Limit laws for the energy of a charged polymer

(1)

www.imstat.org/aihp 2008, Vol. 44, No. 4, 638–672

DOI: 10.1214/07-AIHP120

Limit laws for the energy of a charged polymer

Xia Chen

¹

Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA. E-mail: xchen@math.utk.edu Received 14 July 2006; revised 12 January 2007; accepted 1 March 2007

Abstract. In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy

Hn=

1≤j <k≤n

ω_jω_k1_{_S_j₌_S_k_}

of the polymer{S₁, . . . , S_n}equipped with random electrical charges{ω₁, . . . , ω_n}. Our approach is based on comparison of the moments betweenH_nand the self-intersection local time

Q_n=

1≤j <k≤n 1_{_S_j₌_S_k_}

run by thed-dimensional random walk{Sk}. As partially needed for our main objective and partially motivated by their independent interest, the central limit theorems and exponential integrability forQnare also investigated in the cased≥3.

Résumé. Cet article est consacré à l’étude du théorème central limite, des déviations modérées et des lois du logarithme itéré pour l’énergie

Hn=

1≤j <k≤n

ω_jω_k1_{_S_j₌_S_k_}

du polymère{S1, . . . , Sn}doté de charges électriques{ω1, . . . , ωn}. Notre approche se base sur la comparaison des moments de Hnet du temps local de recoupements

Qn=

1≤j <k≤n 1_{_S_j₌_S_k_}

de la marche aléatoired-dimensionelle{Sk}. L’étude du théorème central limite et de l’intégrabilité exponentielle deQn(dans le casd≥3) est également menée, tant pour comme outil pour notre principal objectif que pour son intérêt intrinsèque.

MSC:60F05; 60F10; 60F15

Keywords:Charged polymer; Self-intersection local time; Central limit theorem; Moderate deviation; Laws of the iterated logarithm

1Research partially supported by NSF Grant DMS-0704024.

(2)

1. Introduction

In the physics literature, the geometric shape of certain polymers is often described as an interpolation line segment with the vertices given as then-step lattice (simple) random walk

{S1, S2, . . . , Sn}.

By placing independent, identically distributed electric chargesω_k= ±1 to each vertex of the polymer, Kantor and Kardar [16] consider a model of polymers with random electrical charges associated with the Hamiltonian

H_n=

1≤j <k≤n

ω_jω_k1_{Sj=Sk}. (1.1)

In the physics literature,H_nis called the energy of the polymer. To understand the physics intuition ofH_n, we assign an electrical chargeω_k to the random siteS_k for allk=1,2, . . . .Assume that when two charges meet, the pair with opposite signs gives negative contribution while the pair with the same sign gives positive contribution. Thus,H_n represents the total electrical interaction charge of the polymer{S1, S2, . . . , S_n}.

We point out some other works by physicists in this direction. In [10], the charges are i.i.d. Gaussian variables.

In [11], the charges take 0–1 values. We also refer the reader to [4,18] for the continuous versions of the polymer with random charges. Finally, we mention the survey paper by van der Hofstad and König [12] for a long list of mathematical models connected to polymers.

As for other connections, we cite the comment by Martínez and Petritis [18]: “It is argued that a protein molecule is very much like a random walk with random charges attached at the vertices of the walk; these charges are interacting through local interactions mimicking Lennard–Jones or hydrogen-bond potentials”.

We study the asymptotic behaviors ofH_n given in (1.1). In the rest of the paper,{S_n}n≥1is a symmetric random walk onZ^d with covariance matrixΓ (or varianceσ²as d=1). We assume that the smallest group that supports {S_n}n≥1isZ^d. Throughout,{ω_k}k≥1is an i.i.d. sequence of symmetric random variable with

Eω²₁=1 and Ee^λ⁰^ω²¹<∞ for someλ₀>0. (1.2)

Our first result is on the central limit theorems.

