H.-K. Hwang, Th'eoremes limites pour les structures combinatoires et les fonctions arithmetiques, PhD these,  Ecole Polytechnique, Dec. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is completely the same as that of Theorem 10. We just have to take care of the additional analytic parameter u. Interestingly there is a strong relation to random variables that are asymptotically Gaussian. We state here a slightly modi ed version of a quite general theorem due to H. K. Hwang [10] that usually referred as the Quasi Power Theorem. Similar theorems can be found in [1, 2] Theorem 16. Let Xn be a random variable with the property that n a(u) b(u) holds uniformly in a complex neighborhoud of u = 1where n and n are sequences of positive real numbers with n ....

H.-K. Hwang, Th'eoremes limites pour les structures combinatoires et les fonctions arithmetiques, PhD these,  Ecole Polytechnique, Dec. 1994.

....(according to the saddlepoint of the integrand) and then evaluating the contribution of the integral. The underlying idea, which consists of expanding the integrand at the saddlepoint, is a rather fruitful one and has been applied in many di#erent contexts with satisfactory estimates (cf. [40, 37, 44, 22]) In addition to the two classical distributions, our methods can also be applied to the many existing Poisson and binomial variants, mixtures, and convolutions, cf. 25, Chaps. 3 and 4] and to other discrete distribution functions. Our techniques for deriving numerical bounds are also suitable ....

....lemma in Section 2.3) our expansions would hold in a wider range for the second parameter. This is so, for example, when f(z) 1 #(z) in the case of the Stirling numbers of the first kind (cf. 23, 14] A great deal of related combinatorial and arithmetical problems can be found in [22]. Integrals of the form 1 z m 1 L(#z) f(z)dz (# with L(z) j (j 1) arising in many combinatorial and arithmetic instances (cf. 22, Chaps. 6,10] can also be dealt with along the lines of this article, using known analytic properties of the modified Bessel functions. ....

H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole polytechnique, 1994.

....estimate for each of these distances. Our results are thus approximation theorems rather than limit theorems . The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]) Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall discuss some of these, including Halasz ....

....of a class of combinatorial structures. This line of investigation, initiated by Bender [13] in the early seventies and then continued by Bender, Canfield, Richmond, Williamson and Gao (cf. 20, 14, 15, 40] was recently further developed, most notably, by Flajolet and Soria [36, 37] and Hwang [53]. In particular, the uniformity provided by the powerful singularity analysis of Flajolet and Odlyzko [35] played an important role. In this section, we start with the bivariate generating functions of several di#erent types of the parameter number of components in partitional complex and ....

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Hwang, H.-K. (1994) Theoremes limites pour les structures combinatoires et les fonctions arithmetiques. These, Ecole polytechnique.

....1 n # 2 n, the first term A k is easily treated by singularity analysis (cf. 10] and the second term by the saddle point method (cf. 7, 14, 20, 21] From there a local limit theorem as above can be obtained. 13 5 Examples Let us consider some typical examples. More examples can be found in [17, 19] and the references cited there. Example 1. Connected components in random mappings. By random mapping (cf. 21] we mean a random single valued mapping of the set 1, 2, n into itself. Structurally, any such mapping can be viewed as a set (partitional complex) of connected components ....

....e #(n) v(r) e 2k (r) # for some polynomials e 2k (r) of r = log a. In this case, it is more convenient to work with the probability generating function n and apply Selberg s method (cf. 29] 30, Ch. II.6] which is a variant of the usual saddle point method; see also [16, 19] for details. 7 Conclusion The model that we developed in [17, 18, 19] and in this paper may be termed an analytic scheme for moment generating functions with which the similarity of the statistical properties of many apparently di#erent structures (like the number of cycles in permutations and ....

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H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole Polytechnique, 1994.

....of the di#erential equation above is the explicit form given in the theorem. Extracting coe#cients in exact form from there is quite di#cult. However, as Philippe Flajolet kindly pointed to us, asymptotic information and most notably, the limiting probability distribution can be established [8, 15]. In this case, it follows that A n converges in distribution (converges in law) to a Gaussian distribution, i.e. P # # A n 12 7 log n # 300 343 log n x # #= 1 # 2# # x # e t 2 2 dt O # 1 # log n # . This result follows from the asymptotic estimation for the average ....

