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The non‐homogeneous Poisson process (NHPP) and the renewal process (RP) are two stochastic point process models that are commonly used to describe the pattern of repeated occurrence data. An inhomogeneous Gamma process (IGP) is a point process model that generalizes both the NHPP and a particular RP, commonly referred to as a Gamma renewal process, which has interarrival times that are i.i.d. gamma...
Consider an irreducible time-homogeneous Markov chain with discrete time. The recurrence-time moments of the states of such stochastic processes are studied. We point out that if the recurrence time of one state has its first k moments finite, then the recurrence times of all the other states have their first k moments finite. We then specialize and investigate the recurrence-time moments of random...
Murray Rosenblatt’s interest in random walks on compact semigroups probably came from his work on representations of stationary processes as shifts of functions of independent random variables described in [11], where products of matrices were studied. His papers [12],[5],[13],[14] generalized the work of Lévy [2] on random walks on the circle and the work of Kawada and Itô [4] on random walks on...
Summary Let xn, n = 0, ±1, ±2, …, be a strictly stationary process. Two closely related problems are posed with respect to the structure of strictly stationary processes. In the first problem we ask whether one can construct a random variable ξn = g(xn, xn-1, …), a function of xn, xn-1, …, that is independent of the past, that is, independent of xn-1, xn-2, …. Such a sequence of random variables {ξn}...
Summary Limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range...
Estimates of the regression coefficients which are unbiased and linear in the observations are discussed in this paper. The residual is assumed to be a stationary process. Two specific estimates are discussed, the least-squares estimate and the Markov estimate. I call the estimate which is computed under the assumption that the residual is an orthogonal process the least-squares estimate. The Markov...
Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.
Summary Recently interest has arisen in statistical applications of the bispectrum of stationary random processes. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The principal area of statistical harmonic analysis to receive attention previous to this time has been second-order (i.e. spectral) theory on which there is a vast literature...
Periodic and aperiodic solutions of the Burgers equation $$u_t + uu_x=\mu u_{xx}, \ \mu > 0,$$ are studied in this paper. A harmonic analysis of the solutions is carried out and the form of the spectrum is estimated for large time. Corresponding estimates of energy decay are also made, In Burgers’ work on this equation. the case in which $$\mu \downarrow 0$$ with t fixed, and one...
The bispectrum, or third order spectrum, of a stationary process has been around at least since the early 50s, for example it appeared in the paper Tukey (1953). It was studied in some detail in John Van Ness’s thesis, “Estimates of the Bispectrum of Stationary Random Processes”, supervised by Murray Rosenblatt. A 1965 Annals of Mathematical Statistics naner. “Estimation of the bispectra”, Rosenblatt...
Let F(λ)={Fik(λ); j, k = 1, …, s} be an s × s matrix-valued function of bounded variation on [—π, π]. By this we mean that every complex-valued element Fik(λ) is of bounded variation. Further, let every difference F(λ1) — F(λ2) be Hermitian. It will be convenient and in no way less general to take F(—π) = 0, the null matrix.
Asymptotic normality is proven for spectral density estimates assuming strong mixing and a limited number of moment conditions for the process analyzed. The result holds for a large class of processes that are not linear and does not require the existence of all moments.
Estimates of the density function of a population based on a sample of independent observations have been considered in a number of papers [1,6-7]. Questions of bias, variance and asymptotic distribution of the estimates have been dealt with at greatest length. Our object is to look at such estimates of the density function when the observations are dependent. The results will not be dealt with in...
Summary A certain class of stationary processes is discussed. It is shown that each process in the class has an absolutely continuous spectrum. Under some moment conditions, it is shown that such processes satisfy the central limit theorem.
Summary We consider time series which are realizations of a stochastic process. From the time series we construct various estimates of the spectral distribution function of the process (Section 3) and we study the sampling distributions of some functionals of these estimates (Sections 4-7). We then obtain confidence regions for the spectral distribution function and various tests of hypotheses in...
This is a brief history of the Rosenblatt process, how it came about, the role it played, its properties and a detailed description of its various representations.
F. J. Breidt, R. A. Davis, K.-S. Lii, and M. Rosenblatt. Maximum likelihood estimation for noncausal autoregressive processes. J. Multivariate Anal., 36(2): 175–198, 1991. Reprinted with permission of Elsevier Inc.
Let g(λ), —π∆λ∆π, be a p×p (p — 1, 2, …) matrix-valued Hermitian function. Further g(λ) is bounded on [—π, π], that is, its elements are bounded on [—π, π]. The Fourier coefficients 1 $$\begin{array}{lll}{*{20}{c}} {{\alpha _k} = \frac{1}{{2\pi }}\int_{ - \pi }^{ - \pi } {{e^{ik{\rm{\lambda }}}}g(\lambda )d\lambda,} } & {k = 0, \pm 1, \cdot \cdot \cdot } \\\end{array}$$ are then bounded...
The results obtained in this chapter may be of some interest from the point of view of analysis. However, they have an immediate interpretation in terms of certain representation theorems for stationary random processes on a finite time interval, and this provided part of the motivation for the investigation. Our interest is in finite interval translation kernel integral equation eigenvalue problems,...
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