Time-reversibility of linear stochastic processes

1975 ◽  
Vol 12 (4) ◽  
pp. 831-836 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Probability Distribution ◽  
Stochastic Processes ◽  
Gaussian Processes ◽  
Moving Average ◽  
Joint Probability ◽  
Unique Property ◽  
Joint Probability Distribution ◽  
Autoregressive Moving Average ◽  
Time Reversibility ◽  
Moving Average Processes

Time-reversibility is defined for a process X(t) as the property that {X(t1), …, X(tn)} and {X(– t1), …, X(– tn)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes.

Time-reversibility of linear stochastic processes

1975 ◽  
Vol 12 (04) ◽  
pp. 831-836 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Probability Distribution ◽  
Stochastic Processes ◽  
Gaussian Processes ◽  
Moving Average ◽  
Joint Probability ◽  
Unique Property ◽  
Joint Probability Distribution ◽  
Autoregressive Moving Average ◽  
Time Reversibility ◽  
Moving Average Processes

Time-reversibility is defined for a process X(t) as the property that {X(t 1), …, X(tn )} and {X(– t 1), …, X(– tn )} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes.

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.

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