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Theses

Large deviations for products of random matrices

Abstract : The purpose of this Ph.D. thesis is to study precise large and moderate deviation asymptotics for products of independent and identically distributed random matrices. In the first part, we establish Bahadur-Rao type and Petrov type exact asymptotics of large deviation probabilities for the norm cocycle «dollard»\log|G_nx|»dollard», where «dollard»G_n = g_n\ldots g_1»dollard» is the product of independent and identically distributed random «dollard»d\times d»dollard» matrices «dollard»g_i»dollard», «dollard»x»dollard» is a unit vector in «dollard»\mathbb R^d»dollard». The second part is devoted to establishing Bahadur-Rao type and Petrov type large deviations for the «dollard»(i,j)»dollard»-th entries «dollard»G_n^{i,j}»dollard» of «dollard»G_n»dollard». In particular, our result improves significantly the large deviation bounds established recently. In the third part, we investigate the Berry-Esseen bound and Cramér type moderate deviation expansion for the norm cocycle of products of random matrices. These results are proved by elaborating a new approach based on a smoothing inequality in the complex plane and on the saddle point method. The fourth part is devoted to studying Berry-Esseen bounds and Cramér type moderate deviation expansions for the operator norm «dollard»\|G_n\|»dollard», the entries «dollard»G_n^{i,j}»dollard» and the spectral radius «dollard»\rho(G_n)»dollard», for positive matrices. In the fifth part, we study the Berry-Esseen type bounds and moderate deviation principles for the operator norm «dollard»\|G_n\|»dollard» and the spectral radius «dollard»\rho(G_n)»dollard», for invertible matrices. We also prove the moderate deviation expansions in the normal range «dollard»[0, o(n^{1/6})]»dollard». The sixth part is devoted to the Cramér type moderate deviation expansion for the entries «dollard»G_n^{i,j}»dollard» of products of invertible matrices.
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Hui Xiao. Large deviations for products of random matrices. Statistics [math.ST]. Université de Bretagne Sud, 2020. English. ⟨NNT : 2020LORIS559⟩. ⟨tel-03258881⟩

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