persistence of gaussian processes non summable correlations suppose the auto correlations of real valued centered gaussian process z cdot are non negative and decay as rho s t for some rho cdot regularly varying at infinity of order alpha in 1 0 with i rho t int 0 t rho s ds its primitive we show that the persistence probabilities decay rate of log mathbb p sup t in 0 t z t 0 is precisely of order t i rho t log i rho t thereby closing the gap between the lower and upper bounds of newell and rosenblatt ann math stat 33 1306 1313 1962 which stood as such for over fifty years we demonstrate its usefulness by sharpening recent results of sakagawa adv appl probab 47 146 163 2015 about the dependence on d of such persistence decay for the langevin dynamics of certain nabla phi interface models on mathbb z d

density solutions to a class of integro differential equations abstract we consider the integro differential equation i 0 f x m f on the half line we show that there exists a density solution which is then unique and can be expressed in terms of the beta distribution if and only if m these density solutions extend the class of generalized one sided stable distributions introduced in 29 and more recently investigated in 27 we study various analytical aspects of these densities and we solve the open problems about infinite divisibility formulated in 27

matrix orthogonal polynomials on the unit circle the properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas the techniques of scattering theory and banach algebras are used in the investigation a matrix generalization of a theorem of baxter is proved

some applications of the hermite matrix polynomials series expansions this paper deals with hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems properties of hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well known methods results are applied to compute accurate approximations of certain differential systems in terms of hermite matrix polynomials

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

on the hypergeometric matrix function abstract this paper deals with the study of the hypergeometric function with matrix arguments f a b c z conditions for matrices a b c so that the series representation of the hypergeometric function be convergent for z 1 and satisfies a matrix differential equation are given after the study of beta and gamma matrix functions an integral representation of f a b c z is obtained for the case where b c and c b are positive stable matrices with bc cb

first passage times of levy flights coexisting with subdiffusion we investigate both analytically and numerically the first passage time fpt problem in one dimension for anomalous diffusion processes in which l evy flights and subdiffusion coexist we analyze the fpt for three subclasses of l evy stable motions i symmetric l evy motions characterized by l evy index ensuremath mu 0l ensuremath mu l2 and skewness parameter ensuremath beta 0 ii one sided l evy motions with ensuremath mu 0l ensuremath mu l1 and skewness ensuremath beta 1 and iii two sided skewed l evy motions the extreme case 1l ensuremath mu l2 and skewness ensuremath beta ensuremath 1 in all three cases the waiting times between successive jumps are heavy tailed with index ensuremath alpha we show that in all three cases the fpt distributions are power laws our findings extend earlier studies on fpts of l evy flights by considering the interplay between long rests and the l evy long jumps

on semi markov processes and their kolmogorov s integro differential equations abstract semi markov processes are a generalization of markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution this paper provides an integro differential form of the kolmogorov s backward equations for a large class of homogeneous semi markov processes having the form of an abstract volterra integro differential equation an equivalent evolutionary differential form of the equations is also provided fractional equations in the time variable are a particular case of our analysis weak limits of semi markov processes are also considered and their corresponding integro differential kolmogorov s equations are identified

odd character correspondences in solvable groups abstract let g be a finite solvable group p be some prime let p be a sylow p subgroup of g and let n be its normalizer in g assume that n has odd order then we prove that there exists a bijection from the set of all irreducible characters of g of degree prime to p to the set of all the irreducible characters of degree prime to p of n such that it preserves the degree modulo p the field of values and the schur index over every field of characteristic zero this strengthens a more general recent result a turull character correspondences in solvable groups j algebra 295 2006 157 178 but only for the case under consideration here in addition we prove some other strong character correspondences that have very good rationality properties as one consequence we prove that a solvable group g has a non trivial rational irreducible character with degree prime to p if and only if the order of the normalizer of a sylow p subgroup of g has even order

