NOVELIST estimator of large correlation and covariance matrices and their inverses
1
Supplementary: additional lemmas and proofs
Firstly, we briefly introduce two lemmas that will be used in the proof of Proposition 1.
R1
Lemma 1 If F satisfies 0 exp( t)dGj (t) < 1, for 0 < | | < 0 , for some 0 > 0, where Gj is the cdf
2
of X1j
, R = {⇢ij } and ⌃ = { ij } are the true correlation and covariance matrices, 1  i, j  p, and
p
0
0
for
sufficiently
large
M
,
if
=
M
log p/n and log p/n = o(1),
ii  M , where M is a constant, then,
p
we have max |ˆ
⇢ij ⇢ij | = Op ( log p/n), for 1  i, j  p.
1i,jp
Proof of Lemma 1: By the sub-multiplicative norm property ||AB||  ||A|| ||B|| (Golub & Van Loan,
1989), we write
max |ˆ
⇢ij
1i,jp
⇢ij |
= max |ˆij /(ˆii ˆjj )
1/2
ij /( ii jj )
1i,jp
 max |ˆii
1/2
+ max |ˆii
1/2
1/2
ii
1ip
ij |
1i,jp
1/2
ii
1ip
| max |ˆij
| max (|ˆij ||
1i,jp
+ max |ˆij
ij | max | ˆii
1i,jp
1ip
p
=Op ( log p/n)
1/2
1/2
|
1/2
1/2
|
jj
| max |
1ip
1/2
|
jj
max |ˆjj
1jp
+ | ˆii
1/2
ii
1/2
||
ij |)
|
(27)
p
1
1
The last equality holds as we have max |ˆij
ij | = Op ( log p/n) = max |ˆij
ij | (Bickel
1i,jp
1i,jp
p
& Levina, 2008b), and max |ˆij | = Op ( log p/n) = max |ˆij 1 |, and ii  M , 1  i, j  p. ⌅
1i,jp
1i,jp
R1
Lemma 2 If F satisfies 0 exp( t)dGj (t) < 1, for 0 < | | < 0 , for some 0 > 0, where Gj is the
2
cdf of X1j
, R = {⇢ij } is the true correlation matrix, 1  i, j  p, then, uniformly on V(q, s0 (p), "0 ), for
p
sufficiently large M 0 , if = M 0 log p/n and log p/n = o(1),
||T (R̂, )
R|| = Op (s0 (p)(log p/n)(1
q)/2
).
(28)
where T is any kind of generalised thresholding estimator.
Lemma 2 is a correlation version of Theorem 1 in Rothman et al. (2009) and follows in a straightˆ ⌃, U (q, c0 (p), M, ✏0 ) and c0 (p) by R̂, R, V(q, s0 (p), "0 ) and s0 (p) in the
forward way by replacing ⌃,
proof of the theorem.
Proof of Proposition 1:
We first show the result for R̂N . By the triangle inequality,
||R̂N
R|| = ||(1
 (1
)R̂ + T (R̂, )
)||R̂
= I + II.
R||
R|| + ||T (R̂, )
R||
(29)
Using Lemma 2, we have
II = Op { s0 (p)(log p/n)(1
q)/2
}.
(30)
2
Na Huang, Piotr Fryzlewicz
For symmetric matrices M , Corollary 2.3.2 in Golub & Van Loan (1989) states that
||M ||  (||M ||(1,1) ||M ||(1,1) )
1/2
= ||M ||(1,1) = max
1ip
p
X
j=1
|mij |.
(31)
Then by Lemma 1,
||R̂
R||  max
1ip
p
X
j=1
Rij |  p max |ˆ
⇢ij
|R̂ij
⇢ij | = Op (p
1i,jp
p
log p/n).
(32)
Thus, we have
I = (1
R||  Op ((1
)||R̂
)p
p
log p/n).
(33)
Combining formulae (30) and (33) yields the first equality. The second equality follows because
||(R̂N )
1
R
1
|| ⇣ ||R̂N
(34)
R||
uniformly on V(q, s0 (p), "0 ).
ˆ N estimator, recalling that T = T (R̂, ) and D = (diag(⌃))1/2 , we have
For the ⌃
ˆN
||⌃
⌃|| = ||D̂R̂N D̂
= ||D̂((1
ˆ
)||⌃
 (1
DRD||
)R̂ + T )D̂
DRD||
⌃|| + ||D̂T D̂
DRD||
= III + IV.
p
Similarly as in 33, we obtain III = Op ((1
)p log p/n). For IV , we write
||D̂T D̂
||D̂
+||T
(35)
DRD||
D|| ||T
R|| ||D̂
R|| ||D̂|| ||D||
=Op ((1 + s0 (p)(log p/n)
D|| + ||D̂
q/2
)
D||(||T || ||D|| + ||D̂|| ||R||)
p
log p/n).
(36)
p
The last equality holds as we have ||T R|| = Op (s0 (p)(log p/n)(1 q)/2 ), ||D̂ D|| = Op ( log p/n),
||D̂|| = Op (1) = ||T ||, and ||D|| = O(1) as ii < M . Because (log p/n)q/2 (s0 (p)) 1 is bounded from
ˆ N ) 1 ⌃ 1 || ⇣ ||⌃
ˆN
above by the assumption that log p/n = o(1) and ||(⌃
⌃|| uniformly on
V(q, s0 (p), "0 ), the result follows. ⌅
Proof of Corollary 2:
Substituting log p by C1 n↵ in (8), we get
˜=
C2 pn(↵ 1)q/2
,
s0 (p) + C2 pn(↵ 1)q/2
(37)
where C2 is a constant. If p = o(n(1 ↵)q/2 ), we have pn(↵ 1)q/2 ! 0, which implies ˜ ! 0, since
s0 (p)  C. On the other hand, if n = o(p2/(1 ↵)q ), we have pn(↵ 1)q/2 ! 1 and ˜ ! 1 as n ! 1.
Additionally, if p ⇣ n(1 ↵)q/2 , then pn(↵ 1)q/2 is of a constant order, which yields ˜ 2 (0, 1), as
required. ⌅
Proof of Corollary 3:
NOVELIST estimator of large correlation and covariance matrices and their inverses
3
Firstly, noting that
Z
p+1
K
q
1
p1
1
we have
Pp
K=1
K
q
dK <
= O(p1
q
q
q
<
p
X
K=1
p
X
K
K
q
q
<
<
K=1
Z
p
K
q
(38)
dK
0
(p + 1)1 q
1
q
1
,
). For the long-memory correlation matrix, we can write
s0 (p) = max
1ip
p
X
j=1
|i
j|
q
= O(p1
q
).
(39)
By substituting log p by C1 n↵ and s0 (p) by (39) in (8), we get
˜=
p
C2 n(↵ 1)q/2
.
q + C n(↵ 1)q/2
2
(40)
Again ˜ depends on p and n. The remaining part of the proof is analogous to that of Corollary 2 and is
omitted here. ⌅