Theorem 1.1. Asd=1, 1

n^3/4H_n−→^d (2σ )⁻^1/2 _∞

−∞L²(1, x)dx 1/2

U, (1.3)

where U is a random variable with standard normal distribution, L(t, x) is the local time of the 1-dimensional Brownian motionW (t )such thatUandW (t )are independent.

Asd=2,

√ 1

nlognHn

−→d 1

√2π√⁴

detΓU. (1.4)

Asd≥3,

√1

nH_n−→^d √

γ U, (1.5)

where

γ=^∞

k=1

P{S_k=0}. (1.6)

(3)

Here is our explanation on the dimensional dependence appearing in Theorem 1.1. The higher the dimension is, the less likely the random walk is to have long-range interaction (self-intersection). In the multi-dimensional case (d≥2), therefore,H_nis a sum of random variables with weak dependence and yields a Gaussian limit when properly normalized. It should be pointed that the low level of long-range interaction is vital for the chaos

1≤j <k≤n

aj,kωjωk

to have a Gaussian limit when properly normalized. A simple example is whena_j,k≡1. In this case

1≤j <k≤n

ω_jω_k=1 2

_n

j=1

ω_j 2

− n j=1

ω²_j .

By the classic law of large numbers and classic central limit theorem, 1

n

1≤j <k≤n

ωjωk

−→d 1 2

U²−1

which sharply contrasts to the statements in Theorem 1.1.

Our next theorem describes the moderate deviation behaviors ofH_n. Theorem 1.2. Asd=1,

nlim→∞

1 b_nlogP

±H_n≥λ(nb_n)^3/4

= −1

2σ^2/3(3λ)^4/3, λ >0, (1.7)

for any positive sequence{b_n}satisfying bn→ ∞ and bn=o√₇

n

, n→ ∞. (1.8)

Asd=2,

nlim→∞

1 b_nlogP

±Hn≥λ

n(logn)bn

= −π

det(Γ )λ², λ >0, (1.9)

for any positive sequence{bn}satisfying

b_n→ ∞ and b_n=o(logn), n→ ∞. (1.10)

Asd≥3,

nlim→∞

1 bn

logP

±H_n≥λ nb_n

= −λ²

2γ, λ >0, (1.11)

for any positive sequence{bn}satisfying bn→ ∞ and bn=o

n logn

1/4

, n→ ∞. (1.12)

Our moderate deviations applied to the law of the iterated logarithm:

Theorem 1.3. Asd=1, lim sup

n→∞

±Hn

(nlog logn)^3/4=2^3/4

3 σ⁻^1/2 a.s. (1.13)

(4)

Asd=2, lim sup

n→∞

±H_n

√nlognlog logn= 1

√πdet(Γ )^1/4 a.s. (1.14)

Asd≥3, lim sup

n→∞

±Hn

√nlog logn=

2γ a.s. (1.15)

We compare the results and treatments between the present paper and some recent works on self-intersection local times such as [3,6]. On the one hand, we shall see that the asymptotic behaviors ofHndescribed in our main theorems are closely related to those of the self-intersection local time

Q_n=

1≤j <k≤n

1_{Sj=Sk}. (1.16)

Indeed, our approach is based on the moment comparisons betweenHnandQn (see Proposition 2.1). In particular, the difference in limiting distribution between the cased =1 and the cased ≥2 in Theorem 1.1 is caused by the fact that in the cased=1,Qnconverges (in distribution) to the Brownian self-intersection local times when properly normalized (see [8]), while asd≥2,Qnis asymptotically close to its expectation (see [3] ford=2 and the Section 5 ford≥3).