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

biorthogonal matrix polynomials related to jacobi matrix polynomials in this paper we define the pair of biorthogonal matrix polynomials suggested by the jacobi matrix polynomials biorthogonality property matrix generating functions and matrix recurrence relations are given

jacobi matrix differential equation polynomial solutions and their properties in this paper jacobi matrix polynomials are introduced starting from the hypergeometric matrix function the differential equation satisfied by them is presented a rodrigues formula orthogonality and a three terms matrix recurrence relationship are then developed for jacobi matrix polynomials

mellin transforms and asymptotics harmonic sums this survey presents a unified and essentially self contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics discrete probabilistic models and the average case analysis of algorithms it relies on the mellin transform a close relative of the integral transforms of laplace and fourier the method applies to harmonic sums that are superpositions of rather arbitrary harmonics of a common base function its principle is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of the transformed function the main applications are in the area of digital data structures probabilistic algorithms and communication theory

weak convergence of positive self similar markov processes and overshoots of levy processes using lamperti s relationship between levy processes and positive self similar markov processes pssmp we study the weak convergence of the law x of a pssmp starting at x 0 in the skorohod space of cadlag paths when x tends to 0 to do so we first give conditions which allow us to construct a cadlag markov process x 0 starting from 0 which stays positive and verifies the scaling property then we establish necessary and sufficient conditions for the laws x to converge weakly to the law of x 0 as x goes to 0 in particular this answers a question raised by lamperti z wahrsch verw gebiete 22 1972 205 225 about the feller property for pssmp at x 0

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

extinction time of non markovian self similar processes persistence annihilation of jumps and the frechet distribution we start by providing an explicit characterization and analytical properties including the persistence phenomena of the distribution of the extinction time mathbb t of a class of non markovian self similar stochastic processes with two sided jumps that we introduce as a stochastic time change of markovian self similar processes for a suitably chosen time change we observe for classes with two sided jumps the following surprising facts on the one hand all the mathbb t s within a class have the same law which we identify in a simple form for all classes and reduces in the spectrally positive case to the frechet distribution on the other hand each of its distribution corresponds to the law of an extinction time of a single markov process without positive jumps leaving the interpretation that the time change has annihilated the effect of positive jumps the example of the non markovian processes associated to levy stable processes is detailed

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

law of the absorption time of some positive self similar markov processes let x be a spectrally negative self similar markov process with 0 as an absorbing state in this paper we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density we provide a power series and a contour integral representation of this density then by means of probabilistic arguments we deduce some interesting analytical properties satisfied by these functions which include for instance several types of hypergeometric functions we also give several characterizations of the kesten s constant appearing in the study of the asymptotic tail distribution of the absorbtion time we end the paper by detailing some known and new examples in particular we offer an alternative proof of the recent result obtained by bernyk dalang and peskir ann probab 36 2008 1777 1789 regarding the law of the maximum of spectrally positive levy stable processes

on a multivariable extension of jacobi matrix polynomials the classical jacobi matrix polynomials only for commutative matrices were first studied by defez et al e defez l jodar a law jacobi matrix differential equation polynomial solutions and their properties comput math appl 48 2004 789 803 the main aim of this paper is to construct a multivariable extension with the help of the classical jacobi matrix polynomials jmps generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived furthermore general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented

field of values and brauer characters of q degree abstract we study the existence of non trivial real valued respectively rational valued irreducible p brauer characters of degree not divisible by q

on generalized stable and related laws abstract a probabilistic characterization is given for schneider s generalized stable laws they belong to the larger family of laws which are invariant under length biasing followed by a random beta scaling questions of infinite divisibility and self decomposability are pursued moment determinacy of reciprocal generalized stable laws are investigated

log convex solutions to the functional equationf x 1 g x f x γ type functions abstract in this paper we discuss log convex solutionsf r r to the functional equation with initial condition given by f x 1 g x f x forx 0 andf 1 1 whereg r r our main result a generalization of the bohr mollerup artin classical characterization of the gamma function is that ifgis eventually log concave with the property that for eachw 0 g x w g x 1 asx then has a unique eventually log convex solution determined by the formula f x lim n g n g 1 g x n g n x g x forx 0 a functionfarising thus is called a type function two examples of which are the gamma function and theq gamma function q 0 r r to certain functional equations of the type f x f x 1 m f x m 1 m h x 1 the backdrop of type functions serves to place classical theory into perspective throughout the concepts of log convexity and log concavity play central roles

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

hitting times for gaussian processes this is the publisher s version also available electronically from http www jstor org stable 25450587 seq 1 page scan tab contents

uniform asymptotic expansions for the barnes double gamma function we develop uniform asymptotic approximations for the barnes double gamma function for various parameter values this involves the use of the method of matched asymptotic expansions to solve the defining functional difference equation in various asymptotic regions the resulting solutions provide extremely accurate approximations to the double gamma function we also give applications of our results to functional difference equations that arise in the theory of moving contact lines and acoustic wave diffraction