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1.
In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function 2F1(a, b, c; z) for large b and c with c > b + 1 .  相似文献   

2.
The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F2(a,b,b,c,c;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F2(a,b,b,c,c;x,y) for large b, b, c and c. We also consider a double integral representation of the fourth Appell function F4(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F4(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.  相似文献   

3.
Expansions in terms of Bessel functions are considered of the Kummer function 1 F 1(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.  相似文献   

4.
We derive higher-order expansions of L-statistics of independent risks X 1, …,X n under conditions on the underlying distribution function F. The new results are applied to derive the asymptotic expansions of ratios of two kinds of risk measures, stop-loss premium and excess return on capital, respectively. Several examples and a Monte Carlo simulation study show the efficiency of our novel asymptotic expansions.  相似文献   

5.
6.
An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n?2, for the 0F0 function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.  相似文献   

7.
The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.  相似文献   

8.
Some properties of the Appell polynomials are studied and analyzed. Various formulas and expressions for the Appell quotient are derived and connection with asymptotic expansions is presented.  相似文献   

9.
New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.  相似文献   

10.
LetB be a real separable Banach space and letX,X 1,X 2,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS n =n ?1/2(X 1+...X n ). Let π:BR be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n ), α<p/2, for the distribution function of the ω-statistic, it is sufficient thatq is nontrivial, i.e., mes{t∈(0, 1):q(t)≠0}>0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).  相似文献   

11.
Given a suitable function Fn we define a class of estimators called asymptotic Fn-estimators (i.e., estimators which approximate the solution of Fn(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that Fn-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators).  相似文献   

12.
A transformation of the triple series T related to the GrassmanianG 2,4 into a series of the same structure type is obtained. This transformation generalizes the reduction formula of Gelfand, Graev, and Retakh taking the series T to the Gauss function under two additional conditions and two more general reduction formulas taking the series T to the Appell function F 1 and to the Horn function G 2 under one of the additional conditions. The approach used to analyze the series T is based on the representation of the initial series T in terms of series with convenient computational properties.  相似文献   

13.
Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.  相似文献   

14.
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli–Euler's one.  相似文献   

15.
We use a specialization of Ramanujan??s 1 ?? 1 summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell?CLerch sums and theta functions.  相似文献   

16.
An asymptotic expansion including error bounds is given for polynomials {P n, Qn} that are biorthogonal on the unit circle with respect to the weight function (1?e)α+β(1?e?iθ)α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2 F 1(a, b; c; z/b), asb→±∞,zb, withz=0 as “transition” point in the uniform expansion.  相似文献   

17.
In this paper we establish new asymptotic expansions for the nth Landau constant Gn. An open problem is indicated as well.  相似文献   

18.
For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.  相似文献   

19.
We establish a general analytic theory of asymptotic expansions of type 1 $$f(x) = a_1 \varphi _1 (x) + \cdots + a_n \varphi _n (x) + o(\varphi _n (x)) x \to x_0 ,$$ , where the given ordered n-tuple of real-valued functions (? 1, ..., ? n ) forms an asymptotic scale at x 0 ?? . By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that (*) hold true. Our theory is concerned with functions which are differentiable (n ? 1) or n times and the presented conditions involve integro-differential operators acting on f, ? 1, ..., ? n . We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach starts from simple geometric considerations and gives conditions expressed as the existence of finite limits, as x ?? x 0, of certain Wronskian determinants constructed with f, ? 1, ..., ? n . There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ? 1, ? 2 because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial expansions.  相似文献   

20.
This paper deals with asymptotic expansions for the non-null distributions of certain test statistics concerning a correlation matrix in a multivariate normal distribution. For this purpose an asymptotic expansion is given for the distribution of a function of the sample correlation matrix. As special cases of the resulting expansion, asymptotic expansions for the distributions of the sample correlation coefficient, Fisher's z-transformation and arcsine transformation are also given.  相似文献   

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