Linear processes and bispectra

1980 ◽  
Vol 17 (1) ◽  
pp. 265-270 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Frequency Response Function ◽  
Moving Average ◽  
Random Variables ◽  
Linear Process ◽  
Linear Filter ◽  
Linear Phase ◽  
Linear Processes ◽  
Spectral Estimates ◽  
Non Gaussian ◽  
Independent Identically Distributed

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

Linear processes and bispectra

1980 ◽  
Vol 17 (01) ◽  
pp. 265-270 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Frequency Response Function ◽  
Moving Average ◽  
Random Variables ◽  
Linear Process ◽  
Linear Filter ◽  
Linear Phase ◽  
Linear Processes ◽  
Spectral Estimates ◽  
Non Gaussian ◽  
Independent Identically Distributed

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

The distributional structure of finite moving-average processes

1988 ◽  
Vol 25 (02) ◽  
pp. 313-321 ◽  
Author(s):  
ED McKenzie
Keyword(s):  
Moving Average ◽  
Markov Property ◽  
Covariance Structure ◽  
Time Reversibility ◽  
Linear Processes ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Standard Models ◽  
Non Gaussian ◽  
Analysis Of Time Series

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

Random filters which preserve the stability of random inputs

1988 ◽  
Vol 20 (2) ◽  
pp. 275-294 ◽  
Author(s):  
Stamatis Cambanis
Keyword(s):  
Transfer Function ◽  
Moving Average ◽  
Random Processes ◽  
Time Shift ◽  
Linear Filter ◽  
Random Inputs ◽  
The Stability ◽  
Non Gaussian ◽  
Random Gain

A stationary stable random processes goes through an independently distributed random linear filter. It is shown that when the input is Gaussian or harmonizable stable, then the output is also stable provided the filter&s transfer function has non-random gain. In contrast, when the input is a non-Gaussian stable moving average, then the output is stable provided the filter&s randomness is due only to a random global sign and time shift.

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (2) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

scholarly journals THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

2002 ◽  
Vol 18 (1) ◽  
pp. 119-139 ◽  
Author(s):  
Qiying Wang ◽  
Yan-Xia Lin ◽  
Chandra M. Gulati
Keyword(s):  
Wiener Process ◽  
Invariance Principle ◽  
Random Variables ◽  
Linear Process ◽  
Standard Wiener Process ◽  
Test Statistic ◽  
Linear Processes ◽  
Partial Sum ◽  
Unit Root Testing ◽  
Partial Sum Process

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

scholarly journals On The Maximum of the Geometric Moving Average

1969 ◽  
Vol 9 (1-2) ◽  
pp. 100-108
Author(s):  
A. M. Hasofer
Keyword(s):  
Stochastic Process ◽  
Real Number ◽  
Moving Average ◽  
Random Variables ◽  
Image Position ◽  
Independent Identically Distributed

By the geometric moving average of the independent, identically distributed random variables {Xn}, we mean the stochastic process , where a is a real number such that 0≦a≦1.

The distributional structure of finite moving-average processes

1988 ◽  
Vol 25 (2) ◽  
pp. 313-321 ◽  
Author(s):  
ED McKenzie
Keyword(s):  
Moving Average ◽  
Markov Property ◽  
Covariance Structure ◽  
Time Reversibility ◽  
Linear Processes ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Standard Models ◽  
Non Gaussian ◽  
Analysis Of Time Series

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

scholarly journals ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES

2013 ◽  
Vol 30 (1) ◽  
pp. 252-284 ◽  
Author(s):  
Karim M. Abadir ◽  
Walter Distaso ◽  
Liudas Giraitis ◽  
Hira L. Koul
Keyword(s):  
Asymptotic Normality ◽  
Long Memory ◽  
Moving Average ◽  
Regression Function ◽  
Linear Process ◽  
Triangular Array ◽  
Weighted Sums ◽  
Martingale Differences ◽  
Linear Processes ◽  
Arch Models

We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel-type estimators of a nonparametric regression function with short or long memory moving average errors.

scholarly journals Asymptotic distribution with random indices for linear processes

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3925-3935
Author(s):  
Yu Miao ◽  
Qinghui Gao ◽  
Shuili Zhang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Linear Process ◽  
Partial Sums ◽  
Real Numbers ◽  
Linear Processes ◽  
The Central Limit Theorem ◽  
Dependent Sequence ◽  
Random Indices

In this paper, we consider the following linear process Xn = ?? i=-? Ci?n-i, n ? Z, and establish the central limit theorem of the randomly indexed partial sums Svn := X1 +... + Xvn, where {ci,i?Z} is a sequence of real numbers, {?n,n?Z} is a stationary m-dependent sequence and {vn;n?1} is a sequence of positive integer valued random variables. In addition, in order to show the main result, we prove the central limit theorems for randomly indexed m-dependent random variables, which improve some known results.

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (02) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

LetUdenote the set of all integers, and suppose thatY= {Yu;u∈U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.)G(·). Leta= {au;u∈U} be a sequence of real numbers with Σuau2= 1. ThenXu= ΣwawYu–wdefines a stationary linear processX= {Xu; u ɛ U} withE(Xu) = 0,E(Xu2) = 1 foru∊U. LetF(·) be the d.f. ofX0.We prove that if maxu|au| is small, then (i) for eachw, Xwis close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy≦gmaxu|au| where Φ(·) is the standard Gaussian d.f., andgdepends only onG(·); (ii) for each finite set (w1, …wn), (Xw1, …Xwn) is close to Gaussian in a similar sense; (iii) theprocess Xis close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filterais studied.

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