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Summary

Chapter 20: This chapter is about some remarkable properties of positive matrices, by which we mean square matrices with real positive entries. Positive matrices are found in economic models, genetics, biology, team rankings, network analysis, Google's PageRank, and city planning. The spectral radius of any matrix is the absolute value of an eigenvalue, but for a positive matrix the spectral radius itself is an eigenvalue, and it is positive and dominant. It is associated with a positive eigenvector, whose ordered entries have been used for ranking sports teams, priority setting, and resource allocation in multicriteria decision-making. Since the spectral radius is a dominant eigenvalue, an associated positive eigenvector can be computed by the power method. Some properties of positive matrices are shared by nonnegative matrices that satisfy certain auxiliary conditions. One condition that we investigate in this chapter is that some positive power has no zero entries.

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

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20 - Positive and Nonnegative Matrices

Summary

An asymptotic formula for the Bayes risk in discriminating between two Markov chains

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2 results

20 - Positive and Nonnegative Matrices

Summary

An asymptotic formula for the Bayes risk in discriminating between two Markov chains