The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives

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by  I-Shing Hu

THE COMMON LIMIT OF THE LINEAR STATISTICS OF ZEROS
OF RANDOM POLYNOMIALS AND THEIR DERIVATIVES
7
1 I-SHING HU AND CHIH-CHUNG CHANG
0
2
n Abstract. Let pn(x) be a random polynomial of degree n and {Zj(n)}nj=1 and
a Xn,k n−k,k <n,bethezerosofp andp(k),thekthderivativeofp ,respectively.
J { j }j=1 n n n
4 We show that if the linear statistics 1 f Z1(n) + +f Zn(n) associated
1 an " bn ! ··· bn !#

] with Z(n) hasalimitasn atsomemodeofconvergence,thelinearstatistics
R { j } →∞
associated with Xn,k converges to the same limit at the same mode. Similar
P { j }
statement also holds for the centered linear statistics associated with the zeros of
h. p andp(k), providedthe zeros Z(n) andthe sequences a and b of positive
at nnumbersnsatisfy some mild cond{itiojns}. { n} { n}
m
[

1
v 1. Introduction
6
4 Fix a probability space (Ω, ,P) and let p (z) ∞ be a sequence of random
9 F { n }n=1
3 polynomials such that deg pn = n. We observe that the randomness of polynomials
0 can be introduced in several different ways.
.
1
Type 1: Given a triangular array of random zeros Z(n) n ,n = 1,2,..., let
0 { j }j=1
7
1 (1.1) p (z) = (z Z(n)) (z Z(n)).
: n − 1 ··· − n
v
i Here the coefficient of zn is set to be 1 for simplicity.
X
Type 2: Given a triangular array of random coefficients a(n) n ,n = 1,2,...,
ar let { j }j=0
(1.2) p (z) = a(n)zn + +a(n)z +a(n).
n n ··· 1 0
Type 3: Given a sequence of random matrices A(n) ∞ , let
{ }n=1
(1.3) p (z) = det(zI A(n)),
n
−
the characteristic polynomial of A(n). Here I, the identity matrix, and A(n)
are square matrices of size n.
2010 Mathematics Subject Classification. Primary: 30C15, 60B10; secondary: 60B20, 60G57,
60F05, 60F15, 60F25.
1
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 2
Nomatter howp isconstructed, we always denotethezerosofp byZ(n),...,Z(n).
n n 1 n
Next, foreach positive integer k < n, let p(k) bethe kthderivative of p , andXn,k,j =
n n j
1,2,...,n k, be the zeros of p(k). In particular, when k = 1, Xn,1, ,Xn,1 are
− n 1 ··· n−1
called the critical points of p .
n
The relation between the zeros and the critical points of polynomials has been
much studied. For example, the Gauss-Lucas theorem asserts that all critical points
of a non-constant polynomial f lie inside the closed convex hull formed by the zeros
of f. It follows by induction that the zeros of f(k),k < deg f, also lie inside the same
closed convex hull. More refinements of Gauss-Lucas theorem can be found in [1]
and the references therein. A recent paper [2] also discussed some related results and
examples.
On the other hand, R. Pemantle and I. Rivin initiated a probabilistic study on the
limit of the critical points of random polynomials of Type 1. Consider the following
probability measures:
n n−k
1 1
(1.4) µ = δ , and µ(k) = δ ,1 k < n,
n n Zj(n) n n k Xjn,k ≤
j=1 − j=1
X X
where δ is the Dirac measure concentrated on z. µ and µ(k) are the empirical mea-
z n n
sures associated with the zeros Z(n) n and Xn,k n−k, respectively. R. Pemantle
{ j }j=1 { j }j=1
and I. Rivin ([3]) showed that, if Z(n) = Z and Z ∞ is a sequence of independent
j j { j}j=1
and identically distributed (i.i.d.) random variables governed by a common law ν,
then µ(1) w ν almost surely (a.s.) as n provided ν satisfies certain energy
n
→ → ∞
w
condition. In this paper means “converges weakly” or “converges in distribution”.
→
In the same i.i.d. setting without any further assumption on the probability law ν,
Z. Kabluchko ([4]) proved in great generality that µ(1) w ν in probability as n .
n
→ → ∞
For the case of higher order derivatives, in the i.i.d. setting, if the probability mea-
sure ν is supported on the unit circle in C, P. L. Cheung et. al. ([5]) showed that
µ(k) w ν a.s. as n . Similar results for the zeros of the generalized derivatives
n
→ → ∞
of polynomials are also obtained in [5].
