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Embedding a Gaussian discrete‐time autoregressive moving average process in a Gaussian continuous‐time autoregressive moving average process
Author(s) -
Huzii Mituaki
Publication year - 2007
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2006.00520.x
Subject(s) - mathematics , autoregressive model , autocovariance , embedding , autoregressive–moving average model , gaussian process , discrete time stochastic process , discrete time and continuous time , gaussian , moving average , process (computing) , mathematical optimization , algorithm , statistics , computer science , continuous time stochastic process , mathematical analysis , artificial intelligence , stochastic differential equation , physics , fourier transform , quantum mechanics , operating system
. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.