Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

On the distribution of the number stranded in bulk-arrival, bulk-service queues of the M/G/1 form

Bulk-arrival queues with single servers that provide bulk service are widespread in the real world, e.g., elevators in buildings, people-movers in amusement parks, air-cargo delivery planes, and automated guided vehicles. Much of the literature on this topic focusses on the development of the theory for waiting time and number in such queues. We develop the theory for the number stranded, i.e., the number of customers left behind after each service, in queues of the M/G/1 form, where there is single server, the arrival process is Poisson, the service is of a bulk nature, and the service time is a random variable. For the homogenous Poisson case, in our model the service time can have any given distribution. For the non-homogenous Poisson arrivals, due to a technicality, we assume that the service time is a discrete random variable. Our analysis is not only useful for performance analysis of bulk queues but also in designing server capacity when the aim is to reduce the frequency of stranding. Past attempts in the literature to study this problem have been hindered by the use of Laplace transforms, which pose severe numerical difficulties. Our approach is based on using a discrete-time Markov chain, which bypasses the need for Laplace transforms and is numerically tractable. We perform an extensive numerical analysis of our models to demonstrate their usefulness. To the best of our knowledge, this is the first attempt in the literature to study this problem in a comprehensive manner providing numerical solutions.     

A hyperfinite flat integral for generalized random fields

In this paper, a nonstandard construction of generalized white noise is established. This provides a (hyperfinite) flat integral representation of probability measures for generalized random fields derived as image probability measures of generalized white noise under certain measurable transformations, including Euclidean random fields obtained as convolution from generalized white noise with Euclidean kernels.     

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ???-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ???-mixing stationary random field, due to the strong mixing condition, doesn??t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(???, ??).     

Solving the random diffusion model in an infinite medium: A mean square approach

This paper deals with the construction of an analytic-numerical mean square solution of the random diffusion model in an infinite medium. The well-known Fourier transform method, which is used to solve this problem in the deterministic case, is extended to the random framework. Mean square operational rules to the Fourier transform of a stochastic process are developed and stated. The main statistical moments of the stochastic process solution are also computed. Finally, some illustrative numerical examples are included.     

A random environment for linearly edge-reinforced random walks on infinite graphs

We consider linearly edge-reinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a mixture of reversible Markov chains, determined by time-independent strictly positive weights on the edges. Furthermore, we prove bounds for the random weights, uniform, among others, in the size of the graph.      

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.     

Random Hermite differential equations: Mean square power series solutions and statistical properties

This paper deals with the construction of random power series solution of second order linear differential equations of Hermite containing uncertainty through its coefficients and initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent. We provide conditions in order to obtain random polynomial solutions and, as a consequence, random Hermite polynomial are introduced. Also, the main statistical functions of the approximate stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples comparing the numerical results with respect to those provided by other available approaches including Monte Carlo simulation.     

Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains

In this paper we study the asymptotic dynamics for stochastic reaction-diffusion equation with multiplicative noise defined on unbounded domains. We investigate the existence of a random attractor for the random dynamical system associated with the equation. The asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far-field values of solutions and a cut-off technique.     

Exact transient analysis of a planar random motion with three directions

Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.     

Asymptotic behavior of roots of random polynomial equations

We derive several new results on the asymptotic behavior of the roots of random polynomial equations, including conditions under which the distributions of the zeros of certain random polynomials tend to the uniform distribution on the circumference of a circle centered at the origin. We also derive a probabilistic analog of the Cauchy-Hadamand theorem that enables us to obtain the radius of convergence of a random power series.     

Coherent random permutations with biased record statistics

We consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens’ distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.     

Lifting Lévy processes to hyperfinite random walks

An internal lifting for an arbitrary measurable Lévy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other - for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of càdlàg modifications of Lévy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic processes”.     

一类推广的随机分形的 Hausdorff维数

该文研究一类推广的${\bf R}^{d}$中具有有限记忆的随机递归模型,引入了一个与该结构有关的函数$\Psi(\beta),\beta\geq 0$,构造了一个随机测度$\mu_\omega$,证明了由该结构产生的随机集 $K(\omega)$的Hausdorff维数是$\alpha:=\inf\{\beta:\Psi(\beta)\leq1\}$.     

PP multivariate random weighting method

According to the Projection Pursuit (PP) method and the random weighting method, we propose a PP random weighting method, and set up the asymptotic distribution theory and strong limit theorem of PP random weighting empirical process. Applying this method, we obtain two kinds of goodness-of-fit test for a multivariate distribution function, i.e., we get the random weighting approximations of PP Kolmogorov Smirnov statistics (PPKS) and PP Smirnov Cramér Von Mises statistics (PPSC), we prove that the asymptotic distribution of PPKS and PPSC are the same as those of their respective random weighting approximations.Supported by the National Natural Science Foundation of China.     

Applications of nonstandard analysis to spin models

Methods of nonstandard analysis are used to construct a Markov semigroup representing the stochastic evolution of an infinite spin system with finite range interaction by means of a hyperfinite spin system. The hyperfinite spin system is then used to derive classical results about phase transitions of the stochastic Ising model without the use of thermodynamic limits.     

相互独立的m值随机变量和的极限分布定理

利用ｍ值随机变量的特征函数,在一定条件下,得到了相互独立的ｍ值随机变量和的极限分布均匀的充要条件;再结合无穷乘积的有关性质,给出了相互独立的ｍ值随机变量和极限分布均匀分布的充分条件,特别当ｍ为素数Ｐ时,所得的充分条件易于验证,且不难满足。     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

Hyperfinite methods applied to the Critical Branching Diffusion

Summary We apply the hyperfinite methods of [Re] to the construction of a version of the Critical Branching Diffusion studied by Dawson et al. Several new sample path properties are derived from this construction.     

Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators

We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.