TostatearesultofType2polynomials,recallthatapolynomialiscalledaKacpoly-
n
nomial ([6]) if it has the form ξ zj, where ξ ∞ is a sequence of non-degenerate
j { j}j=0
j=0
X
i.i.d. random variables. Furthermore, given a sequence of deterministic complex
n
numbers w ∞ , a polynomial of the form ξ w zj is called a Littlewood-Offord
{ j}j=0 j j
j=0
X
random polynomial. Clearly, any kth derivative of a Kac polynomial is a Littlewood-
Offord random polynomial. Z. Kabluchko and D. Zaporozhets ([6] and Theorem 14 of
[1]) proved that both sequences of the empirical measures µ and µ(k) converge
n n
{ } { }
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 3
weakly to the uniform distribution on the unit circle of C centered at the origin in
probability as n , provided that E[log(1 + ξ )] < . See [6] for the explicit
0
→ ∞ | | ∞
statements of the theorems and examples.
Under various settings and assumptions, the eigenvalue statistics of random ma-
trices/sample covariance matrices exhibits various interesting and important limit
behavious, for example the circle law, semicircle law, Marčenko-Pastur law, central
limit theorem, large deviations, and so on. See [7] and [8] for a systematic introduc-
tion. To name a result related to our work, consider a sequence of random Hermitian
matrices A(n) ∞ with p its characteristic polynomial. S. O’Rourke ([1]) showed
{ }n=1 n
that the Lévy distance between µ and µ(1) tends to zero almost surely as n .
n n
→ ∞
This observation implies that a.s. µ(1) converges weakly to the same semicircle law
n
{ }
as µ does. Such phenomenon that µ(1) converges weakly to the same law as that
n n
{ } { }
of µ (at some mode of convergence) was further demonstrated for several compact
n
{ }
classical matrix groups by S. O’Rourke in the same paper. Check [1] (theorem 6,
corollary 7, theorem 9, and remark 10) for explicit statements and references therein.
Here we merely point out that all the limit laws of the eigenvalue statistics of the
matrix models considered are compactly supported in C.
In view of the fact that all the results concerning the relation between µ and
n
{ }
µ(k) reviewed above are of the type of law of large numbers, it is natural to ask
n
{ }
how about other types of limit theorem? The goal of this paper is to show that if
certain limit property, for example law of large numbers, central limit theorem, law
of iterated logarithm, and so on, holds for the linear statistics of Z(n) n (see below
{ j }j=1
for the precise statement), then the same limit property passes to that of Xn,k n−k
{ j }j=1
for any k, provided the zeros Z(n) n satisfy some mild conditions which we now
{ j }j=1
state.
Denote by z the imaginary part of a complex number z.
ℑ
A1. There exists a non-negative constant C 0 independent of n such that
0
≥
(1.5) sup max Z(n) C a.s.
n∈N1≤j≤n|ℑ j | ≤ 0
That is, the imaginary parts of Z(n) are uniformly bounded with probability one.
{ j }
Since every zero of p(k) lies inside the closed convex hull of the zeros of p(k−1) by
n n
Gauss-Lucas theorem, we know by induction that
(1.6) sup max max Xn,k C a.s.
n∈N1≤k<n1≤j≤n−k|ℑ j | ≤ 0
When the zeros are real numbers, we put C = 0.
0
n
Recallthatµ w ν isequivalent to lim f dµ = lim 1 f(Z(n)) = f dν for
n → n→∞ˆ n n→∞ n j ˆ
j=1
X
each bounded continuous function f. It is therefore quite common to study such sums
(namely, linear statistics) for various categories of test functions. [9], [10], [11], [12],
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 4
and [13] are a few examples. In particular, the issue of regularity conditions for the
testfunctionsisdiscussed in[14]and[15]. Inthispaperwerestrictourselves toregular
test functions f such that f and f′ are bounded, fˆexists and f(x) = fˆ(t)eitxdt,
| |
and ´
(1.7) fˆ(t) e3C0|t|dt < ,
ˆ | | ∞
R
where i = √ 1,fˆ(t) = 1 f(u)e−itudu is the Fourier transform of f, and C is the
− 2π 0
same absolute constant ap´peared in (1.5).
