Tensor approach to mixed high-order moments of absorbing Markov chains
نویسنده
چکیده
Moments of absorbing Markov chain are considered. First moments and non-mixed second moments are determined in classical textbooks such as the book of J. Kemeny and J. Snell “Finite Markov Chains”. The reason is that the first moments and the non-mixed second moments can be easily expressed in a matrix form. Since the representation of mixed moments of higher orders in a matrix form is not straightforward, if ever possible, they were not calculated. The gap is filled by this paper. Tensor approach to the mixed high-order moments is proposed and compact closed-form expressions for the moments are discovered. Key-words: Absorbing Markov Chain, high-order moment, tensor in ria -0 04 26 76 3, v er si on 2 9 N ov 2 00 9 Une approche tensorielle pour le calcul des moments mixtes d’ordre supérieur des chaînes de Markov avec absorption Résumé : Cet article s’intŕesse au calcul des moments d’une chaîne de Markov absorbante. Les premiers et les seconds moments non-mixtes sont déterminés dans les livres classiques, tel que le livre de J. Kemeny et J. Snell "Finite Markov Chains". La raison en est que les premiers et les seconds moments non-mixtes s’expriment facilement sous forme matricielle, ce qui n’est pas le cas des moment mixtes d’ordre arbitraire. Le fossé est comblé dans cet article oú grâce á une approche tensorielle des formules explicites pour les moments supérieurs mixtes d’ordre arbitraire sont obtenues. Mots-clés : Chaînes de Markov absorbantes, moments d’ordre supérieur, tenseur. in ria -0 04 26 76 3, v er si on 2 9 N ov 2 00 9 Tensor approach to mixed high-order moments of absorbing Markov chains 3
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