correlation coefficients for random variables on a unit sphere or hypersphere summary a coefficient to measure association between two random variables each distributed on the surface of the unit sphere in p dimensional space is proposed analogous to an existing correlation coefficient for circular data the sample estimate is studied in detail for the case p 3 and comparisons are made with other statistics in current use some applications are given to studying remanent magnetization in rock samples

kernel density estimation for linear processes in this paper we provide a detailed characterization of the asymptotic behavior of kernel density estimators for one sided linear processes the conjecture that asymptotic normality for the kernel density estimator holds under short range dependence is proved under minimal assumptions on bandwidths we also depict the dichotomous and trichotomous phenomena for various choices of bandwidths when the process is long range dependent

random covering of the circle the configuration space of the free deposition process consider a circle of circumference 1 throw at random n points sequentially on this circle and append clockwise an arc or rod of length s to each such point the resulting random set the free gas of rods is a collection of a random number of clusters with random sizes it models a free deposition process on a 1d substrate for such processes we shall consider the occurrence times number of rods and probabilities as n grows of the following configurations those avoiding rod overlap the hard rod gas those for which the largest gap is smaller than rod length s the packing gas those parking configurations for which hard rod and packing constraints are both fulfilled and covering configurations special attention is paid to the statistical properties of each such rare configuration in the asymptotic density domain when ns for some finite density of points using results from spacings in the random division of the circle explicit large deviation rate functions can be computed in each case from state equations lastly a process consisting in selecting at random one of these specific equilibrium configurations called the observable can be modelled when particularized to the parking model this system produces parking configurations differently from renyi s random sequential adsorption model

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

non borel summable φ4 theory in zero dimension a toy model for testing numerical and analytical methods the exact analytical solution for the generating functional of the zero dimensional 4 theory with degenerate minima is obtained in the whole complex coupling parameter plane for testing purposes the efficiency and precision of different computing tools proposed in non borel summable field theories to obtain approximate solutions in both perturbative and nonperturbative regimes are analyzed furthermore a new resummation approach is proposed in order to successfully deal with factorially divergent series it provides a representation of the generating function in terms of an unambiguously defined laplace borel integral on the other hand a recent approach called the generalized borel transform is shown to be an accurate and robust technique to capture non perturbative contributions in the coupling parameter an extension of this approach to path integrals is proposed

asymptotic expansion of m estimators with long memory errors this paper obtains a higher order asymptotic expansion of a class of m estimators of the one sample location parameter when the errors form a long memory moving average a suitably standardized difference between an m estimator and the sample mean is shown to have a limiting distribution the nature of the limiting distribution depends on the range of the dependence parameter if for example 1 3 then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric if 0 1 3 then the corresponding limiting distribution is nonnormal this paper thus goes beyond that of beran who observed in the case of long memory gaussian errors that m estimators t n of the one sample location parameter are asymptotically equivalent to the sample mean in the sense that var t n var x n 1 and t n x n o p var x n

exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard rayleigh schrodinger perturbation theory rspt power series we give a simple algorithm for the calculation of these subseries to any desired order in terms of the rspt coefficients expressed as polynomials in the quantum number and illustrate our results with numerical borel pade summations of the expansion up to third exponentially small order

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

the asymptotic distributions of generalized u statistics with applications to random graphs we consider the random variable s n v f sum limits i 1 i v leqq n f x i 1 x i v y i 1 i 2 y i v 1 i v