To adapt to the different scalings in various limit theorems, we consider different
sequences of positive numbers for different linear statistics to be defined later. Below
we list three groups of assumptions to be used in the three main theorems of this
paper, respectively.
A2. There exists a sequence of positive numbers a ∞ such that
{ n}n=1
(1.8) lim a = ,
n
n→∞ ∞
n a a
(1.9) lim | n−k − n−k−1| = 0, for each fixed k < n 1,
n→∞ an−kan−k−1 −
Z(n)
(1.10) lim sup | j | = 0 a.s. for each fixed k < n.
n→∞1≤j≤n( an−k )
A3. There exists a sequence of positive numbers b ∞ such that
{ n}n=1
(1.11) lim b = ,
n
n→∞ ∞
n
b b
(1.12) lim | n−k − n−k−1| Z(n) = 0 a.s. for each fixed k < n 1,
n→∞ bn−kbn−k−1 " | j |# −
j=1
X
Z(n)
(1.13) lim sup | j | = 0 a.s. for each fixed k < n.
n→∞1≤j≤n( bn−k )
A4. There exist two sequences of positive numbers a ∞ and b ∞ such that
{ n}n=1 { n}n=1
(1.14) lim a = lim b = ,
n n
n→∞ n→∞ ∞
n
b b
(1.15) lim | n−k − n−k−1| Z(n) = 0 a.s. for each fixed k < n 1,
n→∞ an−k−1bn−kbn−k−1 " | j |# −
j=1
X
Z(n)
(1.16) lim sup | j | = 0 a.s. for each fixed k < n.
n→∞1≤j≤n(an−kbn−k)
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 5
Now we are ready to introduce the key objects that we want to study. Consider
the following three linear statistics associated with Z(n)
{ j }
1
L (f) = f Z(n) + +f Z(n) ,
n,1 a 1 ··· n
n
h (cid:16) (cid:17) i
Z(n) Z(cid:0)(n) (cid:1)
L (f) = f 1 + +f n ,
n,2
b ··· b
n ! n !
1 Z(n) Z(n)
L (f) = f 1 + +f n ,
n,3 a b ··· b
n " n ! n !#
and the three linear statistics associated with Xn,k
{ j }
1
L(k)(f) = f Xn,k + +f Xn,k ,
n,1 a 1 ··· n−k
n−k
h (cid:16) (cid:17) (cid:16) (cid:17)i
Xn,k Xn,k
L(k)(f) = f 1 + +f n−k ,
n,2 b ··· b
n−k! n−k !
1 Xn,k Xn,k
L(k)(f) = f 1 + +f n−k .
n,3 a b ··· b
n−k " n−k! n−k !#
It is also necessary to consider the centered (mean zero) linear statistics
L¯ (f) = L (f) E[L (f)], L¯(k)(f) = L(k)(f) E[L(k)(f)], ℓ = 1,2,3.
n,ℓ n,ℓ − n,ℓ n,ℓ n,ℓ − n,ℓ
For example, L (f) with a = n and L¯ (f) with a = √n play the typical roles
n,1 n n,1 n
in law of large numbers and central limit theorem, respectively. In the random matrix
models, one studies L with a = n,b = √n for law of large numbers results. In
n,3 n n
these cases A2 and A4 are valid obviously.
Theorem 1. Let the random zeros Z(n) n and the sequence a of positive num-
{ j }j=1 { n}
bers be given as above such that they satisfy the assumptions A1 and A2. If the linear
statistics L (f) has a limit as n at some mode of convergence, then, for each
n,1
→ ∞
k < n, the linear statistics L(k)(f) converges to the same limit at the same mode of
n,1
convergence. Similar statement holds for L¯ (f) and L¯(k)(f).
n,1 n,1
Theorem 2. Let the random zeros Z(n) n and the sequence b of positive num-
{ j }j=1 { n}
bers be given as above such that they satisfy the assumptions A1 and A3. If the linear
statistics L¯ (f) converges weakly to some probability law ν = ν as n , then
n,2 f
→ ∞
L¯(k)(f) w ν for each fixed k < n.
n,2 → f
Theorem 3. Let the random zeros Z(n) n and the sequences a and b of
{ j }j=1 { n} { n}
positive numbers be given as above such that they satisfy the assumptions A1 and A4.
