In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.
A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.
Consider a Bayesian problem of success probability estimation in a series of conditionally independent trials with binary outcomes. We study the asymptotic behaviour of the weighted differential entropy for posterior probability density function conditional on x successes after n conditionally independent trials when n tends to infinity. Suppose that one is interested to know whether the coin is approximately fair with a high precision and for large n is interested in the true frequency. In other words, the statistical decision is particularly sensitive in small neighbourhood of the particular value γ = 1/2. For this aim the concept of weighted differential entropy used when the frequency γ is necessary to emphasize. It was found that the weight in suggested form does not change the asymptotic form of Shannon, Renyi, Tsallis and Fisher entropies, but changes the constants. The leading term in weighted Fisher Information is changed by some constant which depends on distance between the true frequency and the value we want to emphasize.
В работе предлагается метод синтеза усилительных схем из функциональных элементов (УСФЭ), позволяющий установить асимптотику функции Шеннона для обобщённой глубины УСФЭ – то есть глубины самой «плохой» функции алгебры логики, зависящей от заданных переменных – в специальном базисе (модели глубины), где глубина элемента определяется как его типом, так и степенью ветвления выхода в схеме. Асимптотическое поведение указанной функции Шеннона установлено с точностью до логарифмического по n слагаемого.
Asymptotic expansions of the null distribution of the MANOVA test statistics including the likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai tests are obtained when both the sample size and the dimension tend to infinity with assuming the ratio of the dimension and the sample size tends to a positive constant smaller than one. Cornish-Fisher expansions of the upper percent points are also obtained. In order to study the accuracy of the approximation formulas, some numerical experiments are done, with comparing to the classical expansions when only the sample size tends to innity.
For the stochastic differential equationdX(t)=a X ( t ) + b X ( t - 1 )dt+dW(t),t≥0,
the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter ϑ=(a,b) * . Eleven different cases are possible if ϑ runs through ℝ 2 . Let ϑ ^ T be the maximum likelihood estimator of ϑ based on (X(t), t≥T). Applications to the asymptotic behaviour of ϑ ^ T as T→∞ are given.
The paper is a survey of recent developments in the asymptotic theory of global fields and varieties over them. First, we give a detailed motivated introduction to the asymptotic theory of global fields which is already well shaped as a subject. Second, we treat in a more sketchy way the higher dimensional theory where much less is known and many new research directions are available.
Asymptotic solution for non-linear stage of suspension-colloidal flow in porous media is developed. The expansion is performed behind the concentration front; the zero order approximation coincides with the short-time solution of the linearised system. Using the first order approximation allows significantly enhancing the validity time period for the analytical model if compared with the linearised solution, allowing using the long-term experimental breakthrough concentration history for the model adjustment. Laboratory injection tests for three size colloids are performed. The asymptotic solution is used to tune the model parameters from the breakthrough histories of two size particles; good quality of matching is observed. The breakthrough concentration history for the third size particles is compared with the prediction by the adjusted model; good quality prediction is observed. The above serves for validation of the asymptotic method for the model tuning and prediction of non-linear suspension-colloidal flow in porous media.
We study the efficiency properties of the goodness-of-fit test based on the Q n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.
A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the tongues are asymptotically close to Bessel functions.
The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s > 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer--Siegel theorem. As an application we obtain a limit formula for Euler--Kronecker constants in families of number fields.
In many applications, the real high-dimensional data occupy only a very small part in the high dimensional ‘observation space’ whose intrinsic dimension is small. The most popular model of such data is Manifold model which assumes that the data lie on or near an unknown manifold (Data Manifold, DM) of lower dimensionality embedded in an ambient high-dimensional input space (Manifold Assumption about high-dimensional data). Manifold Learning is a Dimensionality Reduction problem under the Manifold assumption about the processed data, and its goal is to construct a low-dimensional parameterization of the DM (global low-dimensional coordinates on the DM) from a finite dataset sampled from the DM.
Manifold Assumption means that local neighborhood of each manifold point is equivalent to an area of low-dimensional Euclidean space. Because of this, most of Manifold Learning algorithms include two parts: ‘local part’ in which certain characteristics reflecting low-dimensional local structure of neighborhoods of all sample points are constructed via nonparametric estimation, and ‘global part’ in which global low-dimensional coordinates on the DM are constructed by solving the certain convex optimization problem for specific cost function depending on the local characteristics. Both statistical properties of ‘local part’ and its average over manifold are considered in the paper. The article is an extension of the paper (Yanovich, 2016) for the case of nonparametric estimation.
We consider fourth order ordinary differential operators on the half-line and on the line, where the perturbation has compactly supported coefficients. The Fredholm determinant for this operator is an analytic function in the whole complex plane without zero.We describe the determinant at zero.We show that in the generic case it has a pole of order 4 in the case of the line and of order 1 in the case of the half-line.