suppose the sequences k and n satisfy k
g
= −1 + sn where g ≥ 1 is fixed and s = on
13
. Fix primes p
1
, p
2
, . . . , p
t
and positive integers β
1
, β
2
, . . . , β
t
. Define e
n = p
β
1
1
p
β
2
2
. . . p
β
t
t
n .
Suppose k = p
1
p
2
. . . p
t
m → ∞. Then the ESD of en
−12
A
k ,
e n
converges weakly in probability to the LSD which has
1 −
Π
t s=
1
p
β
s
s −1
mass at zero, and rest of the probability mass is distributed as U
1
Q
g i=
1
E
i 12g
where U
1
and {E
i
} are as in Theorem 4.
3.4.2 n = k
g
− 1 for some g ≥ 2
For z
i
, w
i
∈ R, i = 1, 2, .., g, and with {N
i
} i.i.d. N0, 1, define H
g
ω
i
, z
i
, w
i
, i = 1, . . . , g = P Bω
1
, ω
2
, .., ω
g
N
1
, ..., N
2g ′
≤ z
i
, w
i
, i = 1, 2, .., g
′
.
Lemma 5. i H
g
is a bounded continuous in ω
1
, . . . , ω
g
for fixed {z
i
, w
i
, i = 1, . . . , g }.
ii F
g
defined below is a proper distribution function. F
g
z
i
, w
i
, i = 1, . . . , g = Z
1
· · · Z
1
H
g
2πt
i
, z
i
, w
i
, i = 1, . . . , g Y
d t
i
. 3.10
iii If LebC
= 0 then F
g
is continuous everywhere and may be expressed as F
g
z
i
, w
i
, i = 1, .., g =
Z · · ·
Z I
{t≤z
k
,w
k
,k=1,..,g }
h Z
1
· · · Z
1
I
{ Q
f 2πu
i
6=0}
2π
g
Q
g i=
1
[π f 2πu
i
]
g
Y
i= 1
e
−
1 2
t2 2i
−1 +t2
2i π f 2πui
Y du
i
i
dt .
where t = t
1
, t
2
, . . . , t
2g −1
, t
2g
and dt =
Q d t
i
. Further F
g
is multivariate with independent com- ponents if and only if f is constant almost everywhere Lebesgue.
iv If LebC
6= 0 then F
g
is discontinuous only on D
g
= {z
i
, w
i
, i = 1, . . . , g : Q
g i=
1
z
i
w
i
= 0}. The proof of lemma is omitted.
Theorem 5. Suppose Assumptions B and C hold. Suppose n = k
g
− 1 for some g ≥ 2. Then as n
→ ∞, F
n
−12
A
k ,n
converges in L
2
to the LSD Q
g i=
1
G
i 1g
where RG
i
, I G
i
; i = 1, 2, . . . g has the distribution
F
g
given in 3.10.
Remark 6. If {x
i
} are i.i.d, with finite 2 + δ moment, then f ω ≡ 12π and the LSD simpli- fies to U
2
Q
g i=
1
E
i 12g
where {E
i
} are i.i.d. E x p1, U
2
is uniformly distributed over the unit circle independent of
{E
i
}. This agrees with Theorem 4 of Bose, Mitra and Sen [2008].
3.5 Simulations
To demonstrate the limits we did some modest simulations with MA1 and MA2 processes. We performed numerical integration to obtain the LSD. In case of k-circulant n = k
2
+ 1, we have plotted the density of F
2
defined in 3.9. 2473
2 4
6 8
10 0.2
0.4 0.6
0.8 1
1.2 1.4
−− dependent entries −6
−4 −2
2 4
6 0.1
0.2 0.3
0.4 0.5
0.6 0.7
symmetric circulant
Figure 1:
i left dashed line represents the density of F
2
when f ω =
1 2π
1.25 + cos x and the continuous line represents the same with f
≡
1 2π
. ii right dashed line represents the LSD of symmetric circulant matrix with entries x
t
= 0.3ε
t
+ ε
t+ 1
+ 0.5ε
t+ 2
where {ε
i
} i.i.d. N0, 1 and the continuous line represents the kernel density estimate of the ESD of the same matrix of order 5000
× 5000 and same {x
t
}.
−5 5
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45
reverse circulant with N0,1 entries −5
5 0.05
0.1 0.15
0.2 0.25
0.3 0.35
0.4 0.45
reverse circulant with Binomail1,0.5
Figure 2:
i left dashed line represents the LSD of the reverse circulant matrix with entries x
t
= 0.3ε
t
+ε
t+ 1
+0.5ε
t+ 2
where {ε
i
} i.i.d. N0, 1. The continuous line represents the kernel density estimate of ESD of the same matrix of order 5000
× 5000 with same {x
t
}. ii same graphs with centered and scaled Bernoulli1, 0.5.
