#  Eigenvalue Fluctuations of 1-dimensional random Schrödinger operators

Takuto Mashiko, Yuma Marui, Naoki Maruyama, and Fumihiko Nakano  Mathematical
Institute, Tohoku University, Sendai 980-8578, Japan e-mail :
<EMAIL_ADDRESS>
###### Abstract

As an extension to the paper by Breuer, Grinshpon, and White [2], we study the
linear statistics for the eigenvalues of the Schrödinger operator with random
decaying potential with order ${\cal O}(x^{-\alpha})$ ($\alpha>0$) at
infinity. We first prove similar statements as in [2] for the trace of $f(H)$,
where $f$ belongs to a class of analytic functions : there exists a critical
exponent $\alpha_{c}$ such that the fluctuation of the trace of $f(H)$
converges in probability for $\alpha>\alpha_{c}$, and satisfies a CLT
statement for $\alpha\leq\alpha_{c}$, where $\alpha_{c}$ differs depending on
$f$. Furthermore we study the asymptotic behavior of its expectation value.

##  1 Introduction

The study of the one-dimensional Schrödinger operator with random decaying
potential was initiated by [3] where they found that it has various spectral
properties depending on the decay exponent. After the discovery [4] of the
connection between the Jacobi matrices and beta-ensemble, there appear many
papers studying from the RMT-point of view, e.g., the eigenvalue statistics on
the bulk [9, 7, 10, 8], linear statistics [11, 2], and eigenfunction
statistics [13, 12]. In this paper, we consider the following Hamiltonian, to
extend the work by Breuer, Grinshpon, and White [2].

$\displaystyle(H_{\alpha}u)_{n}$ $\displaystyle:=$
$\displaystyle\begin{cases}u_{n+1}+u_{n-1}+V_{\alpha}(n)u_{n}&(n\geq 2)\\\
u_{2}+V_{\alpha}(1)u_{1}&(n=1)\end{cases}$ $\displaystyle V_{\alpha}(n)$
$\displaystyle:=$ $\displaystyle\frac{X_{n}}{n^{\alpha}},\quad n\in{\bf N}$

where $\\{X_{n}\\}$ is a family of i.i.d. random varables. For $N\in{\bf N}$,
let $H_{\alpha,N}$ be the restriction of $H_{\alpha}$ on $[1,N]$. Breuer,
Grinshpon, and White [2] studied the fluctuation of $Tr(P(H_{\alpha,N}))$ and
showed that we can find a critical exponent $\alpha_{c}$ depending on
$P\in{\cal P}_{m}$ such that the behavior of the fluctuation of
$TrP(H_{\alpha,N})$ changes drastically at $\alpha_{c}$ :

$\displaystyle(1)$ $\displaystyle\quad(\alpha>\alpha_{c})\quad
TrP(H_{\alpha,N})-{\bf E}[TrP(H_{\alpha,N})]\stackrel{{\scriptstyle
a.s.}}{{\to}}Q$ $\displaystyle(2)$
$\displaystyle\quad(\alpha\leq\alpha_{c})\quad\frac{1}{g_{\alpha/\alpha_{c}}(N)}\left\\{TrP(H_{\alpha,N})-{\bf
E}[TrP(H_{\alpha,N})]\right\\}\stackrel{{\scriptstyle
d}}{{\to}}N(0,\sigma(P)^{2})$

where $Q$ is a random variable with finite variance, and

$\displaystyle g_{t}(N)$ $\displaystyle:=$
$\displaystyle\begin{cases}\dfrac{N^{1-t}}{1-t}&(0<t<1)\\\ \log
N&(t=1)\end{cases}$

$\sigma(P)\geq 0$ is an explicit constant. Moreover, the space ${\cal P}_{m}$
of polynomials of degree $m$ has a decomposition ${\cal P}_{m}={\cal
Q}_{m}\oplus{\cal E}_{m}\oplus{\cal Q}_{m}^{\perp}$ such that

$\displaystyle\alpha_{c}=\begin{cases}1/2&(P\in{\cal Q}_{m})\\\ 1/4&(P\in{\cal
E}_{m})\\\ 1/6&(P\in{\cal Q}^{\perp}_{m})\\\ \end{cases}$

On the other hand, [11] studied its continuum analogue of $H_{\alpha}$ and
derived the asymptotics of $Tr(1_{(a,b)}(H_{\alpha,N})$ ($0<a<b$).