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

the asymptotic distributions of incomplete u statistics an incomplete u statistic is obtained by sampling the terms of an u statistic this paper derives the asymptotic distribution if the variance is finite depending on the number of sampled terms the resulting distribution is either the same as for the u statistic a normal distribution or something intermediate also the case of a non random sampling of the terms is treated as an example a non parametric test of the independence of two circular random variables is studied the results are generalized to generalized u statistics

on the bootstrap of u and v statistics bootstrap distributional limit theorems for u and v statistics are proved they hold a s under weak moment conditions and without restrictions on the bootstrap sample size as long as it tends to infty regardless of the degree of degeneracy of u and v a testing procedure based on these results is outlined

on the bahadur representation of sample quantiles for dependent sequences we establish the bahadur representation of sample quantiles for linear and some widely used nonlinear processes local fluctuations of empirical processes are discussed applications to the trimmed and winsorized means are given our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables

u statistics of random size samples and limit theorems for systems of markovian particles with non poisson initial distributions abstract limiting distributions of square integrable infinite order u statistics were first studied by dynkin and mandelbaum 1983 and mandelbaum and taqqu 1984 we extend their results to the case of non poisson random sample size multiple integrals of non gaussian generalized fields are constructed to identify the limiting distributions an invariance principle is also established we use these results to study the limiting distribution of the amount of charge left in some set by an infinite system of signed markovian particles when the initial particle density goes to infinity by selecting the initial particle distribution we determine the limiting distribution of charge constructing different non gaussian generalized random fields including laplace alpha stable and their multiple integrals

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

expansion around instantons in quantum mechanics we calculate numerically a few terms of the corrections to the large order behavior of the ground state energy of the o n anharmonic oscillator by analyzing the perturbation series we then generate 94 terms of the perturbative expansion of the difference between the energies of the two low lying states of the double well potential and analyze their large order behavior

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions

on estimation and testing goodness of fit for m dependent stable sequences a class of estimators of the characteristic index of m dependent stable sequences is proposed the estimators are shown to be consistent and asymptotically normal in addition a class of goodness of fit tests for stability is also obtained the performance of the estimators and the goodness of fit tests is evaluated through a simulation study one of the estimators is shown to have a reasonably high relative efficiency which is uniformly superior to that of the regression estimator which is currently most widely used our results have significance for modelling financial data like stock returns which have thick tailed distributions and exhibit non linear behaviour

a central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

a continuous counting observation and posterior quantum dynamics a stochastic model for a continuous photon counting measurement is proposed an equation for the generating functional of the statistics of counts is found a stochastic linear equation for the unnormalised posterior state vector of atoms continuously collapsing without mixing is derived by using quantum stochastic calculus methods

factorization of quasiclassical integrable systems abstract using asymptotic methods one calculates the first temperature correction to the free energy of non linear kink bearing scalar fields the method allows to make a distinct separation between the contribution of the anharmonic optical phonon field and that of the dilute gas of kinks the same technique can be used to calculate the multi kink contributions to the free energy

gaussian and bootstrap approximations for high dimensional u statistics and their applications this paper studies the gaussian and bootstrap approximations for the probabilities of a nondegenerate u statistic belonging to the hyperrectangles in mathbb r d when the dimension d is large a two step gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed subject to mild moment conditions on the kernel we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in mathbb r d that decays polynomially in sample size for a high dimensional scaling limit where the dimension can be much larger than the sample size we also provide computable approximation methods for the quantiles of the maxima of centered u statistics specifically we provide a unified perspective for the empirical bootstrap the randomly reweighted bootstrap and the gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity we show that all three methods are inferentially first order equivalent for high dimensional u statistics in the sense that they achieve the same uniform rate of convergence over all d dimensional hyperrectangles in particular they are asymptotically valid when the dimension d can be as large as o e n c for some constant c in 0 1 7 the bootstrap methods are applied to statistical applications for high dimensional non gaussian data including i principled and data dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals ii simultaneous inference for the covariance and rank correlation matrices in particular for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter we show that for a class of sub gaussian data error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold in addition we also show that the gaussian like convergence rates can be achieved for heavy tailed data which are less conservative than those obtained by the bonferroni technique that ignores the dependency in the underlying data distribution

on the limiting distribution of pair summable potential functions in many particle systems for systems with finite phase space volume the density of states can be viewed as a multiple of the probability density of the energy when the phase space variables are independent uniformly distributed random variables we show that the distribution of a random variable proportional to the sum of pairwise interactions of independent identically distributed random variables converges to a limiting distribution as the number of variables goes to infinity when the interaction satisfies certain homogeneity requirements the moments of this distribution are simple combinations of cyclic integrals of the potential function the existence of this limit gives information about the structure function of some systems in statistical mechanics having pair summable interactions even in the absence of a thermodynamic limit the result is applied to several examples including systems of two dimensional point vortices