On the other hand, the fact thatQnis close to the quadratic form

x∈Z^d

l²(n, x)

of the local time l(n, x)plays a crucial role in the study of the self-intersection local timeQn (see e.g., [3,8]). It allows, for example, some technologies developed along the line of probability in Banach space. Unfortunately, this idea does not work in our setting. Indeed, the fact (in view of Theorem 1.1) that the second term in the decomposition

x∈Z^d

_n

j=1

ω_j1_{Sj=x}

2

=2Hn+ n j=1

ω²_j (1.17)

is the dominating term shows thatHnis not even in the same asymptotic order as the quadratic form on the left-hand side.

The key estimations are carried out in Proposition 2.1. Our approach relies on the following crucial observation.

By (1.17) we have

H_n=1 2

x∈Z^d

_n

j=1

ω_j1_{Sj=x}

2

− n j=1

ω_j1_{Sj=x} . (1.18)

Conditioned on the random walk{Sk}, the random variables _n

j=1

ωj1_{S_j=x}

2

− n j=1

ωj1_{S_j=x}, x∈Z^d,

form an independent family and, for each fixx∈Z^d, _n

j=1

ωj1_{S_j=x}

2

− n j=1

ωj1_{S_j=x} d

= _l(n,x)

j=1

ωj

2

−

l(n,x)

j=1

ωj.

(5)

By a classic estimate for independent sums, and by some combinatorial computation, a conditional moment estimate given in Proposition 2.1 linksH_nwithQ_n. Another fact repeatedly used in this work is that the self-intersection occur- ring at the frequently visited sites does not make a significant contribution to the quantitiesH_nandQ_n. Consequently, the pairsH_nandH_n(defined in (2.3));Q_nandQ_n(defined in (2.4) below) are exchangeable in our setting.

Beyond mathematical technicality, the creation of the present paper is based on our belief thatH_nresembles, in the limiting behaviors described in our main theorems, the random quantity

H_n=

1≤j <k≤n

1_{S_j=S_k}U_j,k,

whereU_j,k are i.i.d. standard normal random variables independent of{S_k}. Notice thatH_n is conditionally normal with conditional varianceQ_n. Our observation explains why and how the limiting behaviors ofH_n depend on its conditional varianceQn. It should be pointed out, however, that the replacement ofωjωkbyUj,kis highly non-trivial and should not be taken for granted, in view of the example given next to Theorem 1.1.

In Section 2, we establish a comparison (Proposition 2.1) between the moments ofHn andQn, and then apply it to prove Theorem 1.1. Our approach relies on combinatorial and conditioning methods. In Section 3, Proposition 2.1 is further applied to prove Theorem 1.2 through a Laplacian argument. In Section 4, the laws of the iterated logarithm given in Theorem 1.3 are proved as a consequence of our moderate deviations. The non-trivial part of this section is a maximal inequality (Lemma 4.1) of Lévy type. In Section 5, we investigate the weak laws and exponential integrabilities for the renormalized self-intersection local timeQn−EQnin the high dimensions (d≥3). The central limit theorem given in Theorem 5.1 and the exponential integrability given in Theorem 5.2 provide sharp bounds onQn−EQn, which constitute the replacement ofQn byEQn carried out in our argument for Theorem 1.1 and for Theorem 1.2 (the estimate ofQn−EQn needed in the cased =2 was established in [3,20]). In addition, the results given in Section 5 are of independent interest as a part of the study of the self-intersection local times in high dimensions and are partially motivated by some recent works of Asselah and Castell [1] and Asselah [2].

2. Moment comparison and laws of weak convergence

We begin with the following classic lemma.

Lemma 2.1. Assume(1.2).Then E

_n

j=1

ω_j 2

− n j=1

ω_j²

2

=2n(n−1). (2.1)

More generally,there is a constantC >0such that for any integersn≥1andm≥2, E

_n

j=1

ωj

2

− n j=1

ω²_j

m

≤m!

Cn(n−1)m/2

. (2.2)

Proof. The first part follows from the following straightforward computation:

E _n

j=1

ω_j 2

− n j=1

ω_j²

2

=4E

1≤j <k≤n

ω_jω_k 2

=4

1≤j <k≤n

E ω²_jω²_k

=2n(n−1).

