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# 5.5 Case Studies and Applications

## 5.5.1 Fundamental Example of Using Large Random Matrix

We follow [279] for this development. Define an M × N complex matrix as

where (Xij)1≤i≤M, 1≤jN are (a number of MN) i.i.d. complex Gaussian variables . x1, x2, …, xN are columns of X. The covariance matrix R is

The empirical covariance matrix is defined as

In practice, we are interested in the behavior of the empirical distribution of the eigenvalues of for large M and N. For example, how do the histograms of the eigenvalues (λi)i=1, …, M of behave when M and N increase? It is well known that when M is fixed, but N increases, that is, is small, the large law of large numbers requires

In other words, ...

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