symmetric statistics poisson point processes and multiple wiener integrals the asymptotic behaviour of symmetric statistics of arbitrary order is studied as an application we describe all limit distributions of square integrable u statistics we use as a tool a randomization of the sample size a sample of poisson size n lambda with en lambda lambda can be interpreted as a poisson point process with intensity lambda and randomized symmetric statistics are its functionals as lambda rightarrow infty the probability distribution of these functionals tend to the distribution of multiple wiener integrals this can be considered as a stronger form of the following well known fact properly normalized a poisson point process with intensity lambda approaches a gaussian random measure as lambda rightarrow infty

strong laws for l and u statistics strong laws of large numbers are given for l statistics linear combinations of order statistics and for u statistics averages of kernels of random samples for ergodic stationary processes extending classical theorems of hoeffding and of helmers for lid sequences examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied and an application is given to estimation of correlation dimension of invariant measures

asymptotic distributions for weighted u statistics we prove limit theorems for weighted u statistics and express the limit by means of multiple stochastic integrals this is a generalisation of the paper by o neil and redner in that paper the method of moments was applied which does not work in the general case hence we had to work out a different method in particular in theorem 4 we describe the limit of a model proposed by o neil and redner in this model the weight functions cause an intricate cancellation and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities

gaussian approximation of nonlinear statistics on the sphere abstract we show how it is possible to assess the rate of convergence in the gaussian approximation of triangular arrays of u statistics built from wavelets coefficients evaluated on a spherical poisson field of arbitrary dimension for this purpose we exploit the stein malliavin approach introduced in the seminal paper by peccati sole taqqu and utzet 2011 we focus in particular on statistical applications covering evaluation of variance in non parametric density estimation and sobolev tests for uniformity

modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model summary the behaviour of nonlinear deterministic vibrations has been studied by many authors and may typically include such features as jump phenomena and limit cycles nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude dependent frequency a discrete time series model is introduced which may be demonstrated to have properties similar to those of nonlinear random vibrations this model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models the model is fitted to the canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self exciting mechanism

on local u statistic processes and the estimation of densities of functions of several sample variables a notion of local u statistic process is introduced and central limit theorems in various norms are obtained for it this involves the development of several inequalities for u processes that may be useful in other contexts this local u statistic process is based on an estimator of the density of a function of several sample variables proposed by frees j amer statist assoc 89 1994 517 525 and as a consequence uniform in bandwidth central limit theorems in the sup and in the lp norms are obtained for these estimators

statistics of two dimensional point vortices and high energy vortex states results from the theory ofu statistics are used to characterize the microcanonical partition function of then vortex system in a rectangular region for largen under various boundary conditions and for neutral asymptotically neutral and nonneutral systems numerical estimates show that the limiting distribution is well matched in the region of major probability forn larger than 20 implications for the thermodynamic limit are discussed vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices vortex states for which clustering is generic in a statistical sense are shown to result from two modeling processes the approximation of a continuous inviscid fluid by point vortex configurations and the modeling of the evolution of a continuous fluid at high reynolds number by point vortex configurations with the viscosity represented by the annihilation of close positive negative vortex pairs this last process with the vortex dynamics replaced by a random walk reproduces quite well the late time features seen in spectral integration of the 2d navier stokes equation