$\displaystyle(1)$ $\displaystyle\quad(\alpha>1/2)\quad
Tr(1_{(a,b)}(H_{\alpha,N})-N\mu(a,b)\stackrel{{\scriptstyle a.s.}}{{\to}}Q$
$\displaystyle(2)$ $\displaystyle\quad(\alpha=1/2)\quad(\log
N)^{-1/2}\left\\{Tr(1_{(a,b)}(H_{\alpha,N}))-N\mu(a,b)-(Const.)\log
N\right\\}\stackrel{{\scriptstyle a.s.}}{{\to}}G(a,b)$ $\displaystyle(3)$
$\displaystyle\quad(\alpha<1/2)\quad
N^{-(1/2-\alpha)}\left\\{Tr(1_{(a,b)}(H_{\alpha,N}))-N\mu(a,b)-\sum_{k=1}^{D+1}C_{k}N^{1-k\alpha}\right\\}\stackrel{{\scriptstyle
a.s.}}{{\to}}G(a,b)$

where $\mu$ is the IDS (in the continuum,
$\mu(a,b)=\pi^{-1}(\sqrt{b}-\sqrt{a})$). $Q$ is a bounded random variable.
$D:=\min\\{d\in{\bf N}\,|\,\frac{1}{2\alpha}<d+1\\}$, and
$\\{G(a,b)\\}_{0<a<b}$, is a Gaussian field. (2), (3) holds in the sense of
joint distribution w.r.t. $a,b$. In view of [2, 11], we consider the following
problems : (1) we consider $Trf(H_{\alpha,N})$ instead of $TrP(H_{\alpha,N})$
where $f$ is an analytic function. (2) For a number of analytic functions
$f_{1},f_{2},\cdots,f_{K}$, we consider the joint limit of
$Trf_{j}(H_{\alpha,N})-{\bf E}[Trf_{j}(H_{\alpha,N})]$, $(j=1,2,\cdots,K)$,
and (3) We study the behavior of ${\bf E}[f(H_{\alpha,N})]$. We possibly would
like to $Tr1_{(a,b)}(H_{\alpha,N})$ but for that we need to study Prüfer
coordinate [6] which is beyond the scope of this paper. It would be difficult
to consider general continuous function, so that we work under the following
condition in this paper.

Assumption
(1) $\\{X_{n}\\}$ is a family of i.i.d. bounded random varables, such that
${\bf E}[X_{n}]=0$, $\eta^{2}:={\bf E}[X_{n}^{2}]>0$, and $|X_{n}(\omega)|\leq
C_{X}$, a.s. for a positive constant $C_{X}$.
(2) $f$ has the Taylor expansion around the origin and its convergence radius
satisfies $r(f)>C_{X}+2$.

$\displaystyle f(x)=\sum_{j=0}^{\infty}c_{j}x^{j},\quad|x|<r(f),\quad
r(f)>C_{X}+2.$

Since $\sigma(H_{\alpha})\subset[-2-C_{X},2+C_{X}]$, the assumption on $r(f)$
is a natural one ensuring the convergence of $f(H)=\sum_{j}c_{j}H^{j}$ in the
norm topology. We state our results in each subsections below.

###  1.1 Analytic extension

Let $f_{k}$, $k=1,2,\cdots$ be the polynomial obtained by truncating higher
order terms in the Taylor expansion of $f$ :

$\displaystyle f_{k}(x)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}c_{j}x^{j}.$

And let

$p^{k}(\beta):=\sharp{\cal P}^{k}(\beta)$ (1.1)

be the number paths where the initial and terminal points coincide and the
number of flat steps on each site is given by a multi-index $\beta$. The
precise definition is given in (2.1).

Then we have an analogues statement as in [2] : we can find a critical
exponent $\alpha_{c}$ such that for $\alpha_{c}<\alpha$ the fluctuation
converges to a random variable, while for $0<\alpha\leq\alpha_{c}$ the
fluctuation is unbounded and satisfies a CLT statement after a suitable
scaling.

Theorem 1
Case A : general case
(1) $1/2<\alpha$ : there exists a random variable $Q_{A}$ with
$var(Q_{A})<\infty$ s.t.

$\displaystyle Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]\stackrel{{\scriptstyle
N\to\infty}}{{\to}}Q_{A},\quad\mbox{ a.s. }$

If $\alpha>1$, $Q_{A}$ is bounded.
(2) $0<\alpha\leq 1/2$ : we have a following CLT statement.

$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{2\alpha}(N)}\stackrel{{\scriptstyle
d}}{{\to}}N(0,\sigma_{A}(f)^{2}),$

where

$\displaystyle\sigma_{A}(f)^{2}:=\left|\sum_{j=0}^{\infty}c_{2j+1}p^{2j+1}(\delta)\right|^{2}\eta^{2},\quad
p^{k}(\delta)=k\binom{k-1}{\frac{k-1}{2}}1(k:\mbox{odd}).$

Case B : Taylor expansion of $f$ have even terms only : $c_{2j-1}=0$,
$j\in{\bf N}$.
(1) $1/4<\alpha$ : there exists a random variable $Q_{B}$ with
$var(Q_{B})<\infty$ s.t.