2474
4 Proofs of main results
Throughout c and C will denote generic constants depending only on d. We use the notation a
n
∼ b
n
if a
n
− b
n
→ 0 and a
n
≈ b
n
if
a
n
b
n
→ 1. As pointed out earlier, to prove that F
n
converges to F say in L
2
, it is enough to show that E
[F
n
t] → Ft and V [F
n
t] → 0 4.1
at all continuity points t of F . This is what we shall show in every case. If the eigenvalues have the decomposition λ
k
= η
k
+ y
k
for 1 ≤ k ≤ n, where y
k
→ 0 in probability then
{λ
k
} and {η
k
} have similar behavior. We make this precise in the following lemma.
Lemma 6. Suppose {λ
n ,k
}
1 ≤k≤n
is a triangular sequence of R
d
-valued random variables such that λ
n ,k
= η
n ,k
+ y
n ,k
for 1
≤ k ≤ n. Assume the following holds: i
lim
n →∞
1 n
P
n k=
1
P η
n ,k
≤ ˜x = F˜x, for ˜x ∈ R
d
, ii
lim
n →∞
1 n
2
P
n k
,l=1
P η
n ,k
≤ ˜x, η
n ,l
≤ ˜y = F˜xF ˜y, for ˜x, ˜y ∈ R
d
iii For any ε 0, max
1 ≤k≤n
P | y
n ,k
| ε → 0 as n → ∞. Then,
1. lim
n →∞
1 n
P
n k=
1
P λ
n ,k
≤ ˜x = F˜x. 2.
lim
n →∞
1 n
2
P
n k
,l=1
P λ
n ,k
≤ ˜x, λ
n ,l
≤ ˜y = F˜xF ˜y. Proof.
We define new random variables Λ
n
with PΛ
n
= λ
n ,k
= 1n for k = 1, . . . , n. Then PΛ
n
≤ ˜x = 1
n
n
X
k= 1
P λ
n ,k
≤ ˜x. Similarly define E
n
with PE
n
= η
n ,k
= 1n for 1 ≤ k ≤ n and Y
n
with PY
n
= y
n ,k
= 1n for 1
≤ k ≤ n. Now observe that Λ
n
= E
n
+ Y
n
and for any ε 0, P
|Y
n
| ε = 1
n
n
X
k= 1
P | y
n ,k
| ε → 0, as n → ∞ by assumption iii. Therefore Λ
n
and E
n
have the same limiting distribution. Now as n → ∞,
PE
n
≤ ˜x = 1
n
n
X
k= 1
P η
n ,k
≤ ˜x → F˜x. by assumption i Therefore as n
→ ∞, 1
n
n
X
k= 1
P λ
n ,k
≤ ˜x = PΛ
n
≤ ˜x → F˜x
2475
and this is conclusion i. To prove ii we use similar type of argument. Here we define new random variables ˜
Λ
n
with P ˜ Λ
n
= λ
n ,k
, λ
n ,l
= 1n
2
for 1 ≤ k, l ≤ n. Similarly define ˜E
n
and ˜ Y
n
. Again ˜
Λ
n
= ˜ E
n
+ ˜ Y
n
and P
kY
n
k ε = 1
n
2 n
X
k ,l=1
P k y
n ,k
, y
n ,l
k ε → 0, as n → ∞. So ˜
Λ
n
and ˜ E
n
will have same limiting distribution and hence conclusion ii holds. We use normal approximation heavily in our proofs. Lemma 7 is a fairly standard consequence
of normal approximation and follows easily from Bhattacharya and Ranga Rao [1976] Corollary 18.1, page 181 and Corollary 18.3, page 184. We omit its proof. Part i will be used in Section
4.1– 4.4 and Part ii will be used in Section 4.4.
Lemma 7. Let X
1
, . . . , X
k
be independent random vectors with values in R
d
, having zero means and an average positive-definite covariance matrix V
k
= k
−1
P
k j=
1
C ovX
j
. Let G
k
denote the distribution of k
−12
T
k
X
1
+ . . . + X
k
, where T
k
is the symmetric, positive-definite matrix satisfying T
2 k
= V
−1 k
, n ≥ 1.
If for some δ 0, EkX
j
k
2+δ
∞, then there exists C 0 depending only on d, such that i
sup
B ∈C
|G
k
B − Φ
d
B| ≤ C k
−δ2
[λ
min
V
k
]
−2+δ
ρ
2+δ
ii for any Borel set A, |G
k
A − Φ
d
A| ≤ C k
−δ2
[λ
min
V
k
]
−2+δ
ρ
2+δ
+ 2 sup
y ∈R
d
Φ
d
∂ A
η
− y where Φ
d
is the standard d dimensional normal distribution function, C is the class of all Borel mea-
surable convex subsets of R
d
, ρ
2+δ
= k
−1
P
k j=
1
E kX
j
k
2+δ
and η = Cρ
2+δ
n
−δ2
.
4.1 Proof of Theorem 1