For the second part, we only need to show E

_n

j=1

ωj

2

− n j=1

ω²_j

m

≤m!C^m/2n^m.

(6)

By the inequality

E

_n

j=1

ω_j 2

− n j=1

ω²_j

m1/m

≤

E _n

j=1

ω_j

2m1/m

+

E _n

j=1

ω_j²

_m1/m

all we need is that E

_n

j=1

ω_j 2m

≤C^m/2m!n^m

and that E

n j=1

ω²_j

m

≤C^m/2m!n^m.

Due to similarity we only prove the first inequality. Notice that by symmetry E

_n

j=1

ω_j 2m

=

k1+···+kn=m k1,...,kn≥0

(2m)!

(2k1)! · · ·(2kn)!Eω^2k¹· · ·Eω^2kⁿ.

By the integrability given in (1.2) there is a constantc₁>0 such that Eω^2k≤k!c^k₁, k=0,1,2, . . . .

Notice also the very rough estimate

c^k₂(k!)²≤(2k)! ≤c^k₃(k!)², k=0,1,2, . . . . So we have

E _n

j=1

ω_j 2m

≤C^m/2m!

k₁+···+k_n=m k₁,...,k_n≥0

m!

k1! · · ·k_n!=C^m/2m!n^m.

Let K_n be a positive sequence which may vary in different settings and will later be specified in each specific setting. Recall thatQ_nis given in (1.16) and define the local time

l(n, x)= n k=1

1_{S_k=x}, x∈Z^d, n=1,2, . . . .

The asymptotic behaviors of the local times of the random walks have been studied extensively. We cite the book by Révész [19] for an overview.

The following two random quantities play important roles in this paper:

Hn=Hn1_{sup_x∈Z_dl(n,x)≤Kn}, (2.3)

Q_n=Q_n1_{sup_x∈Zdl(n,x)≤Kn}. (2.4)

In addition, we introduce the deterministic quantity Am(n)= 1

2^m

(y1,...,ym)∈Bm

E

1_{sup_x_∈Z_dl(n,x)≤K_n}

m k=1

l(n, yk)

l(n, yk)−1 ,

(7)

wherem, n=1,2, . . .and B_m=

(y₁, . . . , y_m)∈ Z^dm

;y₁, . . . , y_mare distinct

. (2.5)

An easy observation gives that A_m(n)≤ 1

2^m

y1,...,ym∈Z^d

E

1_{sup_x_∈Z_dl(n,x)≤Kn}

m k=1

l(n, y_k)

l(n, y_k)−1

=EQ^m_n. (2.6)

Some more substantial comparisons are given in the following.

Proposition 2.1. There is a constantC >0independent ofn,mand the choice ofK_n,such that EH_n^m≤ m!

2^m

[2⁻¹m] l=1

1

l!K_n^m⁻^2l2^lC^(m⁻^2l)/2

m−l−1 m−2l

EQ^l_n. (2.7)

On the other hand,for any integersm, n≥1, EH_n^2m≥(2m)!

2^mm!Am(n), (2.8)

EQ^m_n ≤ m

l=1

m l

lK_n² 2

_m₋_l

Al(n). (2.9)

Proof. Notice that H_n=1

2

x∈Z^d

_n

j=1

ω_j1_{Sj=x}

2

− n j=1

ω_j²1_{Sj=x} =1 2

x∈Z^d

Λ_n(x) (say). (2.10)

Hence,

EH_n^m=2⁻^m

x1,...,xm∈Z^d

E

1_{sup_x∈Z_dl(n,x)≤Kn}

m k=1

Λ_n(x_k)

.

For each 1≤l≤m, let

Al= {(x1, . . . , xm)∈(Z^d)^m;#{x1, . . . , xm} =l}.

