12-FinEcon-Model Sel Page 330 Wednesday, February 4, 2004 12:59 PM
330 The Mathematics of Financial Modeling and Investment Management
EXHIBIT 12.1 Fluctuations of the Variance-Covariance Matrix
It can then be demonstrated that the density of eigenvalues of the ran-
dom matrix tends to the following distribution:
ρλ
Q (λmax – λ)(λmin – λ)
() = ------------- -------------------------------------------------------
2 πσ^2 λ
MN, → , ∞ Q= M N ⁄ ≥ 1
1 1
λmax min , = σ^21 + -----± 2 -----
Q Q
where σ^2 is the average eigenvalue of the matrix. Exhibit 12.2 illustrates
the theoretical function and a sample computed on 500 simulated inde-
pendent random walks. The shape of the distribution of the eigenvalues
is the signature of randomness.