隨機多項式的一個普遍性

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國立臺灣大學理學院數學系碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

隨機多項式的一個普遍性

A Universality of Polynomials with Complex Random Roots

胡亦行 I-Shing Hu

指導教授：張志中博士

Advisor: Chih-Chung Chang, Ph.D.

中華民國 106 年 2 月 Feb 2017

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口試委員會審定書………...………

誌謝………..……….ii

英文摘要………...……….…….iii

Introduction……….………..1

A Comparison Identity………..6

Proofs and Concluding Remarks………...8

Large Deviations………...………..10

Appendix……….12

Reference……….…...… 17

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誌謝

如果誌謝只能打一行，我要感謝我的母親和張志中老師。後者示範了怎麼做數學而前者示範了怎麼做人。另外要感謝口試委員們：陳宏老師、黃啟瑞老師和江金倉老師，提供了寶貴的意見。還要感謝陳俊全老師和下列（基於隱私恕不一一列名）：

LU、q-group、Chen、413、阿斯拉、實習、讀書會(們)、圖書館

希望我有資格引用這句話作結：

「所有的科學著作都應該是某種推理小說—這是一份追尋聖杯過程的報告書」

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iii

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A UNIVERSALITY OF CRITICAL POINTS OF POLYNOMIALS WITH COMPLEX RANDOM ROOTS

I-SHING HU

Abstract. Let pn(x) be a random polynomial of degree n and {Zj⁽ⁿ⁾}ⁿ_j=1 and {X_j^n,k}^n−k_j=1, k < n, be the zeros of pn and p^(k)n , the kth derivative of pn, respectively.

We show that if the linear statistics 1 a_n

"

f Z₁⁽ⁿ⁾ b_n

!

+ · · · + f Zn⁽ⁿ⁾

b_n

!#

associated with {Zj⁽ⁿ⁾}has a limit as n → ∞ at some mode of convergence, the linear statistics associated with {Xj^n,k} converges to the same limit at the same mode. Similar statement also holds for the centered linear statistics associated with the zeros of pn and p^(k)n , provided the zeros {Zj⁽ⁿ⁾}and the sequences {an}and {bn}of positive numbers satisfy some mild conditions.

1. Introduction

Fix a probability space (Ω, F, P) and let {pn(z)}^∞_n=1 be a sequence of random polynomials such that deg pn = n. We observe that the randomness of polynomials can be introduced in several dierent ways.

Type 1: Given a triangular array of random zeros {Z_j⁽ⁿ⁾}ⁿ_j=1, n = 1, 2, . . ., let (1.1) p_n(z) = (z − Z₁⁽ⁿ⁾) · · · (z − Z_n⁽ⁿ⁾).

Here the coecient of zⁿ is set to be 1 for simplicity.

Type 2: Given a triangular array of random coecients {a⁽ⁿ⁾j }ⁿ_j=0, n = 1, 2, . . ., let

(1.2) p_n(z) = a⁽ⁿ⁾_n zⁿ+ · · · + a⁽ⁿ⁾₁ z + a⁽ⁿ⁾₀ .

Type 3: Given a sequence of random matrices {A⁽ⁿ⁾}^∞_n=1, let

(1.3) p_n(z) = det(zI − A⁽ⁿ⁾),

the characteristic polynomial of A⁽ⁿ⁾. Here I, the identity matrix, and A⁽ⁿ⁾ are square matrices of size n.

2010 Mathematics Subject Classication. Primary: 30C15, 60B10; secondary: 60B20, 60G57, 60F05, 60F15, 60F25.

1

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A UNIVERSALITY OF CRITICAL POINTS OF POLYNOMIALS 2

No matter how pnis constructed, we always denote the zeros of pnby Z1⁽ⁿ⁾, . . . , Zn⁽ⁿ⁾. Next, for each positive integer k < n, let p^(k)n be the kth derivative of pn, and X_j^n,k, j = 1, 2, . . . , n − k, be the zeros of p^(k)ⁿ . In particular, when k = 1, X1^n,1, · · · , X_n−1^n,1 are called the critical points of pn. 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 renements 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:

(1.4) µ_n= 1 n

n

X

j=1

δ_Z(n) j

, and µ^(k)_n = 1 n − k

n−k

X

j=1

δ_X^n,k

j , 1 ≤ k < n,

where δz is the Dirac measure concentrated on z. µn and µ^(k)n are the empirical measures associated with the zeros {Z_j⁽ⁿ⁾}ⁿ_j=1 and {X_j^n,k}^n−k_j=1, respectively. R. Pemantle and I. Rivin ([3]) showed that, if Zj⁽ⁿ⁾= Z_j and {Zj}^∞_j=1 is a sequence of independent and identically distributed (i.i.d.) random variables governed by a common law ν, then µ⁽¹⁾ⁿ → ν^w almost surely (a.s.) as n → ∞ provided ν satises certain energy condition. In this paper →^w 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⁾ⁿ → ν^w in probability as n → ∞.