$\displaystyle Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]\stackrel{{\scriptstyle
N\to\infty}}{{\to}}Q_{B}\quad\mbox{ in probability. }$

(2) $0<\alpha\leq 1/4$ :

$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{4\alpha}(N)}\stackrel{{\scriptstyle
d}}{{\to}}N(0,\sigma_{B}(f)^{2})$

where

$\displaystyle\sigma_{B}(f)^{2}$ $\displaystyle:=$
$\displaystyle\left(\sum_{j=1}^{\infty}c_{j}p^{j}(2\delta)\right)^{2}\left({\bf
E}[X_{1}^{4}]-\eta^{4}\right)+\sum_{s=1}^{\infty}\left(\sum_{j=1}^{\infty}c_{j}p^{j}(\delta+\delta^{s})\right)^{2}\eta^{4}.$

Case C : Taylor expansion of $f$ have odd terms only : $c_{2j}=0$, $j\in{\bf
N}$, and $f$ has the form of

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\sum_{j=3}^{\infty}c_{j}\left(x^{j}-p^{j}(\delta)x\right)=\sum_{j=3}^{\infty}c_{j}x^{j}-\left(\sum_{j=3}^{\infty}p^{j}(\delta)\right)x.$

(1) $1/6<\alpha$ : there exists a random variable $Q_{C}$ with
$var(Q_{C})<\infty$ s.t.

$\displaystyle Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]\stackrel{{\scriptstyle
N\to\infty}}{{\to}}Q_{C}\quad\mbox{ in probability. }$

(2) $0<\alpha\leq 1/6$ :

$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{6\alpha}(N)}\stackrel{{\scriptstyle
d}}{{\to}}N(0,\sigma_{C}(f)^{2})$

where

$\displaystyle\sigma_{C}(f)^{2}$ $\displaystyle:=$
$\displaystyle\lim_{k\to\infty}\sigma_{C}(f_{k})^{2}.$

The expression of $\sigma_{C}(f_{k})$, given in [2](2.11), is so complicated
and is omitted. We believe that the convergence in Theorem 1(1) in cases B and
C also holds almost surely, not only in probability. In fact, we can show the
a.s. convergence if we interchange the order of summation in a random series
(Lemma 2.9).

###  1.2 Joint limit

Suppose $f_{t}$ ($t=1,2,\cdots,d$) satisfy Assumption. We assume that they
belong to same cases among A, B, and C, and set $\alpha_{c}:=1/2$ (Case A),
$1/4$ (Case B), and $1/6$ (Case C), and similarly,
$\sigma(f):=\sigma_{\sharp}(f)$ for Case $\sharp$ ($\sharp=A,B,C$). We
consider the joint limit of the following vector :

$\displaystyle{\bf F}_{N}$ $\displaystyle:=$
$\displaystyle(F_{1,N},\cdots,F_{d,N})$ $\displaystyle\mbox{ where }\quad
F_{t,N}$ $\displaystyle:=$ $\displaystyle\frac{Trf_{t}(H_{\alpha,N})-{\bf
E}[Trf_{t}(H_{\alpha,N})]}{g_{\alpha/\alpha_{c}}(N)},\quad t=1,2,\cdots,d.$

Theorem 2  Let $0<\alpha\leq\alpha_{c}$

$\displaystyle{\bf F}_{N}$ $\displaystyle\stackrel{{\scriptstyle d}}{{\to}}$
$\displaystyle{\bf F},\quad where\quad{\bf F}\sim N(0,\Sigma_{\bf
F}),\quad\Sigma_{\bf F}:=\left(\sigma(f_{i})\sigma(f_{j})\right)_{i,j}.$

###  1.3 Behavior of the expectation value

Let $m:=\max\\{k\in{\bf N}\,|\,k\alpha\leq 1\\}$. Then we have the asymptotic
expansion for the expectation value of the trace of $f(H)$.

