Asymptotic properties of the periodogram of a discrete stationary process

1967 ◽  
Vol 4 (3) ◽  
pp. 508-528 ◽  
Author(s):  
Richard A. Olshen
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Stationary Process ◽  
Asymptotic Properties ◽  
Spectral Densities ◽  
Image Position ◽  
Erratic Behavior

Suppose x1,…, xN are indefinitely many observations on a stochastic process which is weakly stationary with spectral density f(λ), – π ≦ λ ≦ π. An asymptotically unbiased, and to that extent plausible, estimate of 4rf(λ)is the periodogram Yet the periodograms of processes which possess spectral densities are notoriously subject to erratic behavior.

Asymptotic properties of the periodogram of a discrete stationary process

1967 ◽  
Vol 4 (03) ◽  
pp. 508-528 ◽  
Author(s):  
Richard A. Olshen
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Stationary Process ◽  
Asymptotic Properties ◽  
Spectral Densities ◽  
Image Position ◽  
Erratic Behavior

Suppose x 1,…, xN are indefinitely many observations on a stochastic process which is weakly stationary with spectral density f(λ), – π ≦ λ ≦ π. An asymptotically unbiased, and to that extent plausible, estimate of 4rf(λ)is the periodogram Yet the periodograms of processes which possess spectral densities are notoriously subject to erratic behavior.

On estimation of the integrals of certain functions of spectral density

1980 ◽  
Vol 17 (01) ◽  
pp. 73-83 ◽  
Author(s):  
Masanobu Taniguchi
Keyword(s):  
Continuous Function ◽  
Spectral Density ◽  
Stationary Process ◽  
Asymptotic Variance ◽  
Asymptotic Properties ◽  
Gaussian Stationary Process

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n –1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.

On estimation of the integrals of certain functions of spectral density

1980 ◽  
Vol 17 (1) ◽  
pp. 73-83 ◽  
Author(s):  
Masanobu Taniguchi
Keyword(s):  
Continuous Function ◽  
Spectral Density ◽  
Stationary Process ◽  
Asymptotic Variance ◽  
Asymptotic Properties ◽  
Gaussian Stationary Process

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n–1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.

Non-Stationary Processes and Spectrum

1968 ◽  
Vol 20 ◽  
pp. 1203-1206 ◽  
Author(s):  
K. Nagabhushanam ◽  
C. S. K. Bhagavan
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Stationary Process ◽  
Spectral Density Function ◽  
Stationary Processes ◽  
Gaussian Stationary Process ◽  
Discrete Parameter ◽  
Image Position

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function1for a non-stationary process {X(t)} denned by1where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.

CONVERGENCE IN DISTRIBUTION OF THE PERIODOGRAM OF CHAOTIC PROCESSES

2002 ◽  
Vol 02 (04) ◽  
pp. 609-624 ◽  
Author(s):  
ARTUR O. LOPES ◽  
SÍLVIA R. C. LOPES
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Fixed Point ◽  
Continuous Function ◽  
Spectral Density ◽  
Stationary Process ◽  
Spectral Density Function ◽  
Convergence In Distribution ◽  
Continuous Transformation ◽  
Distribution Sense

In this work we analyze the convergence in distribution sense of the periodogram function (to the spectral density function) based on a time series of a stationary process Xt = (φ ◦ Tt)(X0) obtained from the iterations of a continuous transformation T invariant for an ergodic probability μ and a continuous function φ taking values in ℝ. We only assume a certain rate of convergence to zero for the autocovariance coefficient of the stochastic process, i.e. we assume there exist C > 0 and β > 2 such that |γX(h)| ≤ C|h|-β, for all h ∈ ℕ, where γX(h) = ∫(φ ◦ Th)(x) φ(x)dμ(x) - (∫ φ(x)dμ(x))2 is the h-autocovariance of the process. Our result applies to the case of exponential decay of correlation (or covariance), as it happens for a continuous expanding transformation T on the circle and a Holder potential φ. It can also be applied to the case when the transformation T has a fixed point with derivative equal to one.

scholarly journals Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series

1964 ◽  
Vol 4 (3) ◽  
pp. 363-384 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Time Series ◽  
Distribution Function ◽  
Spectral Density ◽  
Spectral Distribution ◽  
Asymptotic Properties ◽  
Spectral Density Function ◽  
Absolutely Continuous ◽  
Spectral Distribution Function ◽  
Image Position ◽  
Least Squares Estimates

Let {xt} (t = 0, ±1, ±2 …) be a stationary non-deterministic time series with E(x2t) < ∞, E(xt) = 0, and let its spectrum be continuous (strictly absolutely continuous) so that the spectral distribution function is the spectral density function. It is well known that {xt} then has a unique one-sided movingaverage representation where .

On estimation of parameters of Gaussian stationary processes

1979 ◽  
Vol 16 (03) ◽  
pp. 575-591 ◽  
Author(s):  
Masanobu Taniguchi
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Density ◽  
Stationary Process ◽  
Parametric Family ◽  
Stationary Processes ◽  
Estimation Of Parameters ◽  
Spectral Densities ◽  
Gaussian Stationary Process

In fitting a certain parametric family of spectral densities fθ (x) to a Gaussian stationary process with the true spectral density g (x), we propose two estimators of θ, say by minimizing two criteria D 1 (·), D 2(·) respectively, which measure the nearness of fθ (x) to g (x). Then we investigate some asymptotic behavior of with respect to efficiency and robustness.

On estimation of parameters of Gaussian stationary processes

1979 ◽  
Vol 16 (3) ◽  
pp. 575-591 ◽  
Author(s):  
Masanobu Taniguchi
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Density ◽  
Stationary Process ◽  
Parametric Family ◽  
Stationary Processes ◽  
Estimation Of Parameters ◽  
Spectral Densities ◽  
Gaussian Stationary Process

In fitting a certain parametric family of spectral densities fθ (x) to a Gaussian stationary process with the true spectral density g (x), we propose two estimators of θ, say by minimizing two criteria D1 (·), D2(·) respectively, which measure the nearness of fθ (x) to g (x). Then we investigate some asymptotic behavior of with respect to efficiency and robustness.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

On two integral equations of queueing theory

1967 ◽  
Vol 4 (2) ◽  
pp. 343-355 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Positive Constant ◽  
Mathematical Description ◽  
Queueing Theory ◽  
Image Position

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

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