For the case of higher order derivatives, in the i.i.d. setting, if the probability measure ν is supported on the unit circle in C, P. L. Cheung et. al. ([5]) showed that µ^(k)n

→ νw a.s. as n → ∞. Similar results for the zeros of the generalized derivatives of polynomials are also obtained in [5].

To state a result of Type 2 polynomials, recall that a polynomial is called a Kac polynomial ([6]) if it has the form

n

X

j=0

ξjz^j, where {ξj}^∞_j=0 is a sequence of non-degenerate i.i.d. random variables. Furthermore, given a sequence of deterministic complex numbers {wj}^∞_j=0, a polynomial of the form

n

X

j=0

ξjwjz^j is called a Littlewood-Oord random polynomial. Clearly, any kth derivative of a Kac polynomial is a Littlewood- Oord random polynomial. Z. Kabluchko and D. Zaporozhets ([6] and Theorem 14 of [1]) proved that both sequences of the empirical measures {µn} and {µ^(k)ⁿ } converge weakly to the uniform distribution on the unit circle of C centered at the origin in

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probability as n → ∞, provided that E[log(1 + |ξ0|)] < ∞. See [6] for the explicit statements of the theorems and examples.

Under various settings and assumptions, the eigenvalue statistics of random matrices/sample covariance matrices exhibits various interesting and important limit behavious, for example the circle law, semicircle law, Mar£enko-Pastur law, central limit theorem, large deviations, and so on. See [7] and [19] for a systematic introduction. To name a result related to our work, consider a sequence of random Hermitian matrices {A⁽ⁿ⁾}^∞_n=1 with pn its characteristic polynomial. S. O'Rourke ([1]) showed that the Lévy distance between µn and µ⁽¹⁾ⁿ tends to zero almost surely as n → ∞.

This observation implies that a.s. {µ⁽¹⁾ⁿ }converges weakly to the same semicircle law as {µn}does. Such phenomenon that {µ⁽¹⁾ⁿ }converges weakly to the same law as that of {µn}(at some mode of convergence) was further demonstrated for several compact 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 {µn} and {µ^(k)n } reviewed above are of the type of law of large numbers, it is natural to ask 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_j⁽ⁿ⁾}ⁿ_j=1 (see below for the precise statement), then the same limit property passes to that of {Xj^n,k}^n−k_j=1 for any k, provided the zeros {Zj⁽ⁿ⁾}ⁿ_j=1 satisfy some mild conditions which we now state.

Denote by =z the imaginary part of a complex number z.

A1. There exists a non-negative constant C0 ≥ 0 independent of n such that

(1.5) sup

n∈N

max

1≤j≤n|=Z_j⁽ⁿ⁾| ≤ C₀ a.s.

That is, the imaginary parts of {Zj⁽ⁿ⁾} are uniformly bounded with probability one.

Since every zero of p^(k)n lies inside the closed convex hull of the zeros of p^(k−1)n by Gauss-Lucas theorem, we know by induction that

(1.6) sup

n∈N

max

1≤k<n max

1≤j≤n−k|=X_j^n,k| ≤ C₀ a.s.

When the zeros are real numbers, we put C0 = 0. Recall that µn → νw is equivalent to lim

n→∞

ˆ

f dµ_n = lim

n→∞

1 n

n

X

j=1

f (Z_j⁽ⁿ⁾) = ˆ

f dν for 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], and [13] are a few examples. In particular, the issue of regularity conditions for

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the test functions is discussed in [14] and [15]. In this paper we restrict ourselves to regular test functions f such that f, f⁰ ∈ L^∞, ˆf exists and f(x) =´ f (t)eˆ ^itxdt, and (1.7)

ˆ

R

| ˆf (t)|e^3C⁰^|t|dt < ∞, where i = √

−1, ˆf (t) = _2π¹ ´

f (u)e^−itudu is the Fourier transform of f, and C0 is the same absolute constant appeared in (1.5).

To adapt to the dierent scalings in various limit theorems, we consider dierent sequences of positive numbers for dierent linear statistics to be dened 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 {an}^∞_n=1 such that

n→∞lim an= ∞, (1.8)

n→∞lim

n |a_n−k− a_n−k−1|

a_n−ka_n−k−1 = 0, for each xed k < n − 1, (1.9)

n→∞lim sup

1≤j≤n

(|Z_j⁽ⁿ⁾| a_n−k

)

= 0 a.s. for each xed k < n.

