The central limit theorem for additive functionals of Markov and semi-Markov processes

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

: The main results described in this report include: (1) A discussion of rough surfaces and their modelling by chi-squared processes and fields. Results which describe how the theory of the latter differs from the Gaussian theory; (2) A development of a new methodology for determining deviation from normality in process and field data; (3) New results on the distribution of the maxima of general Gaussian processes; and (4) Development of a new representation of generalized Gaussian fields and those fields subordinated to it via central limit theorems for additive functionals of Markov processes.

We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

Let (Formula presented) be a collection of N(0, 1) random variables forming a real-valued continuous stationary Gaussian field on (Formula presented), and set (Formula presented). Let (Formula presented) be such that (Formula presented), let R be the Hermite rank of φ, and consider (Formula presented) with (Formula presented) compact. Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Yt have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right. The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as t→∞the fluctuations of Yt around its mean are, in general (i.e., except possibly in very special cases), Gaussian when B has short memory, and non-Gaussian when B has long memory and R ≥ 2. We show in this paper that this intuition forged over the last 40 years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where Yt admits Gaussian fluctuations in a long memory context. To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer-Major theorem to situations where (Formula presented) Our main mathematical tools are the Malliavin- Stein method and Fourier analysis techniques.

6 Dependence notions in reliability theory

Conditions are given under which a semi-Markov process $Z(t)$ can be obtained from a Markov process $Y(t)$ by a time change (i.e. $Z(t) = Y(\gamma(t))$). Estimates are given for $P\{\sup_{s\leqq t} |s - \gamma(s)| > \varepsilon\}$ and the construction is used to give conditions under which a sequence of semi-Markov processes will have the same convergence properties as the corresponding sequence of Markov processes.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

A Markov Renewal Process (M.R.P.) is a process similar to a Markov chain, except that the time required to move from one state to another is not fixed, but is a random variable whose distribution may depend on the two states between which the transition is made. For an M.R.P. ofm (<∞) states we derive a goodness-of-fit test for a hypothetical matrix of transition probabilities. This test is similar to the test Bartlett has derived for Markov chains. We calculate the first two moments of the test statistic and modify it to fit the moments of a standard χ2. Finally, we illustrate the above procedure numeerically for a particular case of a two-state M.R.P.

The tail decay of the hitting probabilities on level zero in M/G/1-type Markov renewal processes (MRPs) is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series. Two numerical methods are introduced to compute the convergence radius of the latter power series. Some numerical experiments are presented that show the effectiveness of the proposed algorithms.

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators.

Markov and Semi-Markov Processes with Fuzzy States. Part 1. Markov Processes

We present Poisson,approximation results for additive functionals switched by Markov and semi-Markov processes. The weak convergence results are obtained via semimartingale representations of additive functionals and the convergence of generators for Markov processes and of compensative operator of the extended Markov renewal processes. This is a review paper of our previous results given in [Korolyuk, 2002; Korolyuk, 2002A].KeywordsAdditive functionalPoisson approximationCompound Poisson approximation with driftMarkovsemi-Markov switchingsemimartingalecompensative operatorextended Markov renewal process

In conjunction with the 15th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)

A fundamental identity, due to Miller (1961a), (1962a, b) and Kemperman (1961), is generalized to semi-Markov processes. Thus the identity applies to processes defined on a Markov chain with discrete state space and random walks with Markov dependent steps (Section 2). Wald's identity is discussed briefly in Section 3. Section 4 is a study of the maxima of partial sums, and Section 5 of maxima in a semi-Markov process.