关于位数函数高次均值的计算

针对1993年美国数论专家Smarandache提出了初等数论及集合论中的105个未解决的问题中的5个关于自然数列的性质问题,就自然数列的位数函数问题进行了研究,给出了在一个正整数的n进制表示中的位数函数定义,采用了归纳、猜想的方法得出了位数函数a（m,n）的高次均值的精确计算公式.     

Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

Let {X(t)}_t∈R be a stationary increments Gaussian process satisfying some assumptions. By using the notion of generalized quadratic variation we build a strongly consistent and asymptotically normal estimator of the uniform Hölder exponent of X, over a compact interval. Our estimator is obtained starting from average values of the process over a regular grid.     

Representation of asymptotic values for nonexpansive stochastic control systems

In ergodic stochastic problems the limit of the value function $V_{\lambda }$ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor $\lambda$ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function $\lambda V_{\lambda }$ converges uniformly to a constant as $\lambda \to 0$. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also $V_{\lambda }$ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of $\lambda V_{\lambda }$, as $\lambda \to 0$.     

Conditions for the convergence in distribution of maxima of stationary normal processes

The asymptotic distribution of the maximum M_n=max_1?t?nξ_t in a stationary normal sequence ξ₁,ξ,… depends on the correlation r_t between ξ₀ and ξ_t. It is well known that if r_t log t → 0 as t → ∞ or if Σr²_t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.     

On a class of asymptotically stationary harmonizable processes

We prove that every harmonizable process with σ-finite bimeasure is asymptotically stationary and we give its associated spectral measure.     

On the asymptotic equilibrium of a population system with migration

In the analysis of economic and social issues of a country (or any larger or smaller socio-economic unit) the demographic dynamics of the considered population often play a crucial role. Very current emergencies in this respect are e.g. ageing, longevity risk, state-run healthcare etc. Over the last decade migration between EU countries also became an important issue, and in recent years the uncontrolled migration from non-EU countries is also a major concern. Therefore, the better theoretical understanding of the evolutionary mechanism of age-classified populations interacting via migration, is a timely modelling-methodological task. This paper is a preliminary demographic methodological contribution to a further research in support of socio-economic modelling and decision making concerning migration issues.It is known that in the framework of the classical age-specific Leslie model, under simple demographic conditions, a closed population in the long term tends to an equilibrium age distribution. As the main theoretical result of the paper, a similar convergence is proved for a system of several populations with migration between them, and this long-term behaviour (convergence theorem) is extended to systems of sex-structured populations. Based on the latter model, medium term projections are also analysed concerning the effect of migration among countries on the development of the old-age dependency ratio (the proportion of pensioner age classes to active ones), which is an aggregate scalar indicator of ageing, a major concern in most industrialized countries. Illustrative simulation analysis is carried out with data from three European countries.     

Extremes of Normed Empirical Moment Generating Function Processes

We present a method for deriving the limiting distribution of the maximum of a normed empirical moment generating function process indexed by one parameter. We first extend slightly the results of Csörg et al. (1986b) to provide the rate of convergence for a Gaussian approximation to a non-Donsker empirical process. In cases we consider, the maximum tends to infinity in probability, but when appropriately scaled has a limiting Gumbel extreme value distribution.AMS 2000 Subject Classification Primary—62E20, 60G70*Author for correspondence: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia.     

On the transient behavior of the processor sharing queue

We analyze the transient behavior of the single server queue under the processor sharing discipline. Under fairly general assumptions, we give the rate of growth of the number of customers in the queue as well as the asymptotic behavior of the residual service times described in terms of a renormalized point process.     

On the exceedance point process for a stationary sequence

Summary It is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.This research has been supported by the Air Force Office of Scientific Research Grant No. F 49620 85 C 0144 and the Katholieke Universiteit Leuven     

Non-parametric estimation of the density of a point process

We consider the Parzen—Rosenblatt type estimator of the density of a point process. Sufficient conditions for convergence of this estimator are established. These results extend those obtained for a random sample.     

Convergence in distribution of point processes on Polish spaces to a simple limit

Let ξ,ξ₁,ξ₂,… be a sequence of point processes on a complete and separable metric space (S,d) with ξ simple. We assume that P{ξ_nB=0}→P{ξB=0} and lim sup_n→∞P{ξ_nB>1}≤P{ξB>1} for all B in some suitable class B, and show that this assumption determines if the sequence {ξ_n} converges in distribution to ξ. This is an extension to general Polish spaces of the weak convergence theory for point processes on locally compact Polish spaces found in Kallenberg (1996).     

