## Thesis etd-10112017-102746 |

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Thesis type

Tesi di laurea magistrale

Author

OTTOLINI, MARTINO

URN

etd-10112017-102746

Thesis title

Spectral norm of random matrices with non-identically distributed entries

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Pratelli, Maurizio

Keywords

- random matrix theory

Graduation session start date

27/10/2017

Availability

Full

Summary

A random matrix is by definition a matrix-valued random variable. Among the many random matrix models, we consider two kind of random matrices: the random matrices X whose entries X_{ij} are zero-mean and independent, and the symmetric random matrices X whose entries are zero-mean and independent up to symmetry, i.e. X_{ij}=X_{ji} for all i and j and the entries above and on the main diagonal (the X_{ij} such that j<= i) are independent.

The study of the spectral properties of random matrices traces back to Wigner, who introduced in 1955 the semicircle distribution, which is named after the shape of its density function. Under suitable hypotheses on the distribution of the entries and after a suitable normalization, the spectral distribution of a symmetric random matrix with i.i.d.\! and zero-mean entries is asymptotically equal to the Wigner semicircle distribution, in the limit of large matrix dimension.

Here, a natural question that arises is whether the spectral norm (that in the case of symmetric matrices coincides with the largest among the moduli of the eigenvalues) is asymptotically equal to the radius of the semicircle distribution, which is 2. In this regards, Bai and Yin in 1988 found that the answer is in general negative: while the spectral distribution converges to the Wigner semicircle law, the spectral norm could instead go to infinity. Though, if the tail of the distribution of the entries is thin enough, i.e. the fourth moment exists, then the spectral norm effectively converges to 2. This result is known as the Bai-Yin theorem.

An analogous result was proven by Bai, Yin and Krishnaiah for non-symmetric random matrices with zero-mean and indepedent entries. The non-symmetric case differs from the symmetric case because it is necessary to consider the singular values instead of the eigenvalues.

Meanwhile, Girko in 1979 found out a way to generalize the Wigner semicircle distribution to the case of random symmetric matrices whose entries are zero-mean and independent, but not necessarily identically distributed. He found that, under suitable hypotheses on the distribution of the entries, the spectral distribution of a random symmetric matrix asymptotically satisfies a particular system of equations: the stochastic canonical equations.

The aim of this thesis is to find and prove a generalization of the Bai-Yin theorem to the case of non-identically distributed entries. The idea is that, under suitable hypotheses, the spectral norm of a symmetric random matrix with zero-mean and independent entries is equal to the radius of the distribution described by the stochastic canonical equations.

The proof of the generalization of the Bai-Yin theorem is based on a refinement of the estimates (regarding the spectral norm of symmetric matrices with independent, zero-mean and bounded entries) that Vu obtained in 2005 using the moment method. The moment method, that consists in studying the moments of the spectral distribution, is one of the most important tools in random matrix theory and allows us to transfer the problem of the estimation of the spectral norm to a combinatorial setting.

Besides, in the case of non-identically distributed entries, using the trick of the augmented matrix, it is simple to boil down the study of the spectral norm of non-symmetric random matrices to the study of the spectral norm of symmetric random matrices.

Therefore our results generalize also the Bai-Yin-Krishnaiah theorem.

The study of the spectral properties of random matrices traces back to Wigner, who introduced in 1955 the semicircle distribution, which is named after the shape of its density function. Under suitable hypotheses on the distribution of the entries and after a suitable normalization, the spectral distribution of a symmetric random matrix with i.i.d.\! and zero-mean entries is asymptotically equal to the Wigner semicircle distribution, in the limit of large matrix dimension.

Here, a natural question that arises is whether the spectral norm (that in the case of symmetric matrices coincides with the largest among the moduli of the eigenvalues) is asymptotically equal to the radius of the semicircle distribution, which is 2. In this regards, Bai and Yin in 1988 found that the answer is in general negative: while the spectral distribution converges to the Wigner semicircle law, the spectral norm could instead go to infinity. Though, if the tail of the distribution of the entries is thin enough, i.e. the fourth moment exists, then the spectral norm effectively converges to 2. This result is known as the Bai-Yin theorem.

An analogous result was proven by Bai, Yin and Krishnaiah for non-symmetric random matrices with zero-mean and indepedent entries. The non-symmetric case differs from the symmetric case because it is necessary to consider the singular values instead of the eigenvalues.

Meanwhile, Girko in 1979 found out a way to generalize the Wigner semicircle distribution to the case of random symmetric matrices whose entries are zero-mean and independent, but not necessarily identically distributed. He found that, under suitable hypotheses on the distribution of the entries, the spectral distribution of a random symmetric matrix asymptotically satisfies a particular system of equations: the stochastic canonical equations.

The aim of this thesis is to find and prove a generalization of the Bai-Yin theorem to the case of non-identically distributed entries. The idea is that, under suitable hypotheses, the spectral norm of a symmetric random matrix with zero-mean and independent entries is equal to the radius of the distribution described by the stochastic canonical equations.

The proof of the generalization of the Bai-Yin theorem is based on a refinement of the estimates (regarding the spectral norm of symmetric matrices with independent, zero-mean and bounded entries) that Vu obtained in 2005 using the moment method. The moment method, that consists in studying the moments of the spectral distribution, is one of the most important tools in random matrix theory and allows us to transfer the problem of the estimation of the spectral norm to a combinatorial setting.

Besides, in the case of non-identically distributed entries, using the trick of the augmented matrix, it is simple to boil down the study of the spectral norm of non-symmetric random matrices to the study of the spectral norm of symmetric random matrices.

Therefore our results generalize also the Bai-Yin-Krishnaiah theorem.

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