coding theorems for shannon s cipher system with correlated source outputs and common information source coding problems are treated for shannon s 1949 cipher system with correlated source outputs x y several cases are considered based on whether both x and y only x or only y must be transmitted to the receiver whether both x and y only x or only y must be kept secret or whether the security level is measured by sup 1 spl sol sub k h x sup k spl verbar w sup 1 spl sol sub k h y sup k spl verbar w or sup 1 spl sol sub k h x sup k y sup k spl verbar w where w is a cryptogram the admissible region of cryptogram rate and key rate for a given security level is derived for each case furthermore two new kinds of common information of x and y say c sub 1 x y and c sub 2 x y are considered c sub 1 x y is defined as the rate of the attainable minimum core of x sup k y sup k by removing each private information from x sup k y sup k as much as possible while c sub 2 x y is defined as the rate of the attainable maximum core v sub c such that if one loses v sub c then each uncertainty of x sup k and y sup k becomes h v sub c it is proved that c sub 1 x y i x y and c sub 2 x y min spl lcub h x h y spl rcub c sub 1 x y justifies the author s intuitive feeling that the mutual information represents a common information of x and y

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

non negativity conditions for the hyperbolic garch model abstract in this article we derive conditions which ensure the non negativity of the conditional variance in the hyperbolic garch p d q hygarch model of davidson 2004 the conditions are necessary and sufficient for p 1 and sufficient for p 2 and emerge as natural extensions of the inequality constraints derived in nelson and cao 1992 and tsai and chan 2008 for the garch model and in conrad and haag 2006 for the figarch model as a by product we obtain a representation of the arch coefficients which allows computationally efficient multi step ahead forecasting of the conditional variance of a hygarch process we also relate the necessary and sufficient parameter set of the hygarch to the necessary and sufficient parameter sets of its garch and figarch components finally we analyze the effects of erroneously fitting a figarch model to a data sample which was truly generated by a hygarch process empirical applications of the hygarch 1 d 1 model to daily nyse and dax30 data illustrate the importance of our results

decay of hankel singular values of analytic control systems we show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a hankel operator that belongs to the schatten class sp for all positive p this implies that the hankel singular values converge to zero faster than any polynomial rate this in turn implies fast convergence of balanced truncations as a corollary decay rates for the eigenvalues of the controllability and observability gramians are also provided applications to the heat equation and a plate equation are given

operators associated with soft and hard spectral edges from unitary ensembles abstract using hankel operators and shift invariant subspaces on hilbert space this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices such tracy widom operators are realized as controllability operators for linear systems and are reproducing kernels for weighted hardy spaces known as sonine spaces periodic solutions of hill s equation give a new family of tracy widom type operators this paper identifies a pair of unitary groups that satisfy the von neumann weyl anti commutation relations and leave invariant the subspaces of l 2 that are the ranges of projections given by the tracy widom operators for the soft edge of the gaussian unitary ensemble and hard edge of the jacobi ensemble

on secret sharing systems a secret sharing system permits a secret to be shared among n trustees in such a way that any k of them can recover the secret but any k 1 have complete uncertainty about it a linear coding scheme for secret sharing is exhibited which subsumes the polynomial interpolation method proposed by shamir and can also be viewed as a deterministic version of blakley s probabilistic method bounds on the maximum value of n for a given k and secret size are derived for any system linear or nonlinear the proposed scheme achieves the lower bound which for practical purposes differs insignificantly from the upper bound the scheme may be extended to protect several secrets methods to protect against deliberate tampering by any of the trustees are also presented

randomness in distribution protocols randomness is a useful computation resource due to its ability to enhance the capabilities of other resources its interaction with resources such as time space interaction with provers and its role in several areas of computer science has been extensively studied in this paper we give a systematic analysis of the amount of randomness needed by secret sharing schemes and secure key distribution schemes we give both upper and lower bounds on the number of random bits needed by secret sharing schemes the bounds are tight for several classes of secret sharing schemes for secure key distribution schemes we provide a lower bound on the amount of randomness needed thus showing the optimality of a recently proposed key distribution protocol

generating random bits from an arbitrary source fundamental limits suppose we are given a random source and want to use it as a random number generator at what rate can we generate fair bits from it we address this question in an information theoretic setting by allowing for some arbitrarily small but nonzero deviation from ideal random bits we prove our results with three different measures of approximation between the ideal and the obtained probability distributions the variational distance the d bar distance and the normalized divergence two different contexts are studied fixed length and variable length random number generation the fixed length results of this paper provide an operational characterization of the inf entropy rate of a source defined in han and verdu see ibid vol 39 no 3 p 752 772 1993 and the variable length results characterize the liminf of the entropy rate thereby establishing a pleasing duality with the fundamental limits of source coding a feature of our results is that we do not restrict ourselves to ergodic or to stationary sources

kernels and point processes associated with whittaker functions this article considers whittaker s confluent hypergeometric function w where is real and is real or purely imaginary then x x 1 2w x arises as the scattering function of a continuous time linear system with state space l2 1 2 and input and output spaces c the hankel operator on l2 0 is expressed as a matrix with respect to the laguerre basis and gives the hankel matrix of moments of a jacobi weight w0 x xb 1 x a the operation of translating is equivalent to deforming w0 to give wt x e t xxb 1 x a the determinant of the hankel matrix of moments of we satisfies the form of painleve s transcendental differential equation pv it is shown that gives rise to the whittaker kernel from random matrix theory as studied by borodin and olshanski commun math phys 211 335 358 2000 whittaker kernels are closely related to systems of orthogonal polynomials for a pollaczek jacobi type weight lying outside the usual szego class