limit theorems for a triangular scheme of u statistics with applications to inter point distances the annals of probability 1986 vol 14 no 4 1347 1358 limit theorems for a triangular scheme of u statistics with applications to inter point distances by s rao jammalamadaka1 and svante j anson university of california santa barbara and uppsala university the asymptotic distribution of a triangular scheme of u statistics is studied two limit theorems applicable in different situations are given one theorem yields convergence to a normal distribution the other includes poisson limits and other limit laws applications to statistics based on small interpoint distances in a sample are given 1 introduction there has been a renewed interest in the limit theory relating to u statistics hoe ding 1948 especially focussing on degenerate kernels and nonnormal limits see for instance rubin and vitale 1980 dynkin and mandelbaum 1983 and berman and eagleson 1983 in section 2 of this paper we study a triangular scheme of u statistics and establish their asymptotic normality under suitable conditions it is worth noting that one can obtain normal limit laws even with degenerate kernels other in nitely divisible limit laws for the triangular scheme including the poisson limits are examined in the next section applications to limit distributions of statistics based on interpoint distances are discussed in section 4 these applications which were the source of motivation for the results derived here were suggested by the studies one of us made about statistics based on spacings see for instance holst and rao 1981 they were also sparked by the recent papers of bickel and breiman 1983 and onoyama et al 1983 theorems yielding asymptotic normality in this setting have also been proved by weber 1983 using a martingale approach his theorems yield the same conclusion as theorem 2 1 but under di erent nonequivalent conditions a few words about notation we write gd to denote convergence in distribution an gc to denote complete convergence cf loeve 1963 page 178 we write po to denote the poisson distribution with mean a and np p 2 to denote a p dimensional normal distribution with mean vector p and covariance matrix 2 2 asymptotic normality of a triangular scheme of u statistics let x1 x2 be a sequence of independently and identically distributed i i d random variables we will use x and y to denote two independent random received october 1984 revised april 1985 formerly j s rao ams 1980 subject classi cations primary 60f05 60e07 secondary 62e20 key words and phrases u statistics triangular scheme multigraph limit laws in nitely divisible distributions inter point distances 1347 t institute of mathematical statistics is collaborating with j stor to digitize preserve and extend access to ef 73i the annals ofprobabiiity 5t0 r www jstor org

higher order corrections to instantons the energy levels of the double well potential receive beyond perturbation theory contributions which are non analytic in the coupling strength these are related to instanton effects for example the separation between the energies of odd and even parity states is given at leading order by the one instanton contribution however to determine the energies more accurately multi instanton configurations have also to be taken into account we investigate here the two instanton contributions first we calculate analytically higher order corrections to multi instanton effects we then verify that the difference betweeen numerically determined energy eigenvalues and the generalized borel sum of the perturbation series can be described to very high accuracy by two instanton contributions we also calculate higher order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two instanton effect and with exact data for the first 200 perturbative coefficients

martingale approximations for sums of stationary processes approximations to sums of stationary and ergodic sequences by martingales are investigated necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained it is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a lindeberg feller condition using the explicit construction of the approximating martingales a central limit theorem is derived for the sample means of linear processes the conditions are not sufficient for the functional version of the central limit theorem this is shown by an example and a slightly stronger sufficient condition is given

instantons in quantum mechanics numerical evidence for a conjecture in previous articles we have given a proper definition of multi instanton contributions in quantum mechanics and calculated these contributions to leading order we have also presented a conjecture about the form of the expansion of these multi instanton contributions to all orders in powers of the coupling constant we give here some numerical results which support this conjecture

asymptotic spectral theory for nonlinear time series we consider asymptotic problems in spectral analysis of stationary causal processes limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given instead of the commonly used strong mixing conditions in our asymptotic spectral theory we impose conditions only involving conditional moments which are easily verifiable for a variety of nonlinear time series

non markovian quantum trajectories instruments and time continuous measurements the linear and nonlinear non markovian quantum state diffusion nmqsd equations are well known tools for the description of certain non markovian open quantum systems in this work we systematically investigate whether the normalized linear nmqsd or the nonlinear nmqsd solutions can be generated by means of a time continuous measurement by considering any conceivable measurement scheme we derive a necessary criterion for the measurement interpretation of both equations concrete examples show that the normalized linear nmqsd solutions are realizable only in the markovian limit in general the application of the presented criterion to the nonlinear nmqsd remains an open issue furthermore this criterion is a necessary one not only in the context of the nmqsds but also for a much broader class of stochastic schrodinger equations