Then,

EH_n^m=2⁻^m m l=1

(x1,...,xm)∈Al

E

m k=1

Λn(xk)

. (2.11)

Write Cl=

F ⊂Z^d;#(F )=l and for any{y1, . . . , yl} ∈Cl, set

Al(y1, . . . , yl)=

(x1, . . . , xm)∈ Z^dm

; {x1, . . . , xm} = {y1, . . . , yl} .

(8)

Notice that

1Al(x1, . . . , x_m)=

{y₁,...,y_l}∈Cl

1Al(y1,...,yl)(x1, . . . , x_m).

Thus

(x₁,...,x_m)∈A_l

E

1_{sup_x_∈Zdl(n,x)≤Kn}

m k=1

Λ_n(x_k)

=

{y1,...,yl}∈Cl

(x₁,...,x_m)∈A_l(y₁,...,y_l)

E

1_{sup_x∈Zdl(n,x)≤Kn}

m k=1

Λ_n(x_k)

.

For any(x1, . . . , xm)∈Al(y1, . . . , yl), letikbe the number ofx1, . . . , xmwhich are equal toyk, wherek=1, . . . , l.

Then E

m k=1

Λ_n(x_k)

=E

l k=1

Λ_n(y_k)ⁱ^k .

Consequently,

(x1,...,xm)∈A_l(y1,...,y_l)

E

m k=1

Λ_n(x_k)

=

i1+···+i_l=m i1,...,i_l≥1

m! (i1)! · · ·(i_l)!E

l k=1

Λ_n(y_k)ⁱ^k .

Summarizing the above discussion,

(x1,...,xm)∈Al

E

m k=1

Λn(xk)

=

{y1,...,yl}∈Cl

i₁+···+i_l=m i₁,...,i_l≥1

m! (i₁)! · · ·(i_l)!E

l k=1

Λn(yk)ⁱ^k .

Notice that the quantity f (y1, . . . , y_l)≡

i1+···+il=m i1,...,il≥1

m! (i1)! · · ·(il)!E

l k=1

Λ_n(y_k)ⁱ^k

is invariant under the permutations over{y1, . . . , yl}. So we have

(x₁,...,x_m)∈A_l

E

m k=1

Λ_n(x_k)

=1 l!

(y₁,...,y_l)∈B_l

i1+···+i_l=m i1,...,i_l≥1

m! (i1)! · · ·(i_l)!E

l k=1

Λ_n(y_k)ⁱ^k ,

(9)

whereB_l is defined by (2.5). By (2.11) EH_n^m=2⁻^m

m l=1

1 l!

i₁+···+i_l=m i₁,...,i_l≥1

m! (i1)! · · ·(i_l)!

×

(y1,...,y_l)∈B_l

E

l k=1

Λ_n(y_k)ⁱ^k . (2.12)

We adopt the notation “E^ω” for the expectation with respect to{ω_k}k≥1and for eachy∈Z^d, writeD(y)= {1≤ k≤n;S_k=y}. Then

Λ(y)=

j∈D(y)

ω_j 2

−

j∈D(y)

ω²_j

and for distincty₁, . . . , y_l, the setsD(y₁), . . . , D(y_l)are disjoint. Hence, by independence, E^ω

l k=1

Λn(yk)ⁱ^k= l k=1

E^ωΛn(yk)ⁱ^k.

In particular, the above quantity is zero if any ofi₁, . . . , i_l is 1. Consequently, the terms in (2.12) withl > m/2 are equal to zero,

EH_n=0 (2.13)

and for any integerm≥2, EH_n^m=2^−m

[2⁻¹m]

l=1

1 l!

i₁+···+i_l=m i₁,...,i_l≥2

m! (i1)! · · ·(i_l)!

