schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

some mu ntz orthogonal systems abstract let 0 1 be a given sequence of complex numbers such that i 1 and re i j 1 0 for each i and j for muntz polynomials p x i 0npix i and q x j 0mqjx j we define an inner product p q by p q 0 1 p q x d x x 2 where p q x i 0 n j 0 m p i q j x i j we obtain the muntz polynomials qn x orthogonal with respect to this inner product and connect them with the malmquist rational functions w z v 0 n 1 z 1 v v 0 n z v orthogonal on the unit circle with respect to the inner product u v 1 2 i 1 u z v z d z z several interesting properties of polynomials qn x are given

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

on the convergence of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles are studied we establish some convergence theorems for the orthogonal rational functions we generalize some of the results for szego polynomials to rational orthogonal functions

upper and lower class sequences for minimal uniform spacings letkn be thek th smallest spacing defined byx1 xn independent random variables uniformly distributed on 0 1 wherek 1 is a fixed integer letun be a sequence of positive numbers satisfyingun n2 we show that p n 2 k n leqq u n i o 0 1 when sum limits n u n k n infty infty i begin gathered if in addition u n uparrow then p n 2 k n geqq u n i o 0 1 when hfill sum limits n u n k n exp u n infty infty hfill end gathered ii

szego functions and multipoint pade approximation abstract rational functions analogous to szego polynomials are introduced and their application in multipoint pade approximation is discussed

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

orthogonal rational functions with poles on the unit circle let n be a sequence of not necessarily distinct points on the unit circle t z c z 1 set ln span 1 1 1 1 n l n 0ln where we have used the notation n nk 1 z k let m be a positive linear functional defined on the space l l with m r real for functions that are real on t define r s m r z s 1 z for r s l in particular if m is given as m r r ei d for some measure then r s r ei s ei d let the orthogonal system n be obtained from 1 n by orthogonalization three term recurrence relations quadrature formulas moment theory and interpolation properties connected with the functional m and the system n are discussed

analysis of switched diversity systems on generalized fading channels the performances of switched diversity systems operating on generalized nakagami fading channels are analyzed using a discrete time model the average bit error rate ber of binary noncoherent frequency shift keying ncfsk on slow nonselective nakagami fading channels is derived closed form expressions that can be used to determine optimum switching thresholds in a minimum error rate sense are also derived in addition the use of optimum fixed thresholds is considered it is found that a considerable amount of diversity gain can be obtained using an optimum fixed rather than adaptive switching threshold results are obtained for both independent and correlated nakagami fading branch signals the effects of fading severity and the correlation coefficient on both the ber and on the optimum switching threshold are investigated it is shown that useful diversity gain can be obtained with power correlation coefficients as high as 0 9 when the fading is strong the results for a rayleigh channel are obtained and presented as a special case of generalized fading model

unified analysis of switched diversity systems in independent and correlated fading channels the moment generating function mgf of the signal power at the output of dual branch switch and stay selection diversity ssd combiners is derived the first order derivative of the mgf with respect to the switching threshold is also derived these expressions are obtained for the general case of correlated fading and nonidentical diversity branches and hold for any common fading distributions e g rayleigh nakagami m rician nakagami q the mgf yields the performance bit or symbol error probability of a broad class of coherent differentially coherent and noncoherent digital modulation formats with ssd reception the optimum switching threshold in a minimum error rate sense is obtained by solving a nonlinear equation which is formed by using the first order derivative of the mgf this nonlinear equation can be simplified for several special cases for independent and identically distributed diversity branches the optimal switching threshold in closed form is derived for three generic forms of the conditional error probability for correlated rayleigh or nakagami m fading with identical branches the optimal switching threshold in closed form is derived for the noncoherent binary modulation formats we show previously published results as special cases of our unified expression selected numerical examples are presented and discussed

