Uniform asymptotic formulae for arrays of complex numbers of the form (f, (r), (s)), with r and s nonnegative integers, are provided as r and s converge to infinity at a comparable rate. Our analysis is restricted to ...
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Uniform asymptotic formulae for arrays of complex numbers of the form (f, (r), (s)), with r and s nonnegative integers, are provided as r and s converge to infinity at a comparable rate. Our analysis is restricted to the case in which the generating function F( z, w) := Sigma f(r), (s)z(r)w(s) is meromorphic in a neighborhood of the origin. We provide uniform asymptotic formulae for the coe. cients f(r), (s) along directions in the ( r, s)- lattice determined by regular points of the singular variety of F. Our main result derives from the analysis of a one dimensional parameter- varying integral describing the asymptotic behavior of f(r), (s). We specifically consider the case in which the phase term of this integral has a unique stationary point;however, we allow the possibility that one or more stationary points of the amplitude term coalesce with this. Our results. nd direct application in certain problems associated to the Lagrange inversion formula as well as bivariate generating functions of the form v( z)/( 1 - w . u( z)).
This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field F-q. The usual bit-complexity is ...
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This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field F-q. The usual bit-complexity is defined with respect to the degree of the quotients;we focus here on a notion of 'fine' complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions);some of our costs are even proved to satisfy an asymptotic Gaussian law. We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers 'step by step' each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field. (C) 2014 Elsevier Inc. All rights reserved.
In this paper a unified theory for studying renewal processes in two dimensions is developed. bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theor...
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In this paper a unified theory for studying renewal processes in two dimensions is developed. bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.
This thesis is composed of five chapters, regarding several models for dependence in stochastic processes. We first discuss the class L of selfdecomposable laws, which is a subclass of the class of infinitely divisibl...
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This thesis is composed of five chapters, regarding several models for dependence in stochastic processes. We first discuss the class L of selfdecomposable laws, which is a subclass of the class of infinitely divisible laws and contains all stable laws. We show an example of selfdecomposable law whose selfdecomposability is related to path decomposition of planar Brownian motions. Then we introduce the family of self-similar additive processes, which is known to have a close relationship with the class L of selfdecomposable laws. The discussion is suggested by the scale invariant Poisson spacings theorem, which arose in various contexts including records, extremal processes and random permutations. We are able to show that the range of a self-similar gamma process is a scale invariant Poisson point process (θx−1dx) and also conversely, this distribution of the range characterizes the gamma process among all self-similar additive processes. We then turn to a discussion of counting processes in discrete times. In particular, when the counting process is stationary 1-dependent, its distribution is determined by the bivariate probability generating function in terms of run probability generatingfunctions. A probabilistic explanation is provided, alongside with comparison to other known encodings including the determinantal representation and a combinatorial enumeration formula. We also compare the bivariategenerating function for 1-dependent sequences with similar generatingfunctions derived from other dependence structures. Lastly, we discuss a positivity problem related to a bivariate probability generating function for renewal processes, allowing signed measures. Fascinating graphs and qualitative observational results are provided, as well as natural but challenging open problems to explain these facts.
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