×

(y1,...,y_l)∈B_l

E

l k=1

E^ωΛ_n(y_k)ⁱ^k . (2.14)

Notice that

E^ωΛn(yk)ⁱ^k =E^ω _l(n,x)

j=1

ωj

2

−

l(n,x)

j=1

ω_j²

i_k

. (2.15)

By Lemma 2.1 we have l

k=1

E^ωΛ_n(y_k)ⁱ^k ≤ l k=1

i_k!C_iⁱ^k^/2

k

l(n, y_k)

l(n, y_k)−1i_k/2

=(i1! · · ·i_l!) C_iⁱ¹^/2

1 · · ·C_iⁱ^l^/2

l

^l

k=1

l(n, y_k)

l(n, y_k)−1i_k/2

,

whereCi=1 asi=2 andCi is the constantC given in (2.2) asi≥3. We may assume thatC≥1 in the rest of the proof.

(10)

Hence, EH_n^m≤ m!

2^m

[2⁻¹m] l=1

1 l!

i₁+···+i_l=m i₁,...,i_l≥2

C_iⁱ¹^/2

1 · · ·C_iⁱ^l^/2

l

×

(y₁,...,y_l)∈B_l

E

1_{sup_x∈Zdl(n,x)≤Kn}

l k=1

l(n, y_k)

l(n, y_k)−1i_k/2

≤ m! 2^m

[2⁻¹m]

l=1

1 l!K_n^m−^2l

i₁+···+i_l=m i₁,...,i_l≥2

C_iⁱ¹^/2

1 · · ·C_iⁱ^l^/2

l A_l(n)

≤ m! 2^m

[2⁻¹m] l=1

1

l!K_n^m⁻^2l2^l

i1+···+il=m i1,...,il≥2

C_iⁱ¹^/2

1 · · ·C_iⁱ^l^/2

l EQ^l_n, (2.16)

where the last step follows from (2.6).

For each(i1, . . . , il)withi1+ · · · +il=m, write k=k(i₁, . . . , i_l)=#{1≤j≤l;i_j=2}. We have

m=i₁+ · · · +i_l≥2k+3(l−k) which leads tol−k≤m−2l. Thus,

i1+···+il=m i1,...,il≥2

C_iⁱ¹^/2

1 · · ·C_iⁱ^l^/2

l

=

i1+···+il=m i1,...,il≥2

C^(l⁻^k)/2≤C^(m⁻^2l)/2

i1+···+il=m i1,...,il≥2

1

=C^(m⁻^2l)/2

m−l−1 m−2l

.

Hence, (2.7) follows from (2.16).

To prove (2.8), we come to (2.14) and we notice that the symmetry of{ω_k}implies that for any integerl≥1, E^ωΛ_n(y_k)ⁱ^k=2ⁱ^kE

1≤j1<j2≤l(n,x)

ω_j₁ω_j₂ i_k

≥0. (2.17)

Replacingmby 2min (2.4) and only keeping the term withl=mon the right-hand side, we obtain EH_n^2m≥ (2m)!

2^2mm!2⁻^m

(y1,...,ym)∈Bm

E

m k=1

E^ωΛ_n(y_k)²

=(2m)! 2^mm!A_m(n),

where the second step follows from (2.1) in Lemma 2.1 and (2.15).

(11)

To prove (2.9), we adopt the argument used for (2.12).

EQ^m_n =2⁻^m

x₁,...,x_m∈Z^d

E

m k=1

l(n, xk)

l(n, xk)−1

=2⁻^m m l=1

1 l!

i1+···+il=m i1,...,il≥1

m! i₁! · · ·i_l!

×

(y₁,...,y_l)∈B_l

E

1_{sup_x_∈Z_dl(n,x)≤Kn}

l k=1

l(n, y_k)

l(n, y_k)−1i_k

≤ m! 2^m

m l=1

1 l!

i1+···+il=m i1,...,il≥1

1 i1! · · ·il!

K_n^2(m⁻^l)

×

(y1,...,yl)∈Bl

E

1_{sup_x∈Z_dl(n,x)≤K_n}

l k=1

l(n, yk)

l(n, yk)−1

=m!

m l=1

1

l!K_n^2(m⁻^l)2⁻^(m⁻^l)Al(n)

i₁+···+i_l=m i₁,...,i_l≥1

1 i₁! · · ·i_l!.