(1.10)

A3. There exists a sequence of positive numbers {bn}^∞_n=1 such that

n→∞lim bn= ∞, (1.11)

n→∞lim

|b_n−k − b_n−k−1| bn−kbn−k−1

" _n X

j=1

|Z_j⁽ⁿ⁾|

#

= 0 a.s. for each xed k < n − 1, (1.12)

n→∞lim sup

1≤j≤n

(|Z_j⁽ⁿ⁾| b_n−k

)

(1.13)

A4. There exist two sequences of positive numbers {an}^∞_n=1 and {bn}^∞_n=1 such that

n→∞lim a_n= lim

n→∞b_n= ∞, (1.14)

n→∞lim

n |a_n−k − a_n−k−1|

a_n−ka_n−k−1 = 0, for each xed k < n − 1, (1.15)

n→∞lim

|b_n−k − b_n−k−1| an−k−1bn−kbn−k−1

" _n X

j=1

|Z_j⁽ⁿ⁾|

#

= 0 a.s. for each xed k < n − 1, (1.16)

n→∞lim sup

1≤j≤n

( |Z_j⁽ⁿ⁾| a_n−kb_n−k

)

(1.17)

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Now we are ready to introduce the key objects that we want to study. Consider the following three linear statistics associated with {Zj⁽ⁿ⁾}

L_n,1(f ) = 1 an

h f

Z₁⁽ⁿ⁾

+ · · · + f Z_n⁽ⁿ⁾i ,

L_n,2(f ) = f Z₁⁽ⁿ⁾ b_n

!

+ · · · + f Zn⁽ⁿ⁾

b_n

! ,

L_n,3(f ) = 1 a_n

"

f Z₁⁽ⁿ⁾ b_n

!

+ · · · + f Zn⁽ⁿ⁾

b_n

!#

, and the three linear statistics associated with {Xj^n,k}

L^(k)_n,1(f ) = 1 a_n−k

h f

X₁^n,k

+ · · · + f

X_n−k^n,k i ,

L^(k)_n,2(f ) = f X₁^n,k b_n−k

!

+ · · · + f X_n−k^n,k b_n−k

! ,

L^(k)_n,3(f ) = 1 a_n−k

"

f X₁^n,k b_n−k

!

+ · · · + f X_n−k^n,k b_n−k

!#

. It is also necessary to consider the centered (mean zero) linear statistics

L¯_n,`(f ) = L_n,`(f ) − E[L_n,`(f )], L¯^(k)_n,`(f ) = L^(k)_n,`(f ) − E[L^(k)_n,`(f )], ` = 1, 2, 3.

For example, Ln,1(f ) with an= n and ¯Ln,1(f ) with an =√

n play the typical roles in law of large numbers and central limit theorem, respectively. In the random matrix models, one studies Ln,3 with an = n, b_n = √

n for law of large numbers results. In these cases A2 and A4 are valid obviously.

Theorem 1. Let the random zeros {Zj⁽ⁿ⁾}ⁿ_j=1 and the sequence {an} of positive numbers be given as above such that they satisfy the assumptions A1 and A2. If the linear statistics Ln,1(f ) has a limit as n → ∞ at some mode of convergence, then, for each k < n, the linear statistics L^(k)n,1(f ) converges to the same limit at the same mode of convergence. Similar statement holds for ¯Ln,1(f ) and ¯L^(k)n,1(f ).

Theorem 2. Let the random zeros {Zj⁽ⁿ⁾}ⁿ_j=1 and the sequence {bn} of positive numbers be given as above such that they satisfy the assumptions A1 and A3. If the linear statistics ¯Ln,2(f ) converges weakly to some probability law ν = νf as n → ∞, then L¯^(k)_n,2(f )→ ν^w f for each xed k < n.

Theorem 3. Let the random zeros {Zj⁽ⁿ⁾}ⁿ_j=1 and the sequences {an} and {bn} of positive numbers be given as above such that they satisfy the assumptions A1 and A4.

If the linear statistics Ln,3(f ) has a limit as n → ∞ at some mode of convergence,

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then, for each k < n, the linear statistics L^(k)n,3(f ) converges to the same limit at the same mode of convergence. Similar statement holds for ¯Ln,3(f ) and ¯L^(k)n,3(f ).

Remark 1. Consider the Type 1 random polynomials with Zj⁽ⁿ⁾= Z_j and {Zj}being an i.i.d. sequence. In this case the uniform condition (1.10) in A2 can be weakened to

n→∞lim

|Z₁| + · · · + |Z_n|

an−kn = 0 a.s. for each xed k < n.