If the linear statistics L (f) has a limit as n at some mode of convergence,
n,3
→ ∞
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 6
then, for each k < n, the linear statistics L(k)(f) converges to the same limit at the
n,3
same mode of convergence. Similar statement holds for L¯ (f) and L¯(k)(f).
n,3 n,3
Remark 1. Consider the Type 1 random polynomials with Z(n) = Z and Z being
j j { j}
an i.i.d. sequence. In this case the uniform condition (1.10) in A2 can be weakened
to
Z + + Z
lim | 1| ··· | n| = 0 a.s. for each fixed k < n.
n→∞ an−kn
Remark 2. Again consider the Type 1 random polynomials with Z(n) = Z and Z
j j { j}
being an i.i.d. sequence. This is the setting studied in [3], [4], and [5]. Note that
our Theorem 1 establishes, in addition to law of large numbers result, also central
limit theorem, law of iterated logarithm, and so on, for the linear statistics of Xn,k .
{ j }
However, Theorem 2 is not as interesting since it does not include the central limit
theoremofrandommatrixmodels. Thisisbecause thesize ofthezerosZ(n) isoforder
j
√n = b , and therefore the conditions (1.12) and (1.13) would not hold. Reasonable
n
condition(s) should involve the centered quantities.
In Section 2 we establish a comparison identity. It is elementary, and yet crucial
to our results. In Section 3 we prove three theorems and make some final remarks.
2. A comparison identity
First we state (with some modifications) a theorem of Cheung and Ng ([16]) which
is the starting point of our argument.
Proposition 4 (Theorem 1.1 of [16]). The set of all critical points of p (z) =
n
n (z z ),n 2, is the same as the set of all eigenvalues of the (n 1) (n 1)
k=1 − k ≥ − × −
matrix M :
n−1
Q
1
(2.1) M = D + (z I D )J ,
n−1 n−1 n 1 n−1 − n−1 n−1
where
z 0 0 1 1 1
2
··· ···
0 z 0 1 1 1
3
(2.2) Dn−1 =  ... ... ·.·.·. ... , Jn−1 =  ... ... ·.·.·. ... ,
 0 0 z   1 1 1 
 ··· n   ··· 
   
and I is the identity matrix of size n 1.
n−1
−
Denote by TrM the trace of a square matrix M. Our results rely on the following
observation.
Lemma 5. Let M and D be defined as in Proposition 2 and i = √ 1. Then
n−1 n−1
−
ic
(2.3) Tr eitMn−1 Tr eitDn−1 = n−1Tr J˜ eitDn−1 ,
n−1
− n
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 7
where
t S˜
c = c (t) = exp iu n−1 du,
n−1 n−1 ˆ n
0 !
S˜ = (z z )+ +(z z ), and
n−1 1 2 1 n
− ··· −
z z 0 0
1 2
− ···
0 z z 0
J˜n−1 =  ... 1 −... 3 ·.·.·. ... Jn−1.
 0 0 z z 
 ··· 1 − n 
 
Proof. It can be proved straightforwardly by Taylor expansions. However, to derive
the constant c in a more natural way, we first use the Duhamel formula
n−1
t
e(L1+L2)t eL1t = eL1(t−τ)L e(L1+L2)τ dτ
− ˆ 2
0
with L = iD and L + L = iM . In fact, L = iJ˜ . Since Tr(eA+B) =
1 n−1 1 2 n−1 2 n n−1
Tr(eAeB) by Tr(AB) = Tr(BA), the left hand side of (2.3) equals
i t
(2.4) Tr J˜ eitDn−1 e(iuJ˜n−1)/n du.
n ˆ n−1
0
(cid:16) (cid:17)
One can show by induction that J˜k = S˜k−1J˜ ,k N. After using this fact in
n−1 n−1 n−1 ∈
the Taylor expansion of e(iuJ˜n−1)/n, the integrand within the integral of (2.4) can be
simplified to
(2.5) Tr J˜ eitDn−1 e(iuS˜n−1)/n.
n−1
(cid:16) (cid:17)
This completes the proof. (cid:3)
Remark 3. The D appeared in [16] is a diagonal matrix with diagonal entries
n−1
z ,...,z , while we choose a different one given in (2.2) by the symmetry among
1 n−1
{ }
the zeros. The M in (2.1) is modified accordingly.
n−1
Remark 4. Observe that the terms in (2.5) can be expressed in a more symmetric
way:
S˜ n z
(2.6) n−1 = z j=1 j,
1
n − n
P
1 n eitzj n z eitzj
(2.7) Tr J˜ eitDn−1 = z j=1 j=1 j .
n−1 1
n n − n
P ! P
(cid:16) (cid:17)
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 8
Suppose that z n satisfies the bound max z C , then it is easy get that
{ j}j=1 1≤j≤n|ℑ j| ≤ 0
e2C0|t| 1
(2.8) c (t) − ,
| n−1 | ≤ 2C
0
1 n z
(2.9) Tr J˜ eitDn−1 eC0|t| z + j=1| j| .
n n−1 ≤ | 1| n
(cid:12) (cid:12) P !
