, Volume 77, Issue 1, pp 53–62 | Cite as

A recursive approach for determining matrix inverses as applied to causal time series processes

  • Serge B. ProvostEmail author
  • John N. Haddad


A decomposition of a certain type of positive definite quadratic forms in correlated normal random variables is obtained from successive applications of blockwise inversion to the leading submatrices of a symmetric positive definite matrix. This result can be utilized to determine Mahalanobis-type distances and allows for the calculation of the full likelihood functions in instances where the observations secured from certain causal processes are irregularly spaced or incomplete. Applications to some autoregressive moving-average models are pointed out and an illustrative numerical example is presented.


Matrix inverse Quadratic forms Mahalanobis distance Craig’s theorem Likelihood function ARMA processes 

Mathematics Subject Classification

Primary: 62M10 15A09 Secondary: 15A63 15B05 



We would like to express our sincere thanks to both referees for their thorough reviews, insightful comments and valuable suggestions. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the first author.


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Copyright information

© Sapienza Università di Roma 2019

Authors and Affiliations

  1. 1.Department of Statistical and Actuarial SciencesThe University of Western OntarioLondonCanada
  2. 2.Department of Mathematics and StatisticsNotre Dame University-LouaizeZouk MosbehLebanon

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