Remark 2. Again consider the Type 1 random polynomials with Zj⁽ⁿ⁾= Z_j and {Zj} 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 {X_j^n,k}. However, Theorem 2 is not as interesting since it does not include the central limit theorem of random matrix models. This is because the size of the zeros Z_j⁽ⁿ⁾is of order

√n = bn, and therefore the conditions (1.12) and (1.13) would not hold. Reasonable 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 nal remarks.

2. A comparison identity

First we state (with some modications) 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 pn(z) = Qn

k=1(z − z_k), n ≥ 2, is the same as the set of all eigenvalues of the (n − 1) × (n − 1) matrix Mn−1:

(2.1) M_n−1 = D_n−1+ 1

n (z₁I_n−1− D_n−1) J_n−1, where

(2.2) D_n−1=







z₂ 0 · · · 0 0 z₃ · · · 0 ... ... ... ...

0 0 · · · z_n







, J_n−1 =







1 1 · · · 1 1 1 · · · 1 ... ... ... ...

1 1 · · · 1





 ,

and In−1 is the identity matrix of size n − 1.

Denote by Tr M the trace of a square matrix M. Our results rely on the following observation.

Lemma 5. Let Mn−1 and Dn−1 be dened as in Proposition 2 and i =√

−1. Then (2.3) Tr e^itMⁿ⁻¹ − Tr e^itDⁿ⁻¹ = ic_n−1

n Tr ˜J_n−1e^itDⁿ⁻¹ ,

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where

c_n−1 = c_n−1(t) = ˆ _t

0

exp iuS˜_n−1 n

! du, S˜_n−1 = (z₁− z₂) + · · · + (z₁− z_n), and

J˜_n−1 =







z₁− z₂ 0 · · · 0 0 z₁− z₃ · · · 0 ... ... ... ...

0 0 · · · z1 − zn





 J_n−1.

Proof. It can be proved straightforwardly by Taylor expansions. However, to derive the constant cn−1 in a more natural way, we rst use the Duhamel formula

e^(L¹^+L²^)t− e^L¹^t= ˆ t

0

e^L¹^{(t−τ )}L₂e^(L¹^+L²^)τdτ

with L1 = iD_n−1 and L1 + L₂ = iM_n−1. In fact, L2 = _nⁱJ˜_n−1. Since Tr(e^A+B) = Tr(e^Ae^B)by Tr(AB) = Tr(BA), the left hand side of (2.3) equals

(2.4) i

n ˆ _t

0

Tr ˜J_n−1e^itDⁿ⁻¹e^{(iu ˜}^Jⁿ⁻¹^)/n du.

One can show by induction that ˜J_n−1^k = ˜S_n−1^k−1J˜_n−1, k ∈ N. After using this fact in the Taylor expansion of e^{(iu ˜}^Jⁿ⁻¹^)/n, the integrand within the integral of (2.4) can be simplied to

(2.5) Tr ˜J_n−1e^itDⁿ⁻¹

e^{(iu ˜}^Sⁿ⁻¹^)/n.

This completes the proof.

Remark 3. The Dn−1 appeared in [16] is a diagonal matrix with diagonal entries {z₁, . . . , z_n−1}, while we choose a dierent one given in (2.2) by the symmetry among the zeros. The Mn−1 in (2.1) is modied accordingly.

Remark 4. Observe that the terms in (2.5) can be expressed in a more symmetric way:

S˜_n−1

n = z1− Pn

j=1z_j

n ,

(2.6)

1

nTr ˜J_n−1e^itDⁿ⁻¹

= z₁ Pn

j=1e^itz^j n

!

− Pn

j=1z_je^itz^j

n .

(2.7)

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Suppose that {zj}ⁿ_j=1 satises the bound max

1≤j≤n|=zj| ≤ C0, then it is easy get that

|c_n−1(t)| ≤ e^2C⁰^|t|− 1 2C₀ , (2.8)

1

nTr ˜J_n−1e^itDⁿ⁻¹

≤ e^C⁰^|t| |z₁| + Pn

j=1|zj| n

! (2.9) .

When all zeros are real, C0 = 0and one simply gets |cn−1(t)| ≤ |t|. These are used in the proofs of three theorems.