(cid:16) (cid:17)
(cid:12) (cid:12)
When all zeros ar(cid:12)e real, C = 0 and(cid:12)one simply gets c (t) t . These are used in
(cid:12) 0 (cid:12) | n−1 | ≤ | |
the proofs of three theorems.
3. Proofs and concluding remarks
We now prove Theorem 1.
Proof. Consider the case of L (f) and L(1)(f) (k = 1). To prove the theorem in
n,1 n,1
this case, it suffices to show that L (f) L(1)(f) 0 a.s. as n . Write
n,1 − n,1 → → ∞
L (f) L(1)(f) = W +W , where
n,1 − n,1 n,1 n,2
1 1
W = f Z(n) + +f Z(n) f Z(n) + +f Z(n)
n,1 a 1 ··· n − a 2 ··· n
n n−1
h (cid:16) (cid:17) i h (cid:16) (cid:17) i
(cid:0) (cid:1) (cid:0) (cid:1)
f Z(n) n
1 a a
= + n−1 − n f Z(n) ,
(cid:16)a (cid:17) a a j
n n−1 n
Xj=2 (cid:16) (cid:17)
and
1 1
W = f Z(n) + +f Z(n) f Xn,1 + +f Xn,1 .
n,2 a 2 ··· n − a 1 ··· n−1
n−1 n−1
h (cid:16) (cid:17) i
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
Clearly
f n a a f
W k k∞ + | n−1 − n|k k∞ 0 a.s.
| n,1| ≤ a a a →
n n−1 n
by(1.8)and(1.9)ofA2. NextweapplyLemma3withZ(n) inplaceofz ,j = 1,...,n,
j j
to obtain that
1
W = fˆ(t) Tr eitDn−1 Tr eitMn−1 dt
n,2 a ˆ −
n−1
1 c(cid:2) (cid:0)(t) (cid:1) (cid:0) (cid:1)(cid:3)
= fˆ(t) n−1 Tr J˜ eitDn−1 dt.
a ˆ n n−1
n−1
(cid:16) (cid:17)
Since we assume A1, the estimates (2.8) and (2.9) in Remark 4 can be used to yield
that
1 Z(n) n Z(n)
(3.1) W fˆ(t) e3C0|t|dt | 1 | + j=1| j | .
| n,2| ≤ 2C ˆ | | a na
0 (cid:18) (cid:19) n−1 P n−1 !
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 9
By the regularity condition (1.7) together with (1.8) and (1.10) of A2, we conclude
that W 0 a.s. when n .
n,2
| | → → ∞
For k > 1 we decompose L (f) L(k)(f) = L (f) L(1)(f) + k−1L(j)(f)
n,1 − n,1 n,1 − n,1 j=1 n,1 −
L(j+1)(f) and need to show that each L(j+1)(f) L(j)(f) 0 a.s. We simply follow
n,1 | n,1 − n,1 | → P
the same strategy as above. Two facts can be useful when estimating the difference
between the traces of two matrices. First one is the basic relation between roots and
coefficients:
Z(n) + +Z(n) Xn,k + +Xn,k
1 ··· n = 1 ··· n−k,k < n.
n n k
−
The second relation can be found in [17] and [18] :
Xn,k + + Xn,k Xn,1 + + Xn,1 Z(n) + + Z(n)
| 1 | ··· | n−k| | 1 | ··· | n−1| | 1 | ··· | n |.
n k ≤ ··· ≤ n 1 ≤ n
− −
The rest is easy and is omitted. When demonstrating the weak convergence of L¯(k)
n,1
from that of L¯ , the converging together lemma should be applied to complete the
n,1
proof. (cid:3)
The proofs of Theorem 2 and Theorem 3 are similar to that of Theorem 1. A mean
value inequality and the boundedness of f′ can justify the transference of the scales
| |
from b to b . When estimating the terms like (3.1), the uniform condition
n−k n−k−1
(1.13) in A3 and/or (1.16) in A4 would be helpful. The rest parts are routine and
thus omitted.