Finally, (2.9) follows from the following estimate:

i1+···+il=m i1,...,il≥1

1

i₁! · · ·i_l! ≤

i1+···+il=m i1,...,il≥1

1

(i₁−1)! · · ·(i_l−1)!

=

i1+···+il=m−l i1,...,il≥0

1

i1! · · ·il!= l^m⁻^l (m−l)!.

Proof of Theorem 1.1. We start with the cased=1. Notice that Q_n=1

2

x∈Z

l²(n, x)−n

.

By Theorem 1.2 of [6], n⁻^3/2Qn

−→d 1 2σ

_∞

−∞L²(1, x)dx. (2.18)

Fix 0< δ <1/2 and letKn=n⁽¹^+δ)/2. By the classic fact (see, for example, [19]) that n⁻^1/2sup

x∈Zl(n, x)−→^d σ⁻¹sup

x∈RL(1, x) (2.19)

we have that

n⁻^3/2Q_n−→^d 1 2σ

_∞

−∞L²(1, x)dx, (2.20)

(12)

which gives

nlim→∞n⁻^3m/2EQ^m_n = 1 (2σ )^mE

_∞

−∞L²(1, x)dx m

, m=1,2, . . . . (2.21)

Replacingmby 2m+1 in (2.7) we have EH_n^2m⁺¹≤ (2m+1)!

2^2m⁺¹ m

l=1

1

l!n⁽¹⁺^δ)(m⁻^l)⁺⁽¹⁺^δ)/22^lC^(2m⁻^2l⁺^1)/2

2m−l 2m−2l+1

EQ^l_n

=O _m

l=1

n⁽¹⁺^δ)(m⁻^l)⁺⁽¹⁺^δ)/2n^3l/2

=o

n^3(2m⁺^1)/4

, n→ ∞, (2.22)

for allm=0,1,2, . . . .

Replacingmby 2min (2.7) and by (2.21) we have EH_n^2m≤ (2m)!

2^2m m l=1

1

l!n⁽¹⁺^δ)(m⁻^l)2^lC^m⁻^l

2m−l−1 2m−2l

EQ^l_n

∼(2m)! 2^2m

m l=1

1

l!n⁽¹⁺^δ)(m⁻^l)(2σ )⁻^ln^3l/22^l

×E _∞

−∞L²(1, x)dx l

C^m⁻^l

2m−l−1 2m−2l

, n→ ∞.

Clearly, the right-hand side is dominated by the term withl=m. Consequently, lim sup

n→∞ n⁻^3m/2EH_n^2m≤ 1 (2σ )^m

(2m)! 2^mm!E

_∞

−∞L²(1, x)dx m

(2.23) for allm=1,2, . . . .In particular, combining this with (2.8) we have

Am(n)=O n^3m/2

, n→ ∞, m=1,2, . . . . (2.24)

On the other hand, by (2.24) and (2.21), the right-hand side of (2.9) is dominated by the term withl=m. Hence, lim inf

n→∞ n⁻^3m/2Am(n)≥ 1 (2σ )^mE

_∞

−∞L²(1, x)dx _m

, m=1,2, . . . . (2.25)

From (2.8), lim inf

n→∞ n⁻^3m/2EH_n^2m≥ 1 (2σ )^m

(2m)! 2^mm!E

_∞

−∞L²(1, x)dx _m

. (2.26)

In summary of (2.22), (2.23) and (2.26), and noticing that EU^2m=(2m)!

2^mm! and EU^2m⁺¹=0 we have that for everym=0,1,2, . . . ,

nlim→∞n⁻^3m/4EH_n^m= 1 (2σ )^m/2

EU^m E

_∞

−∞L²(1, x)dx m/2

. (2.27)