3. Proofs and concluding remarks We now prove Theorem 1.

Proof. Consider the case of Ln,1(f ) and L⁽¹⁾n,1(f ) (k = 1). To prove the theorem in this case, it suces to show that Ln,1(f ) − L⁽¹⁾_n,1(f ) → 0 a.s. as n → ∞. Write L_n,1(f ) − L⁽¹⁾_n,1(f ) = W_n,1+ W_n,2, where

W_n,1 = 1 a_n

h f

Z₁⁽ⁿ⁾

+ · · · + f Z_n⁽ⁿ⁾i

− 1

a_n−1 h

f Z₂⁽ⁿ⁾

+ · · · + f Z_n⁽ⁿ⁾i

= f

Z₁⁽ⁿ⁾

an

+a_n−1− a_n an−1an

n

X

j=2

f Z_j⁽ⁿ⁾

, and

Wn,2 = 1 a_n−1

h f

Z₂⁽ⁿ⁾

+ · · · + f Z_n⁽ⁿ⁾i

− 1

a_n−1 f X₁^n,1 + · · · + f X_n−1^n,1 . Clearly

|W_n,1| ≤ kf k∞

an

+n |a_n−1− a_n|kf k∞

an−1an

→ 0 a.s.

by (1.8) and (1.9) of A2. Next we apply Lemma 3 with Zj⁽ⁿ⁾in place of zj, j = 1, . . . , n, to obtain that

W_n,2 = 1 a_n−1

ˆ

f (t)ˆ Tr e^itDⁿ⁻¹ − Tr e^itMⁿ⁻¹ dt

= 1

an−1

ˆ

f (t)ˆ c_n−1(t)

n Tr ˜J_n−1e^itDⁿ⁻¹ dt.

Since we assume A1, the estimates (2.8) and (2.9) in Remark 4 can be used to yield that

(3.1) |W_n,2| ≤ 1 2C₀

ˆ

| ˆf (t)|e^3C⁰^|t|dt

|Z₁⁽ⁿ⁾| a_n−1 +

Pn

j=1|Z_j⁽ⁿ⁾| na_n−1

! .

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By the regularity condition (1.7) together with (1.8) and (1.10) of A2, we conclude that |Wn,2| → 0 a.s. when n → ∞.

For k > 1 we decompose Ln,1(f ) − L^(k)_n,1(f ) = L_n,1(f ) − L⁽¹⁾_n,1(f ) +Pk−1

j=1L^(j+1)_n,1 (f ) − L^(j)_n,1(f ) and need to show that each |L^(j)n,1(f ) − L^(j+1)_n,1 (f )| → 0 a.s. We simply follow the same strategy as above. Two facts can be useful when estimating the dierence between the traces of two matrices. First one is the basic relation between roots and coecients:

Z₁⁽ⁿ⁾+ · · · + Zn⁽ⁿ⁾

n = X₁^n,k+ · · · + X_n−k^n,k

n − k , k < n.

The second relation can be found in [17] and [18] :

|X₁^n,k| + · · · + |X_n−k^n,k |

n − k ≤ · · · ≤ |X₁^n,1| + · · · + |X_n−1^n,1|

n − 1 ≤ |Z₁⁽ⁿ⁾| + · · · + |Zn⁽ⁿ⁾|

n .

The rest is easy and is omitted. When demonstrating the weak convergence of ¯L^(k)n,1

from that of ¯Ln,1, the converging together lemma should be applied to complete the

proof.

The proofs of Theorem 2 and Theorem 3 are similar to that of Theorem 1. A mean value inequality and the boundedness of |f⁰| can justify the transference of the scales from bn−k to bn−k−1. When estimating the terms like (3.1), the uniform condition (1.13) in A3 and/or (1.16) in A4 would be helpful.

Now prove Theorem 2.

Proof. Again, since

L˜_n,2− ˜L⁽¹⁾_n,2 = (

n

X

k=1

f (Z_k⁽ⁿ⁾

b_n ) − Ef (Z_k⁽ⁿ⁾ b_n )) − (

n

X

k=2

f (Z_k⁽ⁿ⁾

b_n−1) − Ef (Z_k⁽ⁿ⁾ b_n−1)) + (

n

X

k=2

f (Z_k⁽ⁿ⁾

b_n−1) − Ef (Z_k⁽ⁿ⁾ b_n−1)) − (

n−1

X

k=1

f (X_k^n,1

b_n−1) − Ef (X_k^n,1 b_n−1)), where (again) rst two summation terms vanishes by A3 and f, f⁰ ∈ L^∞. The last two summation terms (again) are equal to

ˆ

f (t)ˆ Tr eîtDⁿ⁻¹^/bⁿ⁻¹ − Tr eîtMⁿ⁻¹^/bⁿ⁻¹−E Tr eîtDⁿ⁻¹^/bⁿ⁻¹ − Tr eîtMⁿ⁻¹^/bⁿ⁻¹ dt, hence it is bounded by

1 2C₀

ˆ

"

|Z₁⁽ⁿ⁾| b_n−1 +

Pn

j=1|Z_j⁽ⁿ⁾| nb_n−1

!

