Asymptotic normality of finite Fourier transforms of stationary generalized processes

This paper indicates a mixing condition under which a net of Fourier transforms, of a stationary generalized process over an abelian locally compact group, has a limiting normal distribution.     

Characterization of the finite variation property for a class of stationary increment infinitely divisible processes

We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Lévy processes, and also their mixtures. We establish two types of zero–one laws for the finite variation property. We also consider some examples to illustrate our results.     

Multiplicity theory for multivariate wide sense stationary generalized processes

It is shown that the representation theory of a multivariate, purely nondeterministic, wide sense stationary generalized process can be reduced to a study of some isomorphism results established for commutation relations occurring in quantum mechanics. Using this simplification a multiplicity theory is developed. The time domain and spectral representation of the process are investigated in this context, and the concept of a generalized innovations process is introduced.     

Explicit extrapolation of stationary processes with generalized rational spectral density

Translated from Issledovaniya po Prikladnoi Matematike, No. 7, pp. 84–94, 1979.     

Traffic processes in a class of finite Markovian queues

This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled in continuous time as Markov processes. The theory is presented for theM/E _k /φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output, departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works, as well as providing some new results. Several extensions to the model are discussed.     

On the convergence of finite linear predictors of stationary processes

It is shown that the finite linear least-squares predictor of a multivariate stationary process converges to its Kolmogorov-Wiener predictor at an exponential rate, provided that the entries of its spectral density matrix are smooth functions. Also, the same rate of convergence holds for the partial sums of the Kolmogorov-Wiener predictor.     

Generalized Markovian decision processes

Summary A general discrete decision process is formulated which includes both undiscounted and discounted semi-Markovian decision processes as special cases. A policy-iteration algorithm is presented and shown to converge to an optimal policy. Properties of the coupled functional equations are derived. Primal and dual linear programming formulations of the optimization problem are also given. An application is given to Markov ratio decision process.

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Zusammenfassung Es wird ein allgemeiner diskreter Entscheidungsprozeß betrachtet, welcher sowohl die undiskontierten und diskontierten Semi-Markoffschen Entscheidungsprozesse als Spezialfälle enthält. Ein auf der Politik-Iteration basierendes Verfahren wird vorgestellt und die Konvergenz gegen eine optimale Politik wird bewiesen. Wir zeigen einige Eigenschaften des Paares gekoppelter Funktionalgleichungen, die in diesem Modell auftreten. Zum Schluß werden noch die Formulierungen dieses Entscheidungsmodelles als primales und duales lineares Optimierungsproblem angegeben.

     

Lower functions for processes with stationary independent increments

Summary Let {X _t } be aR ¹-valued process with stationary independent increments and . In this paper we find a sufficient condition for there to exist nonnegative and nondecreasing functionh(t) such that lim infA _t /h(t)=C a.s. ast0 andt, for some positive finite constantC whenh(t) takes a particular form. Also two analytic conditions are considered as application.This research is partially supported by Korea Science & Engineering Foundation     

Power variation for Gaussian processes with stationary increments

We develop the asymptotic theory for the realised power variation of the processes X=?•G$X=?•G$, where G$G$ is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G$G$ and certain regularity conditions on the path of the process ?$?$ we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the Hölder index of the path of ?$?$, we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu and Nualart [Y. Hu, D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. (33) (2005) 948–983], Nualart and Peccati [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. (33) (2005) 177–193] and Peccati and Tudor [G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXVIII, in: Lecture Notes in Math, vol. 1857, Springer-Verlag, Berlin, 2005, pp. 247–262], for sequences of random variables which admit a chaos representation.     

An optimality principle for Markovian decision processes

The following optimality principle is established for finite undiscounted or discounted Markov decision processes: If a policy is (gain, bias, or discounted) optimal in one state, it is also optimal for all states reachable from this state using this policy. The optimality principle is used constructively to demonstrate the existence of a policy that is optimal in every state, and then to derive the coupled functional equations satisfied by the optimal return vectors. This reverses the usual sequence, where one first establishes (via policy iteration or linear programming) the solvability of the coupled functional equations, and then shows that the solution is indeed the optimal return vector and that the maximizing policy for the functional equations is optimal for every state.     

Large deviation for stationary processes

A quasi-local variational characterization of the entropy for stationary processes is given. This is used to establish upper and lower large deviation estimates for arbitrary stationary processes. The upper and lower rate functions are shown to coincide for all quasi-local stationary processes. The contents of the paper is the following: 1. Introduction; 2. Notations; 3. Relative entropy of conditional expectations; 4. Relative entropy of a stationary process with respect to a covariant family of conditional expectations; 5. The role of locality and quasi-locality properties; 6. Large deviation upper estimate; 7. The Lower estimate; 8. The variational principle.     

On Markovian decision models with a finite skeleton

We present a class of countable state space Markovian decision models that can be investigated by means of an associated finite-state, finite-action reduced model which we call the skeleton. In particular, we obtain a turnpike theorem for the original model (Theorem 2 in Section 5) from a known turnpike theorem for the reduced finite model. For illustration, we present in detail an application of this approach to an inventory model (re-establishing a known turnpike result) and sketch analogous results for a cash-balance model and a growth model.

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Zusammenfassung Wir führen eine Klasse von Markovschen Entscheidungsmodellen mit abzählbarem Zustandsraum ein, die mittels eines verbundenen, reduzierten Modells mit endlichem Zustands- und Aktionsraum, welches wir das Skelett nennen, untersucht werden können. Insbesondere erhalten wir ein Turnpike Theorem für das ursprüngliche Modell (Theorem 2 im Abschnitt 5) von einem bekannten Turnpike Theorem für das reduzierte endliche Modell. Zur Erläuterung stellen wir im einzelnen eine Anwendung dieses Ansatzes für ein Lagerhaltungsmodell (Wiederherleitung eines bekannten Turnpike Ergebnisses) dar, und wir skizzieren analoge Ergebnisse für ein Kassenhaltungsmodell und ein Wachstumsmodell.

     

Limit theorems for Markovian Hawkes processes with a large initial intensity

Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study the linear Hawkes process with an exponential exciting function in the asymptotic regime where the initial intensity of the Hawkes process is large. We derive limit theorems under this asymptotic regime as well as the regime when both the initial intensity and the time are large.     

Controlled diffusion processes with Markovian switchings for modeling dynamical engineering systems

In this paper the intrinsic complex nature of engineering systems under control is treated by introducing an approach based on Controlled Stochastic Differential Equations with Markovian Switchings (in short CSDEMS). Technical conditions for the existence and uniqueness of the solutions of the CSDEMS are provided. In this context it is not unusual to deal with non-linear CSDEMS that cannot be solved analytically. Therefore, we develop a new two-step, predictor–corrector method for finding numerical approximations to solutions of CSDEMS. This method utilizes the Euler–Maruyama method. An illustrative application to the biochemical engineering area is presented to highlight the usefulness of our approach as a simulation tool.     

On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes for which there is a finite K () such that the conditional distribution of X₁ given the entire past is equal to the conditional distribution of X₁ given only . The least such value of K is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of K, both in the backward sense that we have just described and in the forward sense, where one observes successive values of {Xn} for n?0 and asks for the least value K such that the conditional distribution of Xn+1 given is the same as the conditional distribution of Xn+1 given . We allow for finite or countably infinite alphabet size.     

Alternative models for stationary stochastic processes

We consider some parametric spectral estimators that can be used in a wide range of situations. Assuming the existence of fourth moments, we establish rates of convergence of the estimators, and a central limit theorem.