We study the structure of tensorial products of H-valued stationary processes, where H is a Hilbert space. One of the motivations is autocovariance estimation by using the empirical autocovariance. For convenience we focus on autoregressive (ARH) and moving average (MAH) standard processes with innovations that are martingale increments. The obtained model is processes (possibly non-standard), where denotes the space of Hilbert–Schmidt operators on H. We also deal with the real case, we give some examples and we provide criteria for standardness of the tensorial products.
On étudie la structure des produits tensoriels de processus stationnaires à valeurs dans un espace de Hilbert H. L'une des motivations est l'estimation de l'autocovariance d'un tel processus à l'aide de l'autocovariance empirique. Pour simplifier on se limite aux processus autorégressifs (ARH) et moyennes mobiles (MAH) d'ordre un, standards, et dont l'innovation est une différence de martingale. Les processus obtenus sont alors du type , éventuellement non-standard, où est l'espace des opérateurs de Hilbert–Schmidt sur H. On s'intéresse aussi au cas réel, on donne des exemples et on fournit des critères assurant que le processus obtenu est standard.
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@article{CRMATH_2009__347_7-8_419_0, author = {Bosq, Denis}, title = {Produits tensoriels de processus {ARMA} fonctionnels}, journal = {Comptes Rendus. Math\'ematique}, pages = {419--423}, publisher = {Elsevier}, volume = {347}, number = {7-8}, year = {2009}, doi = {10.1016/j.crma.2009.01.031}, language = {fr}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.01.031/} }
TY - JOUR AU - Bosq, Denis TI - Produits tensoriels de processus ARMA fonctionnels JO - Comptes Rendus. Mathématique PY - 2009 SP - 419 EP - 423 VL - 347 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.01.031/ DO - 10.1016/j.crma.2009.01.031 LA - fr ID - CRMATH_2009__347_7-8_419_0 ER -
Bosq, Denis. Produits tensoriels de processus ARMA fonctionnels. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 419-423. doi : 10.1016/j.crma.2009.01.031. http://www.numdam.org/articles/10.1016/j.crma.2009.01.031/
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