+ E |Z₁⁽ⁿ⁾| b_n−1 +

Pn

j=1|Z_j⁽ⁿ⁾| nb_n−1

!#

which also vanishes by A1 and A3.

Finally prove Theorem 3.

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Proof. Similarly

L_n,3= 1 a_n

n

X

k=1

f (Z_k⁽ⁿ⁾

b_n ) − 1 a_n−1

n

X

k=2

f (Z_k⁽ⁿ⁾

b_n−1) + 1 a_n−1

n

X

k=2

f (Z_k⁽ⁿ⁾

b_n−1) − 1 a_n−1

n−1

X

k=1

f (X_k^n,1 b_n−1), where (again) rst two summation terms vanishes by A4 and f, f⁰ ∈ L^∞. The last two summation terms (again) are equal to

1 a_n−1

ˆ

f (t)ˆ Tr e^itDⁿ⁻¹^/bⁿ⁻¹ − Tr e^itMⁿ⁻¹^/bⁿ⁻¹ dt hence it is bounded by

1 2a_n−1C₀

ˆ

|Z₁⁽ⁿ⁾| b_n−1 +

Pn

j=1|Z_j⁽ⁿ⁾| nb_n−1

!

which also vanishes by A1 and A4.

Remark 5. In the proofs of the theorems we deal with the strongest mode of convergence, namely the almost sure convergence, of Ln,`(f ) − L^(k)_n,`(f ) → 0, ` = 1, 2, 3. If the original Ln,`(f ), ` = 1, 3, converges at a weaker mode, an alternative argument using weaker estimates should be considered. Consequently, it is conceivable that the assumptions weaker than A1 and A2 (or A4) might be sucient 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 pn, under appropriate scaling, obeys a large deviations principle, so does the empirical measure associated with the zeros of p^(k)n for each k. This is treated elsewhere ([21]).

4. Large Deviations

But what can we say if we only have large deviation principle?

Observe that: if {xn}_n∈N, {y_n}_n∈N⊂ R and lim

n→∞x_n− y_n= 0, then lim inf

n→∞ x_n= lim inf

n→∞ y_n and

lim sup

n→∞

x_n = lim sup

n→∞

y_n.

So if

n→∞lim Pr {X_n ∈ A} − Pr {Y_n ∈ A} = 0

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for any A is Borel measurable and {Xn}_n∈N satisfy large deviation principle, then {Y_n}_n∈N also satisfy large deivation principle with the same rate function.

Obviously, the total variation sup

A

| Pr {X_n ∈ A} − Pr {Y_n ∈ A} | vanishes as n → ∞ implies

n→∞lim Pr {Xn ∈ A} − Pr {Yn ∈ A} = 0.

Nevertheless, even

n→∞limX_n− Y_n = 0

almost surely does not imply the total variation vanishes as n → ∞. One way to guarantee is that

Lemma 6. If there is another measure m such that Pr {Xn ∈ A} = ´

A

f_ndm and Pr {Y_n ∈ A} = ´

A

g_ndm for all A is measurable, where fn, g_n are integrable w.r.t.

measure m, then

n→∞limsup

A

| Pr {Xn ∈ A} − Pr {Yn ∈ A} | = 0 ⇔ lim

n→∞

ˆ

|fn− gn|dm = 0.

Proof: One side is trivial sinceˆ

|f_n− g_n|dm ≥ ˆ

A

|f_n− g_n|dm ≥ | ˆ

A

f_n− g_ndm|, for any A is measurable. Conversely, because

sup

A

| Pr {X_n ∈ A}−Pr {Y_n ∈ A} | = max





 ˆ

{fn−gn≥0}

f_n− g_ndm, − ˆ

{fn−gn<0}

f_n− g_ndm







→ 0

as n → ∞, we have ˆ

|f_n− g_n|dm = ˆ

{fn−gn≥0}

f_n− g_ndm − ˆ

{fn−gn<0}

f_n− g_ndm → 0 as n → ∞. Done.

This is the end of our main results.

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5. Appendix

Proof of Proposition 4: Since (here pn(z) = (z − Z₁) · · · (z − Z_n))

0 = p⁰_n(w) p_n(w) =

n

X

j=1

1 w − Z_j

if w is a critical point of pn and w is not equal to any of roots Zj. We have 1

w − Z_n =

n−1

X

j=1

1 Z_j− w, so

n

X

j=1

Z_j − Z_n Z_j− w =

n

X

j=1

Z_j − w + w − Z_n Z_j− w = n.