high precision numerical determination of eigenvalues for a double well potential related to the zinn justin conjecture a numerical method of high precision is used to calculate the energy eigenvalues and eigenfunctions for a symmetric double well potential the method is based on enclosing the system within two infinite walls with a large but finite separation and developing a power series solution for the schrodinger equation the obtained numerical results are compared with those obtained on the basis of the zinn justin conjecture and found to be in excellent agreement

a note on the behaviour of the correlation integral in the presence of a time series summary the behaviour of the correlation integral is well known when applied to simple chaotic regimes and to pure random noise the asymptotic behaviour of a chaotic system is represented by an infinite set of points called an attractor the dimension of which can be estimated by the correlation integral essentially the correlation integral of a time series is a normalized count of the number of close pairs of points of the series lying in phase space we examine the expected behaviour of the integral for some stochastic process models which place a weak structure on the noise such as simple autoregressive and moving average formulations

statistical mechanics approach to some problems in conformal geometry a weak law of large numbers is established for a sequence of systems of n classical point particles with logarithmic pair potential in rn or sn n n which are distributed according to the configurational microcanonical measure e h or rather some regularization thereof where h is the configurational hamiltonian and e the configurational energy when n with non extensive energy scaling e n2e the particle positions become i i d according to a self consistent boltzmann distribution respectively a superposition of such distributions the self consistency condition in n dimensions is some nonlinear elliptic pde of order n pseudo pde if n is odd with an exponential nonlinearity when n 2 this pde is known in statistical mechanics as poisson boltzmann equation with applications to point vortices 2d coulomb and magnetized plasmas and gravitational systems it is then also known in conformal differential geometry where it is the central equation in nirenberg s problem of prescribed gaussian curvature for constant gauss curvature it becomes liouville s equation which also appears in two dimensional so called quantum liouville gravity the pde for n 4 is paneitz equation and while it is not known in statistical mechanics it originated from a study of the conformal invariance of maxwell s electromagnetism and has made its appearance in some recent model of four dimensional quantum gravity in differential geometry the paneitz equation and its higher order n generalizations have applications in the conformal geometry of n manifolds but no physical applications yet for general n interestingly though all the paneitz equations have an interpretation in terms of statistical mechanics

large order perturbation theory for the o 2 anharmonic oscillator with negative anharmonicity and for the double well potential for the o 2 anharmonic oscillator with negative anharmonicity and one dimensional double well potential u x x2 r2 2 8r2 r is a large parameter for both problems conventional rayleigh schrodinger expansions in power series of 1 r2 for the energy eigenvalues agree the authors studied asymptotic expansions for the eigenvalues which contain in addition to the perturbation series exponentially small terms in r2 due to the tunnelling through corresponding potential barriers numerical calculations are also performed and comparisons with the asymptotic formulae are given

rigorous statistical procedures for data from dynamical systems various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations this includes symmetry tests and the determination of dimension coefficients the paper contains the theoretical results and several simulations explain the methods

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

the empirical process of some long range dependent sequences with an application to u statistics let x j infty j 1 be a stationary mean zero gaussian process with covariances r k ex k 1 x 1 satisfying r 0 1 and r k k d l k where d is small and l is slowly varying at infinity consider the two parameter empirical process for g x j bigg f n x t frac 1 n sum lbrack nt rbrack j 1 lbrack 1 g x j leq x p g x 1 leq x rbrack infty x infty 0 leq t leq 1 bigg where g is any measurable function noncentral limit theorems are obtained for f n x t and they are used to derive the asymptotic behavior of some suitably normalized von mises statistics and u statistics based on the g x j s the limiting processes are structurally different from those encountered in the i i d case