....1 #(# 1) 2n O # 1 n 2 # # we get uniformly in the circle v 1 1 4 [z n ]A z (z, v) z n ] v 2 (1 2v)# (# 4v 3) 1 z) # 1) 2 O(n) v 2 (# 4v 3) 1 2v)##( # 1 2 ) n # 3 2 # 1 O # 1 # n # # . Applying the following quasi power theorem of Hwang [15, 7] leads immediately to the above given result. Theorem 7.3. Quasi power theorem [H. K. Hwang] Assume that the Laplace transforms # n (s) E # e sXn # of a sequence of random variables X n are analytic in a disc s #, for some # 0, and satisfy there an expansion of the form # n (s) e # ....

H.-K. Hwang. Theoremes limites pour les structures combinatoires et les fonctions arithmetiques. PhD thesis, Ecole Polytechnique, 1994.

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H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole Polytechnique, December, 1994.

....the Taylor expansion of f . The notation [u ]f(z, u) is then defined as [u ] z ]f(z, u) The Vinogradov symbol is used as a synonym for Landau s O( symbol. Gao and Richmond [15] established, also as a special case of their general results, a local limit theorem n . Hwang [26] studied a general exp(log) analytic scheme of which (2) is a special case. Many limit and approximation theorems n were also derived. For example, if m = log n x # log n, where x = o( # log n) then e 2 x log nV (x # log n) # 2# log n x , 4) uniformly in x, where V (t) ....

....of Poisson approximation to a class of discrete distributions, see [23] The result (5) states that the distribution n is well approximated by a Poisson distribution with mean H n . Since H n we have from (5) a central limit theorem n with a BerryEsseen error of order (log n) 1 2 (cf. [26]) Our purpose of this paper is to show that if instead of a Poisson approximation (5) we choose a Poisson negative binomial convolution approximation to the distribution n , then for almost all values n of interest, m we have an asymptotic expression with an error term essentially of ....

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H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole polytechnique, 1994.

....it is useful to establish general theorems from which one can conclude certain asymptotic results (especially, statistical properties) for parameters in the underlying structure by verifying some basic analytic properties of the generating function. Bender initiated this line of investigation; see [1, 2, 3, 7, 13, 14, 15, 19]. Applying singularity and probabilistic analysis on bivariate generating functions, Flajolet and Soria [13, 14] proved a series of central limit theorems for the parameter number of components in a wide class of combinatorial structures issuing principally from combinatorial constructions. ....

....exp log class covers such problems as cycles in permutations, cycles in random mappings and random mapping patterns, irreducibles in polynomials over finite fields (and, more generally, over an additive arithmetic semigroup under Knopfmacher s axiom A ) profiles of increasing trees, etc. See [13, 23, 19]. Recall that a generating function C(z) analytic at 0 is called logarithmic [13] if there exists a constant a 0, such that for z #, # 0 being the radius of convergence of C, C(z) a log 1 H(z) where the function H(z) is analytic in the region # : z : z # # # and arg(z ....

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H.-K. Hwang. Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole polytechnique, 1994.

....are proved by classical analytic methods in probability theory: Fourier inversion formula, Berry Essen inequality, Cramer s method and the saddle point method. These methods have been systematically applied to statistical parameters of combinatorial structures and number theoretic functions in [7] in which a rather general (Gaussian) scheme is studied. Although the problem considered here does not fit into that scheme, the methods used there do apply. It is of interest to note that the basic parameter, namely, x # Var # n , and the coe#cients in the results of this paper are all ....

H. K. Hwang, Theoremes limite pour les structures combinatoires et les fonctions arithmetiques, PhD Thesis, Ecole Polytechnique, 1994.

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H.-K. Hwang. Theoremes limites pour les structures combinatoires et les fonctions arithmetiques. These, Ecole polytechnique, (1994).

.... with Oppenheim s problem of factorisatio numerorum (the unordered counterpart of Kalmar s problem) having bivariate generating function m#1# n,m z zn s can be dealt with using the methods developed in [1, 13] and asymptotic properties of the modified Bessel functions; cf. [10, 14, 20, 23, 25]. In particular, we have a Bessel geometric convolution law for the quantity n,m (properly normalized) For related materials, see [3] and page 969 (fourth paragraph) of the Unsolved Problems column of The American Mathematical Monthly, volume 104, number 10 (1997) Notation. Throughout this ....

H.-K. Hwang, "Theoremes limites pour les structures combinatoires et les fonctions arithmetiques," Ph.D. Thesis, Ecole polytechnique, 1994.

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H.-K. Hwang. Theoremes limites pour les structures combinatoires et les fonctions arithmetiques. These, Ecole polytechnique, 1994.

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H.-K. Hwang, Theoremes limites pour les structures combinatoires et les fonctions arithmetiques, These, Ecole polytechnique, 1994.

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