Conversely this observation reveals that if a number λ /∈ {Zj}ⁿ⁻¹_j=1 and Pⁿ

j=1 Zj−Zn

Zj−λ = n, then λ 6= Zn and p⁰n(λ) = 0.

First we would show any critical point w of pn, w /∈ {Zj}ⁿ_j=1, is an eigenvalue of M with eigenvector

Z1−Zn

n(Z1−w), · · · ,_n(Z^Zⁿ⁻¹^−Zⁿ

n−1−w)

T

via

M







Z1−Zn

n(Z1−w), ...

Zn−1−Zn

n(Zn−1−w)





=







Z1(Z1−Zn) n(Z1−w)

...

Zn−1(Zn−1−Z_n) n(Zn−1−w)





− 1 n²

n−1

X

j=1

Z_j − Z_n Z_j− w D



 1...

1





=







Z1(Z1−Z_n) n(Z1−w)

...

Zn−1(Zn−1−Zn) n(Zn−1−w)





−







Z1(Z1−Z_n) n...

Zn−1(Zn−1−Zn) n





= w







Z1(Z1−Z_n) n(Z1−w)

...

Zn−1(Zn−1−Zn) n(Zn−1−w)





.

If w is equal to one of roots saying Zi, then w = Zi = Z_j, i 6= j. In particular if i = n, then w = Zn is an eigenvalue of M with eigenvector

e_j + 1, · · · , 1T

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by showing

M





 1...

2...

1







=





 Z₁

...

2Z_j ...

Z_n−1







−







Z₁− Z_n ...

0...

Z_n−1− Z_n







= Z_n





 1...

2...

1





 ,

where ej is the standard basis, i.e. , 2 only occurs at j^th component.

For w = Zi = Z_j, i, j ≤ n − 1, we have

M (e_i− e_j) = D(e_i− e_j) = w(e_i− e_j).

Therefore

{z C | p⁰n(z) = 0} ⊆ σ(M ).

Conversely if λ ∈ σ(M) with eigenvector v1, · · · , v_n−1T

, i.e.





(Z₁− λ)v₁ ...

(Zn−1− λ)vn−1



= 1 n(

n−1

X

j=1

v_j)





Z₁ − Z_n ...

Zn−1− Zn



.

If ⁿ⁻¹P

j=1

vj = 0, then at least two of {vj}ⁿ⁻¹_j=1, saying v1 and v2, are non-zero. Hence λ = Z₁ = Z₂. Now assume ⁿ⁻¹P

j=1

v_j 6= 0. If λ = Zj, j ≤ n − 1, then Zn = Z_j = λ. If λ /∈ {Zj}ⁿ⁻¹_j=1, then

v_j

n−1

P

i=1

v_i

= Z_j − Z_n n(Z_j− λ). Summing up all j from 1 to n − 1 we have

n

X

j=1

Zj − Z_n Z_j − λ = n, as a result,

{z C | p⁰n(z) = 0} ⊇ σ(M ).

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Adopt the proof above we have

Lemma 7 (Gauss-Lucas Theorem). Again pn(x) :=

n

Q

k=1

(x − Z_k), we have {z C | p⁰n(z) = 0} ⊆ Conv {Z_k}ⁿ_k=1.

Proof: Let w ∈ {z C | p⁰n(z) = 0} \ {Z_k}ⁿ_k=1 (otherwise it is trivial), then 0 = p⁰_n(w)

p_n(w) =

n

X

j=1

1 w − Z_j =

n

X

j=1

w − Zj

|w − Z_j|², so we conclude that

w =

n

P

j=1 Zj

|w−Z_j|² n

P

j=1 1

|w−Z_j|²

∈ Conv {Z_k}ⁿ_k=1.

From above we can have

Proposition 8. If {Zk}ⁿ_k=1 ⊆ L^p, p ≥ 1, then

{z C | p⁰n(z) = 0} ⊆ L^p.

Proof: Let w ∈ {z C | p⁰n(z) = 0} \ {Z_k}ⁿ_k=1 (otherwise it is trivial), then (through Jensen's inequality)

E|w|^p = E|

n

X

j=1

Zj

|w−Zj|² n

P

k=1 1

|w−Z_k|²

|^p ≤ sup

1≤k≤n

{E|Zk|^p} < ∞.

The ideas below originally follow from [19]: First it is easy to see that

Proposition 9 ([19], Lemma 2.2 and Lemma 2.3). If every coecients akof a random polynomial Pⁿ

k=0

a_kx^k are in L^p, p ≥ 1, then the polynomial is continuous in L^p, i.e.

x→xlim0

E|

n

X

k=0

a_kx^k−

n

X

k=0

a_kx^k₀|^p = 0 and hence continuous in probability.

