is that for the purposes of inference the use of a truncated series together with an approximately gaussian error term has superior statistical properties and is likely a preferable choice in practice

a version of chevet s theorem for stable processes let x bo y be the injective tensor product of the separable banach spaces x and y and let sx sy and sx bo y be the unit spheres of these spaces the tensor product of two symmetric finite measures 1 on sx and 2 on sy 1 2 is defined in a natural way as a measure on sx bo y it is shown that 1 2 is the spectral measure of a p stable random variable w on x bo y 0 p 2 if and only if 1 and 2 are the spectral measures of p stable random variables u and v on x and y respectively actually upper and lower bounds for e w r 1r in terms of the random variables u and v are obtained when x c s y c t with s t compact metric spaces and 1 and 2 are discrete our results imply that if i ij are i i d standard symmetric real valued p stable random variables 0 p 2 xi c s and yi c t then the series ij ijxi s yj t converges uniformly a s iff the series i ixi s and i iyi t both converge uniformly a s when p 2 this follows from chevet s theorem on gaussian processes several examples are given one of them requires an interesting upper bound on the probability distribution of the maximum of i i d p stable random variables taking values in a general banach space