Asymptotic expansions of the mean values of DirichletL-functions III

Asymptotic formulas for certain mean square of DirichletL-functions and their derivatives are considered. The main tool of our proof is a suitably modified delicate lemma of F. V. Atkinson (Lemma 2.3) which asymptotically evaluates certain exponential integrals. Partially supported by Grant-in-Aid for Scientific Research (No. 03740051), Ministry of Education, Science and Culture     

Asymptotic expansions in mean and covariance structure analysis

Asymptotic expansions of the distributions of parameter estimators in mean and covariance structures are derived. The parameters may be common to, or specific in means and covariances of observable variables. The means are possibly structured by the common/specific parameters. First, the distributions of the parameter estimators standardized by the population asymptotic standard errors are expanded using the single- and the two-term Edgeworth expansions. In practice, the pivotal statistic or the Studentized estimator with the asymptotically distribution-free standard error is of interest. An asymptotic distribution of the pivotal statistic is also derived by the Cornish-Fisher expansion. Simulations are performed for a factor analysis model with nonzero factor means to see the accuracy of the asymptotic expansions in finite samples.     

The mean square of the DirichletL-functions in the critical strip

We study an asymptotic formula of the DirichletL-functions in the critical strip. This is an analogy of the Atkinson-type formula for DirichletL-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 201–213, April–June, 2000.     

Asymptotic expansions of the Robbins-Monro process

Assume that the function values f(x) of an unknown regression function f: ℝ → ℝ can be observed with some random error V. To estimate the zero ϑ of f, Robbins and Monro suggested to run the recursion X _n+1 = X _n − a/n Y _n with Y _n = f(X _n ) − V _n . Under regularity assumptions, the normalized Robbins-Monro process, given by (X _n+1 − ϑ)/√Var(X _n+1, is asymptotically standard normal. In this paper Edgeworth expansions are presented which provide approximations of the distribution function up to an error of order o(1/√n) or even o(1/n). As corollaries asymptotic confidence intervals for the unknown parameter ϑ are obtained with coverage probability errors of order O(1/n). Further results concern Cornish-Fisher expansions of the quantile function, an Edgeworth correction of the distribution function and a stochastic expansion in terms of a bivariate polynomial in 1/√n and a standard normal random variable. The proofs of this paper heavily rely on recently published results on Edgeworth expansions for approximations of the Robbins-Monro process.      

Asymptotic expansions of the binomial coefficients

A general formulae for the asymptotic expansion of not centered binomial coefficients are derived and some useful estimates of the binomial coefficients are presented. The sum of the binomial coefficients is also studied.     

Asymptotic expansions of the Mehler-Fock transform

Asymptotic expansions in the two limitsx → + ∞ andx → 0+ are obtained for the Mehler-Fock transform

Asymptotic expansions using blow-up

The method of matched asymptotic expansions and geometric singular perturbation theory are the most common and successful approaches to singular perturbation problems. In this work we establish a connection between the two approaches in the context of the simple fold problem. Using the blow-up technique [5], [12] and the tools of geometric singular perturbation theory we derive asymptotic expansions of slow manifolds continued beyond the fold point. Our analysis explains the structure of the expansion and gives an algorithm for computing its coefficients.     

Asymptotic expansions using blow-up

The method of matched asymptotic expansions and geometric singular perturbation theory are the most common and successful approaches to singular perturbation problems. In this work we establish a connection between the two approaches in the context of the simple fold problem. Using the blow-up technique [5], [12] and the tools of geometric singular perturbation theory we derive asymptotic expansions of slow manifolds continued beyond the fold point. Our analysis explains the structure of the expansion and gives an algorithm for computing its coefficients.*Research supported by the Austrian Science Foundation under grant Y 42-MAT.Received: February 1, 2001; revised: November 22, 2002