We also observe that

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Proposition 10. Let pn(x) :=

n

P

k=0

akx^k be a random polynomial, then

E [p_n(β) − p_n(α) | σ {p_n(x) : x ≤ α}] = E

ˆ β α

p⁰_n(x)dx | σ {a_k}ⁿ_k=1

, hence pn(x) is a sub- or super-martingale on [a, b] ⊆ R if p⁰_n(x) ≥ or ≤ 0 on [a, b].

Nevertheless we still need to answer a fundamental question: are roots of

p_n(x) :=

n

X

k=0

a_kx^k

measurable if complex coecients {ak}ⁿ_k=0 are measurable?

To answer the question, we need measurable selection Theorem due to Kuratowski and Ryll-Nardzewski, and the proof presented here follows from [20].

Theorem 11 ([20], Theorem 6.9.3). Let (Ω, Σ)be a measurable space, X be a Polish space and F : Ω → X be a mapping with values in the family of non-empty closed subsets of X. If for every open subset U ⊂ X, we have

F (U ) := {ω ∈ Ω : F (ω) ∩ G 6= φ} ∈ Σ.˜ Then F has a Σ−measurable selection f : Ω → X such that

f (ω) ∈ F (ω) , ∀ ω ∈ Ω.

Proof: Let {xn}_n∈N be a countable dense subset of X. Dene f0 : Ω → {x_n}_n∈N as f₀(ω) := x_j,

where

j := inf {n ∈ N : F (ω) ∩ B (xn, 1) 6= φ} . Hence f0 is Σ−measurable since

f₀⁻¹{x_j} = ˜F (B (x_j, 1)) \ ^j−1∪

m=1

F (B(x˜ _m, 1)) ∈ Σ.

Now we construct inductively measurable mappings fk : Ω → {x_n}_n∈N such that d (f_k(ω) , f_k+1(ω)) < 2^−k+1

and

d (f_k(ω) , F (ω)) < 2^−k.

Suppose fk is already constructed. Fix k, then for all disjoint f_k⁻¹{xi} , i ∈ N, dene

f_k+1 : f_k⁻¹{x_i} → {x_n}_n∈N

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as

f_k+1(ω) := x_j, where

j := infn ∈ N : F (ω) ∩ B xi, 2^−k ∩ B x_n, 2^−k−1 6= φ since F (ω) ∩ B xi, 2^−k 6= φ, ∀ ω ∈ f_k⁻¹{x_i}.

So

d (f_k+1(ω) , F (ω)) < 2^−k−1 and

d (f_k(ω) , f_k+1(ω)) < 2^−k + 2^−k−1 < 2^−k+1. As a result, we have a Σ−measurable mapping

lim

k→∞f_k(ω) =: f (ω) ∈ F (ω) , since {fk(ω)}_k≥0 is Cauchy and φ 6= F (ω) is closed.

Now we are able to answer the question.

Theorem 12 ([19], Theorem 2.2). All roots of pn(x) :=

n

X

k=0

akx^k

are Σ−measurable if complex coecients {ak}ⁿ_k=0 are Σ−measurable.

Proof: Let F (ω) := p⁻¹n {0} ⊂ C, which is closed and non-empty in C by Funda- mental Theorem of Algebra, and let D be countable dense subset of C.

Observe that for every open subset G ⊂ X, F (G)˜ ^c= ∪

x∈G{ω ∈ Ω : pn(x) ∈ {0}^c} := ∪

x∈GAx, i.e.,

ω ∈ A_x, x ∈ G ⇐⇒

n

X

k=0

a_k(ω) x^k 6= 0.

Fix such ω ∈ Ax and x ∈ G, then

∃ ˜x ∈ G ∩ D 3

n

X

k=0

ak(ω) ˜x^k ∈ {0}^c

since pn(ω)⁻¹({0}^c)is open (then ˜x is in a neighbourhood of x contained in pn(ω)⁻¹({0}^c) such that Pⁿ

k=0

a_k(ω) ˜x^k 6= 0) and ˜x ∈ D ⊂ C is dense.

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Hence

F (G)˜ ^c = ∪

x∈GAx = ∪

˜

x∈G∩DAx˜ ∈ Σ.

Applying above measurable selection theorem, we have Z₁(ω) ∈ p⁻¹_n {0}

is Σ−measurable.

Take

p_n−1(x) := p_n(x) x − Z₁

(we may assume an 6= 0 a.s.) and then the result follows by backward induction.

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Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, R.O.C.

E-mail address: [email protected]