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

on limiting distributions of intermediate order statistics from stationary sequences abstract the problem of finding the asymptotic distribution of the maximum term from a stationary dependent sequence of random variables r v s has been extensively investigated in the literature of particular interest are the cases in which the concept of approximate independence is formulated mathematically in terms of conditions such as strong mixing or for normal sequences conditions on the rate of decay of the covariances our objective in this paper is to obtain analogous results for so called intermediate order statistics a sequence is given of intermediate order statistics and intermediate rank sequence these conditions parallel those used to obtain the corresponding result in the extreme order statistic problem a primary difference being that certain more rapid mixing rates have to be assumed using our procedure it is convenient to deal directly with an appropriate level exceedance problem and to regard that of asymptotic distributions as a specialization

convergence of orthogonal rational functions outside of the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions outside of the unit circle

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

performance of two dual branch postdetection switch and stay combining schemes in correlated rayleigh and rician fading the performance of binary frequency shift keying bfsk and m ary frequency shift keying mfsk with dual branch postdetection switch and stay combining ssc in correlated rayleigh and rician fading is studied two postdetection ssc receivers are considered and the performances of noncoherent bfsk and mfsk are analyzed closed form expressions are derived for the average bit error rate ber of bfsk and mfsk with postdetection ssc in correlated rayleigh and rician fading optimum switching thresholds that minimize the average ber are obtained extensive monte carlo simulations are presented to test the validity of the analytical results the performance of dual branch postdetection ssc is compared with the performance of dual branch predetection ssc the effects of correlation fading parameter average fading power imbalance and switching threshold on the performances of postdetection ssc receivers are examined

a strong ergodic theorem for extreme and intermediate order statistics abstract we study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences first we provide sufficient dependence conditions under which these order statistics converges almost surely to the left or right endpoint of the population support as in the classical setup of sequences of independent and identically distributed random variables next we derive a generalization of this result valid in the class of all strictly stationary sequences for this purpose we introduce notions of conditional left and right endpoints of the support of a random variable given a sigma field and present basic properties of these concepts using these new notions we prove that extreme and intermediate order statistics from any discrete time strictly stationary process converges almost surely to some random variable we describe the distribution of the limiting variate thus we establish a strong ergodic theorem for extreme and intermediate order statistics

signaling constellations for fading channels the performance of various coherent 8 ary and 16 ary modulations in additive white gaussian noise awgn and slowly fading channels are analyzed new expressions for the exact symbol error rates ser in fading with diversity combining are derived for any two dimensional signaling format having polygonal decision boundaries maximal ratio combining equal gain combining and selection combining are considered the ser formulas obtained make it possible for the first time to optimize parameters of various constellations precisely and to determine which constellation has the lowest probability of error for example a star constellation such as that specified in the ccitt v 29 standard can be improved by adjusting the amplitude ratios of the points in the constellation to save about 0 63 db power in awgn without sacrificing the phase error tolerance while maintaining the same error rate the sensitivity of each constellation to phase error is presented and comparisons are made six 8 ary signal sets and 11 16 ary signal sets are examined using the new symbol error probability formulas to determine best signal sets for fading channels

a rational stieltjes moment problem let a k be a sequence of points on the real axis and let l denote the linear span of the set1 w 0 1 w 1 1 w n where w 0 1 w n z a 1 z a 2 z a n forn 1 a measure m with support in an interval c and with respect to which all functions in l l are integrable induces an inner product in l and hence orthogonal rational functions f n and associated functions s n corresponding to the sequence 1 w n assuming certain constellations of the points a k it is shown that the sequences s 2 m z f 2 m z and s 2 m 1 z f 2 m 1 z are monotonic on a certain real interval a b and convergent in the complex plane outside the interval c to functions f z m 0 and f z m where f z m denotes the stieltjes transform d m t t z of a measure m furthermore for every m giving rise to the same integral values for functions in l l the inequality f x m 0

rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle abstract in this paper multipoint rational approximants to the riesz herglotz transform of a borel measure supported on are considered we give estimates for the rate of convergence of these approximants these estimates are obtained from the asymptotic behaviour of a sequence of rational functions orthogonal with respect to and having prescribed poles 1 k where k 1 is a sequence of complex numbers compactly contained in the open unit disk as an application we also give the rate of convergence of rational szego quadrature formulas for integrals with respect to

performance analysis of multibranch switched diversity systems we investigate the performance of multibranch switched diversity systems specifically we first derive generic formulas for the cumulative distribution function probability density function and moment generating function of the combined signal power for both switch and stay combining ssc and switch and examine combining sec schemes we then capitalize on these expressions to obtain closed form expressions for the outage probability and average error rate for various practical communication scenarios of interest as a byproduct of our analysis we prove that for ssc with identically distributed and uniformly correlated branches increasing the number of branches to more than two does not improve the performance but the performance can be different in the case the branches are not identically distributed and or not uniformly correlated we also show that in general the sec performance improves with additional branches the mathematical formalism is illustrated with a number of selected numerical examples

a general method for calculating error probabilities over fading channels signal fading is a ubiquitous problem in mobile and wireless communications in digital systems fading results in bit errors and evaluating the average error rate under fairly general fading models and multichannel reception is often required predominantly to date most researchers perform the averaging using the probability density function method or the moment generating function mgf method this paper presents a third method called the characteristic function chf method for calculating the average error rates and outage performance of a broad class of coherent differentially coherent and noncoherent communication systems with or without diversity reception in a myriad of fading environments unlike the mgf technique the proposed chf method based on parseval s theorem enables us to unify the average error rate analysis of different modulation formats and all commonly used predetection diversity techniques i e maximal ratio combining equal gain combining selection diversity and switched diversity within a single common framework the chf method also lends itself to the averaging of the conditional error probability involving the complementary incomplete gamma function and the confluent hypergeometric function over fading amplitudes which heretofore resisted to a simple form as an aside we show some previous results as special cases of our unified framework

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

orthogonal rational functions and tridiagonal matrices we study the recurrence relation for rational functions whose poles are in a prescribed sequence of numbers that are real or infinite and that are orthogonal with respect to an hermitian positive linear functional we especially discuss the interplay between finite and infinite poles the recurrence relation will also be described in terms of a tridiagonal matrix which is a generalization of the jacobi matrix of the polynomial situation which corresponds to placing all the poles at infinity this matrix not only describes the recurrence relation but it can be used to give a determinant expression for the orthogonal rational functions and it also allows for the formulation of a generalized eigenvalue problem whose eigenvalues are the zeros of an orthogonal rational function these nodes can be used in rational gauss type quadrature formulas and the corresponding weights can be obtained from the first components of the corresponding eigenvectors

infinite series representations of the bivariate rayleigh and nakagami m distributions despite the usefulness of the bivariate correlated nakagami m cumulative distribution function cdf in communications systems analysis an infinite series for the computation of this function is not known in this letter an infinite series representation of the bivariate cdf is derived bounds on the error resulting from truncation of the infinite series are also derived

a density problem for orthogonal rational functions abstract let n n 1 be a sequence of points in the open unit disk in the complex plane and let b 0 1 and b n z k 0 n k k k z 1 k z n 1 2 k k 1 when k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product on l l find a nondecreasing function on or a positive borel measure on such that f g f e i g e i d for f g l we give a necessary and sufficient condition called n extremality on a solution of the moment problem in order that l is dense in l2

solutions of the strong hamburger moment problem abstract the strong hamburger moment problem for a bi infinite sequence c n n 0 1 2 can be described as follows 1 find conditions for the existence of a positive measure on such that c n t n d t for all n 2 when there is a solution find conditions for uniqueness of the solution 3 when there is more than one solution describe the family of all solutions in this paper a theory concerning question 3 is developed in particular an analog to the nevanlinna parametrization describing the solutions of the classical hamburger moment problem is given