On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes

Let {X(t),t≥0}$\{X\left(t\right),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004)  [23], which we refer to as Piterbarg’s max-discretisation theorem gives the joint asymptotic behaviour (T→∞$T\to \infty$) of the continuous time maximum M(T)=_maxt∈[0,T]X(t)$M\left(T\right)={max}_{t\in [0,T]}X\left(t\right)$, and the maximum M^δ(T)=_maxt∈R(δ)X(t)$M^{\delta }\left(T\right)={max}_{t\in \mathfrak{R}\left(\delta \right)}X\left(t\right)$, with R(δ)⊂[0,T]$\mathfrak{R}\left(\delta \right)\subset [0,T]$ a uniform grid of points of distance δ=δ(T)$\delta =\delta \left(T\right)$. Under some asymptotic restrictions on the correlation function Piterbarg’s max-discretisation theorem shows that for the limit result it is important to know the speed δ(T)$\delta \left(T\right)$ approaches 0 as T→∞$T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.     

The asymptotic form of the stationary separated circumfluence of a body at high reynolds numbers

An asymptotic theory of the stationary separated circumfluence of bodies at high Reynolds numbers, Re, is constructed. It is shown that the length and width of the separated zone (SZ) is proportional to Re and that the drag cofficient is proportional to Re⁻¹. A cyclic boundary layer is located around the separated zone with a constant vorticity. In the scale of the body, the flow tends towards a Kirchhoff flow with a velocity on a free line of flow of the order of Re^−1/2 which satisfies the Brillouin-Villat condition.

A review of the attempts which have been made to describe the two-dimensional separated circumfluence of a body at high Reynolds numbers is given in /1, 2/. Certain features of the asmyptotic structure of the solution based on qualitative arguments were pointed out in /3, 4/. The corresponding shape of the separated zone was calculated in /5/. However, no complete theory was constructed in these papers. The appearance of the numerical calculations in /6, 7/ stimulated further investigations and a model with a non-zero jump in the Bernoulli constant on the boundary of the separated zone was proposed in /8/. A number of hypotheses concerning the limiting structure of the flow were put forward in /9/.

In the solution obtained below the flow in the scale of the body is described as in /1, 2/ but the velocity is of the order of Re^−1/2. The flow characteristics in this zone are correspondingly renormalized. The flow in the scale of the separated zone corresponds to the assumptions made in /3, 4/. Unlike in /1–4/, the flow in the scale of the body is not directly combined with the flow in the scale of the separated zone. There are several embedded zones and the possibility of uniting these ensures the selfconsistency of the expansion. Moreover, the cyclic boundary layer on the boundary of the separated zone plays an important role.     

On asymptotic optimization of a class of nonlinear stochastic hybrid systems

We consider the problem of control for continuous time stochastic hybrid systems in finite time horizon. The systems considered are nonlinear: the state evolution is a nonlinear function of both the control and the state. The control parameters change at discrete times according to an underlying controlled Markov chain which has finite state and action spaces. The objective is to design a controller which would minimize an expected nonlinear cost of the state trajectory. We show using an averaging procedure, that the above minimization problem can be approximated by the solution of some deterministic optimal control problem. This paper generalizes our previous results obtained for systems whose state evolution is linear in the control.This work is supported by the Australian Research Council. All correspondence should be directed to the first author.     

A representation for the posterior distribution of the location of a moving target

A target moves according to a continuous stochastic process in Euclidean R_^H. A search is conducted by choosing a search strategy and by observing the state of a detection process. Methods for representing the posterior distributions for target location given no detection are discussed. In particular, methods for parameterizing Gaussian models for target motion and the transition intensity of the detection process which yield tractable representations are introduced. The methodology is discussed in terms of two examples.     

On the differentiability of a class of stationary gaussian processes

For a stationary Gaussian process either almost all sample paths are almost everywhere differentiable or almost all sample paths are almost nowhere differentiable. In this paper it is shown by means of an example involving a random lacunary trigonometric series that “almost everywhere differentiable” and “almost nowhere differentiable” cannot in general be replaced by “everywhere differentiable” and “nowhere differentiable”, respectively.     

On the angle between past and future for multivariate stationary stochastic processes

A characterization for the positivityof the angle between past and future of multivariate stationary stochastic processes is established. In order to prove the results a lemma is proved which is of independent interest, and which is very useful in other areas of prediction theory as well.