文档介绍：Supelec
Random Matrix Theory
for
munications
Mérouane Debbah
merouane.******@
February, 2008
Presentation
Stieltjes Transform Method
1
Important Lemma
Z. D. Bai and J. W. Silverstein, " No eigenvalue outside the support of the limiting spectral
distribution of large dimensional sample covariance matrices", Ann. Probab. vol. 26, no.
1, pp. 316-345, 1998.
Let A be a deterministic N × plex matrix with uniformly bounded spectral radius
for all N. Let x = √1 [x , ..., x ]T where the {x } are random variables
N 1 N i
with zero mean, unit variance and finite eight moment. Then:
· ¸
1 C
E | xHAx − TraceA |4 ≤
N N 2
where C is constant that does not depend on N or A.
Corollary. This result ensures that
1
xHAx − TraceA → 0
N
almost surely.
2
Definition
Uniformly Bounded Spectral Radius: There exists a real number, independent of N,
which bounds the magnitudes of the eigenvalues of A for all N.
3
Why Eighth Moment Bounded?
The proof proceeds by induction on p on the following inequality:
µ p ¶
³ ´ p
H 1 p 4 H 2 2p H
E | x Ax− Trace(A) | ≤ Kp E | x1 | Trace(AA ) + E | x1 | Trace(AA )2
N
4
An Even Stronger Result holds
. Tse and O. Zeitouni, "Linear multiuser receivers in random environments", IEEE
Transactions on Information Theory, 46(1): 171-188, Jan. 2000
A stronger result was shown to hold:
· ¸
√ 1
N xHAx − TraceA → N (0, σ2)
N
5
Elements of proof for the Marchenko-pastur law
Write · ¸
u
W =
U
where W is of size N × K and u is of size 1 × K with centered elements and
1
variance N .
· ¸−1
³ ´−1 H H
H uu − z uU
WW − zIN = H H
Uu UU − zIN−1
¡ H ¢−1
Element (1, 1) of WW − zIN can be rewritten (using inversion formula of block
matrices):
· ¸
³ ´−1
H 1
WW − zIN = .
H H H −1 H
11 uu − z − uU (UU − zIN−1) Uu
6
Elements of proof for the Marchenko-pastur law
Using the identity:
³ ´−1 ³ ´−1
H H H
IK − U UU − zIN−1 U = −z U U − zIK
W