limit theory for the sample autocorrelations and extremes of a garch 1 1 process the asymptotic theory for the sample autocorrelations and extremes of a garch i 1 process is provided special attention is given to the case when the sum of the arch and garch parameters is close to 1 that is when one is close to an infinite variance marginal distribution this situation has been observed for various financial log return series and led to the introduction of the igarch model in such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values and the sample autocorrelations of the squared time series have nondegenerate limit distributions we discuss the consequences for a foreign exchange rate series

external theory for stochastic processes abstract the purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes the theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts emphasis is placed on stationary cases but other important classes e g mark of sequences are included significant ideas and methods are described rather than details and in particular the nature and role of important underlying point processes such as exceedances and upcrossings are emphasized applications are given to particular classes of process e g normal moving average and connections with related theory such as convergence of sums are indicated

convergence of modified approximants associated with orthogonal rational functions abstract let n be a sequence in the unit disk d z c z consisting of a finite number of points cyclically repeated and let l be the linear space generated by the functions b n z k 0 n k z k k 1 k z let n z be orthogonal rational functions obtained from the sequence bn z orthogonalization with respect to a given functional on l and let n z be the corresponding functions of the second kind with superstar transforms n z and n z respectively interpolation and convergence properties of the modified approximants r n z u n v n u n n z v n n z u n n z v n n z that satisfy u n v n are discussed

strong and weak convergence of orthogonal systems of rational functions on the unit circle abstract pan k strong and weak convergence of orthogonal systems of rational functions on the unit circle journal of computational and applied mathematics 46 1993 427 436 we study weak and strong convergence of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk the erd s measures have been characterized as well

asymptotic behavior of bivariate gaussian powered extremes abstract in this paper joint asymptotics of powered maxima for a triangular array of bivariate gaussian random vectors are considered under the husler reiss condition limiting distributions of powered maxima are derived furthermore the second order expansions of the joint distributions of powered maxima are established under the refined husler reiss condition

infinite series representations of the trivariate and quadrivariate rayleigh distribution and their applications few theoretical results are known about the joint distribution of three or more arbitrarily correlated rayleigh random variables rvs consequently theoretical performance results are unknown for three and four branch equal gain combining egc selection combining sc and generalized sc gsc in correlated rayleigh fading this paper redresses this gap by deriving new infinite series representations for the joint probability density function pdf and the joint cumulative distribution function cdf of three and four correlated rayleigh rvs bounds on the error resulting from truncating the infinite series are derived a classical approach due to miller is used to derive our results unfortunately miller s approach cannot be extended to more than four variates and in fact the quadrivariate case considered in this paper appears to be the most general result possible for brevity we treat only a limited number of applications in this paper the new pdf and cdf expressions are used to derive the outage probability of three branch sc the moments of the egc output signal to noise ratio snr and the moment generating function of the gsc 2 3 output snr in arbitrarily correlated rayleigh fading a novel application of bonferroni s inequalities allows new outage bounds for multibranch sc in arbitrarily correlated rayleigh channels

dual diversity over correlated log normal fading channels a great deal of attention has been devoted in the literature to studying the bit error rate ber performance of diversity combining systems in the presence of rayleigh rice and nagakami m fading by comparison the literature is relatively sparse in comparable analyses over log normal channels which typically characterize shadowing from indoor obstacles and moving human bodies one reason for this disparity stems from the difficulty in evaluating the exact average ber when log normal variates are involved using for example the moment generating function mgf approach due to the inability of expressing the mgf itself in a simple closed form since it is possible to evaluate the marginal and joint statistical moments as well as the cumulative distribution function cdf associated with a log normal distribution in closed form we rather focus here on other performance measures namely average combined output signal to noise ratio amount of fading and outage probability the first two performance measures depend only on the moments whereas the outage probability depends solely on the cdf closed form expressions in terms of known functions single integral representations or upper and lower bounds are obtained for these measures corresponding to maximal ratio combining selection combining and switch and stay combining schemes allowing for the possibility of correlation between the two branches numerical evaluations of these expressions illustrating the performances of each individual diversity type as well as comparisons among them are also presented

