Hyperfinite representation of distributions

Hyperfinite representation of distributions is studied following the method introduced by Kinoshita [2,3], although we use a different approach much in the vein of [4]. Products and Fourier transforms of representatives of distributions are also analysed.     

Hyperfinite Von Neumann games

Let ${}^1\text{[V(}\text{R}\text{)]}$ be an ω₁-saturated enlargement in the sense of Keisler (1977) and let $\text{F}$ be a hyperfinite finite set in ¹$\text{N}$. Following the suggestion of Wesley (1971) we define a class of hyperfinite games of the form: $\text{Γ}{}_{\mathrm{<mtext>F</mtext>}}\text{(}{}^1\text{υ)=〈Φ,}\text{A}\text{(}\text{F}\text{),}{}^1\text{υ〉}$, and show that measure-theoretic analogues of the kernel and bargaining set exist in this nonstandard setting such that their standard parts Loeb-measurable measurable on the Loeb space generated by the internal ¹finitely additive measure $\text{u}{}_{\mathrm{<mtext>F</mtext>}}\text{:}\text{A}\text{(}\text{F}\text{)→}{}^1\text{R}{}_{\mathrm{+}}$.     

A Nonstandard Levy-Khintchine Formula and Levy Processes

We use methods from nonstandard analysis to obtain a short and simple derivation of the Levy-Khintchine formula via an explicit construction of certain laws of the infinitesimal increments. Consequently, any arbitrary Levy process is representable as the standard part of a hyperfinite sum of infinitesimal increments.     

Sampling and Π-sampling expansions

Using the hyperfinite representation of functions and generalized functions this paper develops a rigorous version of the so-called ‘delta method’ approach to sampling theory. This yields a slightly more general version of the classical WKS sampling theorem for band-limited functions.     

Small time central limit theorems for semimartingales with applications

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews.     

超有限Ⅱ_1型因子中Cartan双模代数上等距和2-局部等距

设M是超有限Ⅱ1型因子．D是M的Cartan子代数,T是对角为D的M 的σ-弱闭的子代数(简称Cartan双模代数)并且生成M．设φ是T到T上的σ-弱连续满线性等距,则Φ可扩张成从M到M上的等距．设φ是T到T上的映射(没假设线性),满足任给a,b∈T,T上存在σ-弱连续满线性等距φa,b(与n,b有关),使得φa,b(a)=φ(a),φa,b(b)=φ(b),则φ是线性等距．     

非标准分析的产生与早期发展

非标准分析是微积分在20世纪的新发展,并且对现代数学的多个分支及层面产生了深刻的影响.通过考察相关资料,研究了非标准分析的产生、非标准分析的早期发展,进而使人们对非标准分析这一工具有所了解.     

Value-distribution of the Lerch zeta-function with algebraic irrational parameter. I

We prove a limit theorem in the sense of weak convergence of probability measures on the complex plane for the Lerch zeta-function with algebraic irrational parameter. The limit measure in this theorem is explicitly given. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 163–176, April–June, 2007.     

Maximum of partial sums and an invariance principle for a class of weak dependent random variables

The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle and the convergence of the moments in the central limit theorem.

     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

On Zeta-Functions of Finite Abelian Groups

In the paper, the zeta-functions of finite Abelian groups of rank 2 and 3 are considered. One- and two-dimensional limit theorems for these zeta-functions on the complex plane and in the space of meromorphic functions are obtained.     

A two-dimensional limit theorem for Mellin transforms of the Riemann zeta-function

In this paper, we obtain a two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for Mellin transforms of the square and the fourth power of the Riemann zeta-function. Partially supported by the Lithuanian Foundation of Studies and Science.     

一类具非标准增长条件和零阶项的抛物方程的存在性结果

在变指数背景下,我们考虑了一类具零阶项的抛物方程解的存在性结果. 存在性的证明在本质上依赖于选取合适的检验函数,这些检验函数要同时兼顾方程右端源项的可积性和零阶项. 利用先验估计和极限分析,借助于Young测度方法确立了非线性项的弱收敛元.     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. II

In this paper, we obtain a joint limit theorem in the sense of weak convergence of probability measures on the complex plane for Lerch zeta-functions with algebraic irrational parameters and give an explicit form of the limit measure. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 482–496, October–December, 2007.     

On Cesari's property (Q)

In the present paper, we generalize, within the family of convex-valued multifunctions, a selection theorem of Cole (Ref. 1). This then leads us to prove some new results concerning Cesari's property (Q) and to the introduction of the Cesari limit of sets, which is then compared to the Kuratowski-Mosco limit.This work was done while the author was a visiting professor at the University of Pavia, Pavia, Italy. Financial support was provided by CNR and by NSF Grant No. 84-03135.     

Survey on normal distributions,central limit theorem,Brownian motion and the related stochastic calculus under sublinear expectations

This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Itô’s type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.     

Weak convergence of compound stochastic process,I

Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {ξ^v(t):t?0}, v=1,2,…,M. At each epoch of the renewal process {A(t):t?0} we initiate a random number of each of the M types. Let m_l:l?1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is

, where the ξ^v_lj() are independent copies of ξ^v,m_lv is the vth component of m and {τ_l:l?1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t?0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.     

The central limit theorem for time series regression

The central limit problem is considered for a simple regression, where the residuals, x(n), are stationary and the sequence regressed on y^(N)(n), may depend on the number of observations, N, to hand. Two situations are considered, one where the residual is generated by a linear process (i.e. the best linear predictor is the best predictor) and the more general situation where that is not so. Two types of conditions are needed, the first of which limits the contribution of any individual y^(N)(n) and the second of which relates to the mixing properties of x(n). If ε(n) is the linear innovation sequence, in the linear case, with $\text{lim}{}_{\mathrm{k}}\text{→∞}\text{E}\text{(ε(n)}{}^2\text{∥}\text{F}{}_{\mathrm{n?k}}\text{)=}\text{F}{}^2\text{,}\text{F}{}_{\mathrm{n}}$ being the associated family of o-algebra, then the central limit theorem holds under minimal conditions on y^(N)(n). Under sligthly stronger conditions on y^(N)(n) and for x(n) weakly mixing this theorem and associated theorems, are shown to hold under further fairly weak conditions on the dependence of x(n) on its past.     

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.