quantum stochastic approach to the description of quantum measurements in the present paper we consider the problem of the description of an arbitrary generalized quantum measurement with outcomes in a measurable space analysing the unitary invariants of a separable statistical realization of a quantum instrument we present the most general form of a possible integral representation of an instrument we introduce the notion of a stochastic realization of an instrument and establish a one to one correspondence between the class of unitarily and phase equivalent separable statistical realizations and the equivalence class of stochastic realizations of an instrument we further single out the invariant class of unitarily and phase equivalent separable statistical realizations for which the integral representation of an instrument is the same for all statistical realizations from this class and is wholly determined by the invariants of this class we call the special form of this integral representation the quantum stochastic representation of an instrument we show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators satisfying the orthonormality relation this approach gives not only the complete statistical description of any generalized direct quantum measurement but the complete description in a hilbert space of the stochastic behaviour of a quantum system under a generalized direct measurement in the sense of specifying the probabilistic transition law governing the change from the initial state to a final one under a single measurement under this approach a unitary evolution of an isolated quantum system is included as a special case in the frame of the proposed approach which we call quantum stochastic approach all possible schemes of measurements upon a quantum system can be considered in the case of repeated or continuous in time measurements the quantum stochastic approach allows to define in the most general case the notion of the family of posterior pure state trajectories quantum trajectories in discrete or continuous time in the hilbert space of a quantum system and to give their probabilistic treatment

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

estimating densities of functions of observations abstract density estimates such as histograms and more sophisticated versions are important in applied and theoretical statistics in applied statistics a density estimate gives the data analyst a graphical overview of the shape of the distribution this overview allows the data analyst to arrive immediately at a qualitative impression of the location scale and various aspects of the extremes of the distribution in theoretical statistics the shape of the density allows the researcher to link the data to families of curves perhaps indexed parametrically by estimating a density nonparametrically certain aspects of the data can be viewed without the restriction of a priori imposing limitations of a class of parametric curves in this article we introduce density estimation for functions of observations to motivate the study one type of function used is the interpoint distance between observations arising in spatial statistics from the fields of biometry and regional science a second type of func

asymptotic properties of u processes under long range dependence let x i i geq 1 be a stationary mean zero gaussian process with covariances rho k pe x 1 x k 1 satisfying rho 0 1 and rho k k d l k where d is in 0 1 and l is slowly varying at infinity consider the u process u n r r in i defined as u n r frac 1 n n 1 sum 1 leq i neq j leq n 1 g x i x j leq r where i is an interval included in rset and g is a symmetric function in this paper we provide central and non central limit theorems for u n they are used to derive the asymptotic behavior of the hodges lehmann estimator the wilcoxon signed rank statistic the sample correlation integral and an associated scale estimator the limiting distributions are expressed through multiple wiener ito integrals

parametrization and optimization of gaussian non markovian unravelings for open quantum dynamics we derive a family of gaussian non markovian stochastic schr odinger equations for the dynamics of open quantum systems the different unravelings correspond to different choices of squeezed coherent states reflecting different measurement schemes on the environment consequently we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the gaussian process by construction the reduced dynamics of the open system does not depend on the squeezing parameters they determine the non hermitian gaussian correlation a wide range of which are compatible with the markov limit we demonstrate the versatility of our results for quantum information tasks in the non markovian regime in particular by optimizing the squeezing parameters we can tailor unravelings for improving entanglement bounds or for environment assisted entanglement protection

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

normal approximations for wavelet coefficients on spherical poisson fields abstract we compute explicit upper bounds on the distance between the law of a multivariate gaussian distribution and the joint law of wavelet needlet coefficients based on a homogeneous spherical poisson field in particular we develop some results from peccati and zheng 2010 42 based on malliavin calculus and stein s methods to assess the rate of convergence to gaussianity for a triangular array of needlet coefficients with growing dimensions our results are motivated by astrophysical and cosmological applications in particular related to the search for point sources in cosmic rays data

central limit theorems for u statistics of poisson point processes a u statistic of a poisson point process is defined as the sum f x1 xk over all possibly infinitely many k tuples of distinct points of the point process using the malliavin calculus the wiener ito chaos expansion of such a functional is computed and used to derive a formula for the variance central limit theorems for u statistics of poisson point processes are shown with explicit bounds for the wasserstein distance to a gaussian random variable as applications the intersection process of poisson hyperplanes and the length of a random geometric graph are investigated

