Stochastic matrices and a property of the infinite sequences of linear functionals |
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Authors: | Roberto Beneduci |
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Affiliation: | Dipartimento di Matematica, Università della Calabria and Istituto Nazionale di Fisica Nucleare, sezione di Cosenza, 87036 Arcavacata di Rende (CS), Italy |
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Abstract: | Our starting point is the proof of the following property of a particular class of matrices. Let T={Ti,j} be a n×m non-negative matrix such that ∑jTi,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that Ti,l≠Tj,l. Then, there exists a real vector k=(k1,k2,…,km)T,ki≠kj,i≠j;0<ki?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ(·)(i)}i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that |
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Keywords: | 15A51 28Axx |
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