Remark 5. In the proofs of the theorems we deal with the strongest mode of conver-
gence, namely the almost sure convergence, of L (f) L(k)(f) 0,ℓ = 1,2,3. If
n,ℓ − n,ℓ →
the original L (f),ℓ = 1,3, converges at a weaker mode, an alternative argument
n,ℓ
using weaker estimates should be considered. Consequently, it is conceivable that the
assumptions weaker than A1 and A2 (or A4) might be sufficient for the theorems to
hold, and the regularity conditions on the test functions used in the linear statistics
might also be relaxed.
Remark 6. One can show that if the empirical measure associated with the zeros of
p , under appropriatescaling, obeys a largedeviations principle, so does theempirical
n
measure associated with the zeros of p(k) for each k. This is treated elsewhere ([19]).
n
References
1. Sean O’Rourke, Critical points of random polynomials and characteristic polyno-
mials of random matrices. arXiv: 1412.4703v1 (2014).
2. T. R. R. Annapareddy, On critical points of random polynomials and spectrum of
certain products of random matrices. arXiv: 1602.05298v1 (2016)
3. R. Pemantle and I. Rivin, The distribution of zeros of the derivative of a random
polynomial. Advances in Combinatorics, Springer (2013), pp. 259 – 273.
COMMON LIMIT OF THE LINEAR STATISTICS OF RANDOM ZEROS 10
4. Z. Kabluchko, Critical points of random polynomials with independent identically
distributed roots. Proc. Amer. Math. Soc. 143 (2015), no. 2, pp. 695 – 702.
5. P. L. Cheung, T. W. Ng, J. Tsai, and S. C. P. Yam, Higher order, polar and
Sz.-Nagy’s generalized derivatives of random polynomials with independent and
identically distributed zeros on the unit circle. Computational Methods and Func-
tion Theory (2014), pp. 1 – 28.
6. Z. Kabluchko and D. Zaporozhets, Asymptotic distribution of complex zeros of
random analytic functions. Ann. Probab., 42 (4) (2014), pp. 1374 – 1395.
7. G. W. Anderson,A. Guionnet, and O. Zeitouni, An introduction to random ma-
trices. Cambridge studies in advanced mathematics. Cambridge University Press,
2009.
8. Z. D. Bai and J. W. Silverstein, Spectral analysis of large dimensional random
matrices. 2nd edition, Springer, 2010.
9. A. Lytova and L. Pastur, Central limit theorem for linear eigenvalue statistics
of random matrices with independent entries. Ann. Probab. 37 (2009), pp. 1778-
1840.
10. A. Lytova and L. Pastur, Non-Gaussian limiting laws for the entries of regular
functions of the Wigner matrices. arXiv: 1103.2345v2 (2011)
11. Z. D. Bai, X. Y. Wang, and W. Zhou, CLT for linear spectral statistics of Wigner
matrices. Electronic Journal of Probab. vol 14 (2009) pp. 2391 – 2417.
12. I. Jana, K. Saha, and A. Soshnikov, Fluctuations of linear eigenvalue statistics of
random band matrices. arXiv: 1412.2445v2 (2015)
13. L. Y. Li, M. Reed, and A. Soshnikov, Central limit theorem for linear eigenvalue
statistics for submatrices of Wigner random matrices.arXiv: 1504.05933v1(2015)
14. Phil Kopel, Regularity conditions for convergence of linear statistics of GUE.
arXiv: 1510.02988v2 (2015)
15. P. Sosoe and P. Wong, Regularity conditions in the CLT for linear eigenvalue
statistics of Wigner matrices. arXiv: 1210.5666v2 (2015)
16. W. S. Cheung and T. W. Ng, A companion matrix approach to the study of zeros
and critical points of a polynomial. J. Math. Anal. Appl. 319 (2006), no. 2, pp.
690 – 707.
17. N. G. de Bruijn and T. A. Springer, On the zeros of a polynomial and of its
derivative II. Indag. Math. 9 (1947), pp. 264 – 270.
18. P. Erdos and I. Niven, On the roots of a polynomial and its derivative. Bull Amer.
Math. Soc. 54 (1948), pp. 184 – 190.
19. The common limit of the linear statistics of zeros of random polynomials and their
derivatives II. In preparation (2017)