holomorphic functions associated with indeterminate rational moment problems we consider indeterminate rational moment problems on the real line with their associated orthogonal rational functions there exists a nevanlinna type parametrization relating to the problem with associated nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem we prove that the growth at the essential singularities of the four functions in the nevanlinna matrix is the same

quadratures associated with pseudo orthogonal rational functions on the real half line with poles in 0 we consider a positive measure on 0 and a sequence of nested spaces 0 1 2 of rational functions with prescribed poles in 0 let f k k 0 with f 0 0 and f k k k 1 k 1 2 be the associated sequence of orthogonal rational functions the zeros of f n can be used as the nodes of a rational gauss quadrature formula that is exact for all functions in n n 1 a space of dimension 2n quasi and pseudo orthogonal functions are functions in n that are orthogonal to some subspace of n 1 both of them are generated from f n and f n 1 and depend on a real parameter t their zeros can be used as the nodes of a rational gauss radau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of n n 1 where the quadrature is exact the parameter t is used to fix a node at a preassigned point the space where the quadratures are exact has dimension 2n 1 in both cases but it is in n 1 n 1 in the quasi orthogonal case and it is in n n 2 in the pseudo orthogonal case

unique solvability of an extended hamburger moment problem abstract r functions are rational functions with no poles in the extended complex plane outside a given set a1 ap of points on the real axis methods from the theory of orthogonal polynomials can be extended to r functions by this means the author solved an extended hamburger moment problem given sequences of real numbers cn i n 1 2 i 1 p find conditions for a distribution function to exist such that d t 1 d t t a i m c m i m 1 2 i 1 p in this paper these methods are extended to treat conditions for the moment problem to have a unique solution the results are related to the classical limit point limit circle situation

strong and weak convergence of orthogonal polynomials le cercle unite fonctionnelles continues sur des mesures resultats principaux sur le cercle unite corollaires de convergence faible les parties absolument continues et les parties singulieres d une mesure integrabilite uniforme et inegalite de turan series orthogonales avec trous la ligne reelle convergence forte convergence faible convergence forte pour des moyennes arithmetiques

on the ill conditioning of locating transmission zeros in least squares arma filtering abstract the algorithms of levinson schur and nevanlinna pick are briefly reviewed both produce least squares predictive filters by minimizing the least squares error with respect to the interpolation points of the nevanlinna pick algorithm we find the transmission zeros of an arma filter it is shown by some simple examples that this is an ill conditioned problem

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

on decorrelation in dual branch diversity systems the performance of a dual branch decorrelator receiver operating in correlated rayleigh and rician fading channels in conjunction with selection combining sc and switch and stay combining ssc is analyzed and compared to the performance of a conventional sc and ssc receiver for each combining scheme analytical expressions for the average bit error rate ber of several modulation techniques of practical interest the mean output signal to noise ratio snr and the outage probability are derived it is shown that the decorrelator receiver has superior performance over the conventional receiver by as much as 2 83 db in average snr the effects of modulation order correlation coefficient and the severity of fading on the performance of the decorrelator receiver are studied

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

extensions of szego s theory of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions when the measure does not satisfy the szego condition

determinacy of a rational moment problem we discuss a moment problem of stieltjes type that is related to the theory of orthogonal rational functions we obtain results which lead to a sufficient condition of carleman type for determinacy of the moment problem

a universal one sided law of the iterated logarithm we prove that the lim inf of suitably normalized sums of i i d nonnegative and nondegenerate random variables can with probability 1 only be a constant between 2 1 2 and 0 moreover we show that each value within this range is attainable by an appropriate choice of the underlying common distribution function

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2