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

coding theorems on the threshold scheme for a general source in this paper coding theorems on the t m threshold scheme for a general source are discussed where m means the number of the shares and t means a threshold the t m threshold scheme treated in this paper encrypts n source outputs xn to m shares at once and is required to satisfy the two conditions that 1 xn is reproduced from arbitrary t shares and 2 almost no information of xn is revealed from any t 1 shares it is shown that the t m threshold scheme must satisfy certain inequalities including the limit inferiors in probability one of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the t m threshold scheme in addition it is shown that a certain variation of shamir s threshold scheme meets the two conditions the same approach can be taken to the problems of shannon s cipher system with the perfect secrecy and fixed length source coding with vanishing decoding error probability it is shown that the same kind of inequalities which indicate the converse coding theorems hold in both two cases

bootstrap refinements for qml estimators of the garch 1 1 parameters a b s t r a c t this paper reconsiders a block bootstrap procedure for quasi maximum likelihood estimation of garch models based on the resampling of the likelihood function as proposed by goncalves and white 2004 maximum likelihood and the bootstrap for nonlinear dynamic models journal of econometrics 119 199 219 first we provide necessary conditions and sufficient conditions in terms of moments of the innovation process for the existence of the edgeworth expansion of the garch 1 1 estimator up to the k th term second we provide sufficient conditions for higher order refinements for equally tailed and symmetric test statistics in particular the bootstrap estimator based on resampling the likelihood has the same higher order improvements in terms of error in the rejection probabilities as those in andrews 2002 higher order improvements of a computationally attractive k step bootstrap for extremum estimators econometrica 70 119 162

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

estimating semiparametric arch models by kernel smoothing methods we investigate a class of semiparametric arch models that includes as a special case the partially nonparametric pnp model introduced by engle and ng 1993 and which allows for both flexible dynamics and flexible function form with regard to the news impact function we show that the functional part of the model satisfies a type ii linear integral equation and give simple conditions under which there is a unique solution we propose an estimation method that is based on kernel smoothing and profiled likelihood we establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model we also discuss efficiency of both the parametric part and the nonparametric part we investigate the performance of our procedures on simulated data and on a sample of s p500 index returns we find evidence of asymmetric news impact functions consistent with the parametric analysis

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

a goodness of fit test for arch models a goodness of fit test in the class of conditional heteroscedastic time series models is examined due to the nonstandard limiting distribution of the test we propose to bootstrap the test showing its asymptotic validity moreover we illustrate the finite sample performance of the test by a small monte carlo study

channel coding and lossy source coding using a generator of constrained random numbers stochastic encoders for channel coding and lossy source coding are introduced with a rate close to the fundamental limits where the only restriction is that the channel input alphabet and the reproduction alphabet of the lossy source code are finite random numbers which satisfy a condition specified by a function and its value are used to construct stochastic encoders the proof of the theorems is based on the hash property of an ensemble of functions where the results are extended to general channels sources and alternative formulas are introduced for channel capacity and the rate distortion region since an ensemble of sparse matrices has a hash property we can construct a code by using sparse matrices

pseudo maximum likelihood estimation of arch models strong consistency and asymptotic normality of the gaussian pseudo maximum likelihood estimate of the parameters in a wide class of arch 1 processes are established we require the arch weights to decay at least hyperbolically with a faster rate needed for the central limit theorem than for the law of large numbers various rates are illustrated in examples of particular parameteriza tions in which our conditions are shown to be satis ed

a general resampling scheme for triangular arrays of alpha mixing random variables with application to the problem of spectral density estimation in 1989 kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the m dimensional joint distribution of stationary and alpha mixing observations the modification amounts to resampling whole blocks of consecutive observations or deleting whole blocks one at a time liu and singh independently proposed in 1988 the same technique for observations that are m dependent however many time series statistics notably estimators of the spectral density function involve parameters of the whole infinite dimensional joint distribution and hence do not fit in this framework in this report we generalize the moving blocks resampling scheme of kunsch and liu and singh a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution we then apply this result to the problem of estimation of the spectral density

coding theorems for a 2 2 threshold scheme with detectability of impersonation attacks coding theorems on a 2 2 threshold scheme with an opponent are discussed in an asymptotic setup where the opponent tries to impersonate one of the two participants a situation is considered where n secrets s n from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to s n with permitting negligible decoding error we introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a 2 2 threshold scheme that is secure against the impersonation attack by the opponent it is shown that if the correlation level between the two shares equals to ell geq 0 the minimum attainable rates coincide with h s ell where h s denotes the entropy of the source and the maximum attainable exponent of the success probability of the impersonation attack equals to ell it is also shown that a simple scheme using an ordinary 2 2 threshold scheme attains all the bounds as well

hankel operators that commute with second order differential operators abstract suppose that is a continuous and self adjoint hankel operator on l 2 0 with kernel x y and that l f d d x a x d f d x b x f x with a 0 0 if a and b are both quadratic hyperbolic or trigonometric functions and satisfies a suitable form of gauss s hypergeometric differential equation or the confluent hypergeometric equation then l l the paper catalogues the commuting pairs and l including important cases in random matrix theory there are also results proving rapid decay of the singular numbers of hankel integral operators with kernels that are analytic and of exponential decay in the right half plane

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

integrable operators and the squares of hankel operators integrable operators arise in random matrix theory where they describe the asymptotic eigenvalue distribution of large self adjoint random matrices from the generalized unitary ensembles this paper gives sufficient conditions for an integrable operator to be the square of a hankel operator and applies the condition to the airy associated laguerre modified bessel and whittaker functions

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field