Statistical Korovkin Theory for Multivariate Stochastic Processes

In this article, we study Korovkin-type approximation theorems for multivariate stochastic processes via the concept of A-statistical convergence. A non-trivial example expressing the importance of our results is also presented.     

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of R ^d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.     

Recursive computation of invariant distributions of Feller processes

This paper provides a general and abstract approach to compute invariant distributions for Feller processes. More precisely, we show that the recursive algorithm presented in Lamberton and Pagès (2002) and based on simulation algorithms of stochastic schemes with decreasing steps can be used to build invariant measures for general Feller processes. We also propose various applications: Approximation of Markov Brownian diffusion stationary regimes with a Milstein or an Euler scheme and approximation of a Markov switching Brownian diffusion stationary regimes using an Euler scheme.     

Mutual subordination of multivariate stationary processes over any locally compact abelian group

Summary Our purpose is to extend Kolmogorov's theorem [5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper [10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.In Section 1 of the paper we shall define and prove theorems on the concept of E-subordination, where E is a projection-valued measure. In Section 2 we shall examine the structure of stationary processes over an lca group. In Section 3 we shall consider the concept of subordination of stationary processes. Finally in Section 4, we shall apply our subordination theorems to deduce that matrixvalued functions in L ₂ on the unit circle having no negative frequencies have a constant rank a.e. (Lebesgue) (4.2), a theorem of F. and M. Riesz (4.3), and a theorem on wandering subspaces due to Robertson [9] (4.4).The author is grateful to Prof. P. R. Masani for his generous help in formulating this paper.     

Stochastic processes indexed by hypergroups II

Further results on weakly stationary processes indexed by hypergroups are presented. The concept of translation operators is developed; processes on orbit spaces and double coset spaces are constructed. It is shown that every weakly stationary process indexed by a hypergroupK with centerC contains a maximalK//C-weakly stationary component. New examples forK-weakly stationary processes are continuous estimates of the mean of a weakly stationary process, isotropic random fields, andK-oscillations.     

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

We investigate the extremal behavior of stationary mixed MA processes for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and sup _{t ∈ [0,1]} Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.     

Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

Another Set of Conditions for Strong n (n = −1, 0) Discount Optimality in Markov Decision Processes

Abstract

In this paper we study discrete-time Markov decision processes with average expected costs (AEC) and discount-sensitive criteria in Borel state and action spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions on the system's primitive data, and under which we prove (1) AEC optimality and strong ? 1-discount optimality are equivalent; (2) a condition equivalent to strong 0-discount optimal stationary policies; and (3) the existence of strong n (n = ?1, 0)-discount optimal stationary policies. Our conditions are weaker than those in the previous literature. In particular, the “stochastic monotonicity condition” in this paper has been first used to study strong n (n = ?1, 0)-discount optimality. Moreover, we provide a new approach to prove the existence of strong 0-discount optimal stationary policies. It should be noted that our way is slightly different from those in the previous literature. Finally, we apply our results to an inventory system and a controlled queueing system.     

排队系统的渐近泊松话务过程

在文中,我们首先给出由马氏过程的一些跳跃时刻形成的简单点过程的有限维分布族弱收敛到泊松过程的相应分布族的条件,并讨论了有限维分布族弱收敛到泊松过程相应分布族的平稳马氏排队系统的话务过程,其次,我们证明了ＧＩ／Ｍ／１排队系统的离去过程的有限维分布族在重话务的情况下弱收敛到泊松过程的相应分布族。     

On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

On non-stationary solutions to MSDDEs: Representations and the cointegration space

In this paper we study solutions to multivariate stochastic delay differential equations (MSDDEs) and their relation to the discrete-time cointegrated VAR model. In particular, we observe that an MSDDE can always be written in an error correction form and, under suitable conditions, we argue that a process with stationary increments is a solution to the MSDDE if and only if it admits a certain Granger type representation. A direct implication of these results is a complete characterization of the cointegration space. Finally, the relation between MSDDEs and invertible multivariate CARMA equations is used to introduce the cointegrated MCARMA processes.     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

Bootstrapping the Empirical Distribution Function of a Spatial Process

In this article, we consider a stationary α-mixing random field in IR ^d. Under a large-sample scheme that is a mixture of the so-called “infill” and “increasing domain” asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block for some 0 < β < 1, where λ _n is the growth rate in the increasing domain asymptotics, we show that the bootstrapped empirical process converges weakly to the same limiting Gaussian process almost surely. Extension to multivariate random fields and application to differentiable statistical functionals are also given. A spatial version of the Bernstein’s inequality is developed, which may be of some independent interest. In final form 13 December 2004     

On a multivariate central limit theorem for stationary bilinear processes

In 1957, Parzen proved a central limit theorem for a class of scalar processes which he called multilinear processes. In the present paper only stationary bilinear processes are considered, but the theory is generalized to the multivariate case.     

On a zero-crossing probability

Let {X(t), 0E{exp (–sX(t))}=exp (–t(s)), where (s)=(1–(s)), is the intensity of the Poisson process, and (s) is the Laplace transform of the distribution of nonnegative jumps. Consider the zero-crossing probability =P{X(t)–t=0 for some t,0<t<}. We show that =() where is the largest nonnegative root of the equation (s)=s. It is conjectured that this result holds more generally for any stochastic process with stationary independent increments and with sample paths that are nondecreasing step functions vanishing at 0.     

Emergent dynamics of the first‐order stochastic Cucker‐Smale model and application to finance

In this paper, we study stochastic aggregation properties of the financial model for the N‐asset price process whose dynamics is modeled by the weakly geometric Brownian motions with stochastic drifts. For the temporal evolution of stochastic components of drift coefficients, we employ a stochastic first‐order Cucker‐Smale model with additive noises. The asset price processes are weakly interacting via the stochastic components of drift coefficients. For the aggregation estimates, we use the macro‐micro decomposition of the fluctuations around the average process and show that the fluctuations around the average value satisfies a practical aggregation estimate over a time‐independent symmetric network topology so that we can control the differences of drift coefficients by tuning the coupling strength. We provide numerical examples and compare them with our analytical results. We also discuss some financial implications of our proposed model.     

Stationary measures and hydrodynamics of zero range processes with several species of particles

We study general zero range processes with different types of particles on a d-dimensional lattice with periodic boundary conditions. A necessary and sufficient condition on the jump rates for the existence of stationary product measures is established. For translation invariant jump rates we prove the hydrodynamic limit on the Euler scale using Yaus relative entropy method. The limit equation is a system of conservation laws, which is hyperbolic and has a globally convex entropy. We analyze this system in terms of entropy variables. In addition we obtain stationary density profiles for open boundaries.     

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on -compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an almost periodic Fourier transform we recover the discrete part of the spectral measure and construct a natural, consistent estimator for the latter from samples of the process.     

A new class of multivariate counting processes and its characterization

In this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.     

On the exponential process associated with a CARMA-type process

We study the correlation decay and the expected maximal increments of the exponential processes determined by continuous-time autoregressive moving average (CARMA)-type processes of order (p, q). We consider two background driving processes, namely fractional Brownian motions and Lévy processes with exponential moments. The results presented in this paper are significant extensions of those very recent works on the Ornstein–Uhlenbeck-type case (p = 1, q = 0), and we develop more refined techniques to meet the general (p, q). In the concluding section, we discuss the perspective role of exponential CARMA-type processes in stochastic modelling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.