arbitrarily accurate eigenvalues for one dimensional polynomial potentials we show that the riccati form of the one dimensional schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function g when g and the potential as for anharmonic oscillators are polynomials the solutions of these two equations are entire functions l and k and the zeros of k are identical to those of the wavefunction requiring such a zero at a large but finite value of the argument yields low energy eigenstates with exponentially small errors approximate formulae for these errors are provided we explain how to choose g in order to dramatically improve the numerical treatment the method yields many significant digits with modest computer means we discuss the extension of this method in the case of several variables

limit theorems for von mises statistics of a measure preserving transformation for a measure preserving transformation t of a probability space x mathcal f mu and some d ge 1 we investigate almost sure and distributional convergence of random variables of the form begin aligned x rightarrow frac 1 c n sum 0 le i 1 ldots i d n f t i 1 x ldots t i d x n 1 2 ldots end aligned where c 1 c 2 ldots are normalizing constants and the kernel f belongs to an appropriate subspace in some l p x d mathcal f otimes d mu d we establish a form of the individual ergodic theorem for such sequences using a filtration compatible with t and the martingale approximation we prove a central limit theorem in the non degenerate case for a class of canonical totally degenerate kernels and d 2 we also show that the convergence holds in distribution towards a quadratic form sum m 1 infty lambda m eta 2 m in independent standard gaussian variables eta 1 eta 2 ldots

on maxima of periodograms of stationary processes we consider the limit distribution of maxima of periodograms for stationary processes our method is based on m dependent approximation for stationary processes and a moderate deviation result

nonparametric renewal function estimation abstract the renewal function is a basic tool used in many probabilistic models and sequential analysis based on a random sample of size n a nonparametric estimator of the renewal function is introduced asymptotic properties of the estimator such as the almost sure consistency and local asymptotic normality are developed a discussion of an application of the estimator is also provided keywords u statistics reverse martingales warranty analysis

symmetric polynomials of random variables attracted to an infinitely divisible law letx i n i 1 n be a triangular array in the domain of attraction of a levy processx t then the symmetric polynomial of orderk in thex i n sum limits 1 leqq i 1 i 2 ldots i k leqq nt x i 1 n x i 2 n ldots x i k n converges weakly to thek th order multiple integral with respect tox t also sums of powers of thex i n converge jointly to the variations of the processx t

limit theorems for functionals of moving averages let x n i 1 a i e n 1 where the e i are i i d with mean 0 and finite second moment and the a i are either summable or regularly varying with index 1 1 2 the sequence x n has short memory in the former case and long memory in the latter for a large class of functions k a new approach is proposed to develop both central n rate and noncentral non n rate limit theorems for s n n 1 n k x n ek x n specifically we show that in the short memory case the central limit theorem holds for s n and in the long memory case s n can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non central limit theorem respectively further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems connections with the usual hermite expansion in the gaussian setting are also explored

langer cherry derivation of the multi instanton expansion for the symmetric double well the multi instanton expansion for the eigenvalues of the symmetric double well is derived using a langer cherry uniform asymptotic expansion of the solution of the corresponding schrodinger equation the langer cherry expansion is anchored to either one of the minima of the potential and by construction has the correct asymptotic behavior at large distance while the quantization condition amounts to imposing the even or odd parity of the wave function this method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi instanton expansion and with trivial modifications can also be used for nonsymmetric double wells

reduced dynamics need not be completely positive the reduced dynamics of a quantum system in contact with a reservoir is generally thought to be completely positive this is certainly true if product initial conditions are used to define the dynamics we show that with correlated initial conditions it need not be so in this case the dynamics can properly be defined only on a subset of initial system states extension by linearity to all possible initial states is trivially possible but the extension may not be physically realizable and may not even be positive let alone completely positive

complete and symmetrically complete families of distributions on utilise la polarisation pour unifier et etendre les resultats classiques concernant la completude des statistiques d ordre generalisees avec des distributions dominees et non dominees