Rescaled Lévy-Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE(κ) as limits.     

Large Cells in Poisson–Delaunay Tessellations

It is proved that the shape of the typical cell of a Delaunay tessellation, derived from a stationary Poisson point process in d-dimensional Euclidean space, tends to the shape of a regular simplex, given that the volume of the typical cell tends to infinity. This follows from an estimate for the probability that the typical cell deviates by a given amount from regularity, given that its volume is large. As a tool for the proof, a stability result for simplices is established.     

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z(^k)(t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.     

Interaction Processes for Unions of Facets,the Asymptotic Behaviour with Increasing Intensity

In the series of models with interacting particles in stochastic geometry, a further contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L ₂ expansion of the covariance of functionals of Poisson process is developed for U-statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U-statistics and submodels of the facet process.     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

An Inequality for the Sum of Independent Bounded Random Variables

We give a simple inequality for the sum of independent, bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.     

Non-homogeneous space-time fractional Poisson processes

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.     

On the Poisson distribution of lengths of lattice vectors in a random lattice

We prove that the volumes determined by the lengths of the non-zero vectors ±x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity ${\frac{1}{2}}$ . This generalizes earlier results by Rogers (Proc Lond Math Soc (3) 6:305?C320, 1956, Thm. 3) and Schmidt (Acta Math 102:159?C224, 1959, Satz 10).     

Nonsynchronous covariation process and limit theorems

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.     

Fractional Poisson process

A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process.As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.     

Networks of $$\cdot /G/\infty $$ queues with shot-noise-driven arrival intensities

We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by a shot-noise process. A shot-noise rate emerges naturally in cases where the arrival rate tends to exhibit sudden increases (or shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable to analysis. In particular, we perform transient analysis on the number of jobs in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of jobs in the system by using a linear scaling of the shot intensity. First we focus on a one-dimensional setting in which there is a single infinite-server queue, which we then extend to a network setting.     

Optimal global approximation of stochastic differential equations with additive Poisson noise

We consider strong global approximation of SDEs driven by a homogeneous Poisson process with intensity λ > 0. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson process. We consider two classes of methods using equidistant or nonequidistant sampling of the Poisson process, respectively. We provide a construction of optimal schemes, based on the classical Euler scheme, which asymptotically attain the established minimal errors. It turns out that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.     

Self-Interacting Markov Chains

Abstract

In this article we study a class of self-interacting Markov chain models. We propose a novel theoretical basis based on measure-valued processes and semigroup techniques to analyze its asymptotic behavior as the time parameter tends to infinity. We exhibit different types of decays to equilibrium, depending on the level of interaction. We illustrate these results in a variety of examples, including Gaussian or Poisson self-interacting models. We analyze the long-time behavior of a new class of evolutionary self-interacting chain models. These genetic type algorithms can also be regarded as reinforced stochastic explorations of an environment with obstacles related to a potential function.     

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process ?? on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of ??. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener?CIt? chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris?CFKG-inequalities for monotone functions of ??.     

The Distributions of the Frequency of Occurrence of Nucleotide Subsequences

The objective is to derive the probability distribution of the frequency of occurrence of a subsequence within a nucleotide sequence under the hypothesis that the four nucleotides occur at random and with equal probability. We also consider the Compound Poisson approximation for the same distribution. The exact probability distribution can be obtained by the finite Markov chain imbedding technique introduced by Fu and Koutras (1994), however we can manage the case as well if the probabilities are not all equal. The compound Poisson approximation by Stein-Chen's method can be used to develop an approximate probability distribution with proper setting of the definition of the sets of dependence. Such structure gives a bound on the total variation distance, which tends to get relatively larger as the frequency goes up. AMS 2000 Subject Classification: Primary: 60E05; Secondary: 60J10     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Further modified forms of binomial and poisson distributions

Summary Some new type of modifications of binomial and Poisson distributions, are discussed. First, we consider Bernoulli trials of lengthn with success ratep up to time whenm times of successes occur, and then, changing the success rate to γp, we continue the remaining trial. The distribution of number of successes is called the modified binomial distribution. The Poisson limit (n tends to infinity andp tends to 0, keepingnp=λ) of the modified binomial is called the modified Poisson distribution. The probability functions of modified binomial and Poisson distributions are given (Section 1). A new concept of (m, γ)-modification is introduced and fundamental theorem which gives the relations between the factorial moments of any probability function and the factorial moments of its (m, γ)-modification, is presented. Then some lower order moments of the modified binomial and Poisson distributions are given explicitly (Section 2). The modified Poisson ofm=2 is fitted to the distribution of number of children for Japanese women in some age group. The fitting procedure is also presented (Section 3). Some historical sketch concerning the modification and generalization of binomial and Poisson distributions is given in Appendix. The Institute of Statistical Mathematics     

Relaxation limit and initial layer to hydrodynamic models for semiconductors

We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in “momentum equilibrium”, an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.     

Stein estimation of Poisson process intensities

We construct superefficient estimators of Stein type for the intensity parameter λ > 0 of a Poisson process, using integration by parts and superharmonic functionals on the Poisson space.