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Stable Laws and Products of Positive Random Matrices

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Abstract

Let S be the multiplicative semigroup of q×q matrices with positive entries such that every row and every column contains a strictly positive element. Denote by (X n ) n≥1 a sequence of independent identically distributed random variables in S and by X (n)=X n ⋅⋅⋅ X 1,  n≥1, the associated left random walk on S. We assume that (X n ) n≥1 satisfies the contraction property

$$\mathbb {P}\biggl(\bigcup_{n\geq1}[X^{(n)}\in S{^{\circ}}]\biggr)>0,$$

where S° is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix X 1 which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products X (n), n≥1, are in the domain of attraction of a stable law.

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Authors and Affiliations

  1. Institut Mathématiques de Rennes, Université de Rennes I, Campus de Beaulieu, 35042, Rennes-Cedex, France

    H. Hennion

  2. I.R.M.A.R, Institut National des Sciences Appliquées, Campus de Beaulieu, 35042, Rennes-Cedex, France

    L. Hervé

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  1. H. Hennion
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  2. L. Hervé
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Hennion, H., Hervé, L. Stable Laws and Products of Positive Random Matrices. J Theor Probab 21, 966–981 (2008). https://doi.org/10.1007/s10959-008-0153-y

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  • DOI: https://doi.org/10.1007/s10959-008-0153-y

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