Theorem 3

$\displaystyle{\bf E}[Trf(H_{\alpha,N})]$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}$
$\displaystyle\left(\sum_{l}c_{l}C_{0,l}\right)N+\left(\sum_{l}c_{l}C_{2,l}\right)S_{2}(N)+\cdots+\left(\sum_{l}c_{l}C_{m,l}\right)S_{m}(N)+C_{N}(f)$

where

$\displaystyle
S_{j}(N):=\sum_{n=1}^{N}\frac{1}{n^{j\alpha}}=\begin{cases}\dfrac{N^{1-j\alpha}}{1-j\alpha}+{\cal
O}(1)&(j\alpha<1)\\\ \log N+{\cal O}(1)&(j\alpha=1)\end{cases}\quad
j=1,2,\cdots,m$

$C_{0,l},\cdots,C_{m,l}$ are explicit constants, and $C_{N}(f)$ is convergent
as $N\to\infty$.

In later sections, we prove these theorems. In Section 2, we first recall the
argument in [2] which relates $Tr(H^{k})$ to the path counting on ${\bf Z}$.
Then we prepare some lemmas to prove Theorem 1. In Section 3, we prove Theorem
1. Proof of the CLT statement is a simple application of a limit theorem on
the triangular array of random variables. However, to show Theorem 1(1) in
Cases B and C, one needs complicated computations and estimates of the
variance. In Section 4, we prove Theorem 2 using Cramer-Wald’s method. In
Section 5 (resp. in Section 6), we prove Theorem 3 for $f(x)=x^{k}$ (resp. for
analytic functions).

##  2 Preliminaries

###  2.1 Paths with flat steps

Since $TrH^{k}_{\alpha,N}$ is a polynomial of $V(1),\cdots,V(N)$, it is
important to derive the coefficients of that. In [2], they express them in
terms of the lattice path on ${\bf Z}$ with certain conditions. We briefly
recall this argument in [2], not only for completeness but also to fix the
notation.

Definition 2.1
(1) A multi-index is a map $\beta:{\bf Z}\to{\bf N}\cup\\{0\\}$ such that
$\beta(h)=0$ but finitely many $h$’s. We sometimes write $\beta_{h}$ instead
of $\beta(h)$. For a multi-index $\beta$, let $V^{\beta}$ be the monimial
given by

$\displaystyle V^{\beta}:=\prod_{h\in{\bf Z}}V(h)^{\beta(h)}.$

We set $V(h)=0$ for $h\leq 0$, and also set $0^{n}:=1$ for convenience. The
multiplicity of $V^{\beta}$ is denoted by $|\beta|:=\sum_{j=1}^{N}\beta(j)$.
The translation $\beta^{i}$ of $\beta$ for $i\in{\bf Z}$ is defined by

$\displaystyle(\beta^{i})(h):=\beta(h-i),\quad h\in{\bf Z}.$

(2) $\delta=\delta_{0}$ is a multi-index with

$\displaystyle\delta(h):=1(h=0).$

Definition 2.2
(1) Path on ${\bf Z}$ of length $k$ is a sequence ${\bf
y}:=(y_{0},y_{1},\cdots,y_{k})$ $(y_{i}\in{\bf Z})$ such that
$|y_{t}-y_{t-1}|\leq 1$. We say a pair $(y_{t-1},y_{t})$ is a up step (resp.
down step, flat step) if $y_{t}-y_{t-1}=1$ (resp. $y_{t}-y_{t-1}=-1$,
$y_{t}-y_{t-1}=0$). We say $(y_{t-1},y_{t})$ is a flat step of level $h\in{\bf
Z}$ if $y_{t-1}=y_{t}=h\in{\bf Z}$. Let ${\cal Q}^{k}$ be the set of paths of
length $k$ starting from the origin :

$\displaystyle{\cal Q}^{k}$ $\displaystyle:=$ $\displaystyle\left\\{{\bf
y}=(y_{0},y_{1},\cdots,y_{k})\,\middle|\,y_{t}\in{\bf Z},\;y_{t}-y_{t-1}=\pm
1,0\;(t=1,2,\cdots,k),\;y_{0}=0\right\\}.$

For a given multi-index $\beta$, let ${\cal Q}^{k}(\beta)$ be the set of paths
in ${\cal Q}^{k}$ such that the number of flat steps of which is counted by a
suitable translation of $\beta$ :

$\displaystyle{\cal Q}^{k}(\beta)$ $\displaystyle:=$
$\displaystyle\left\\{{\bf y}=(y_{0},y_{1},\cdots,y_{k})\in{\cal
Q}^{k}\,\middle|\,\exists i\in{\bf
Z}\;s.t.\;\beta^{i}(h)=\sharp\\{t\,|\,y_{t-1}=y_{t}=h\\},\;\forall h\right\\}$

We note that, for ${\bf y}\in{\cal Q}^{k}(\beta)$, the corresponding $i\in{\bf
Z}$ is uniquely determined.

We aim to relate each terms in $Tr(H^{k}_{\alpha,N})$ (as a polynomial of $V$)
to the paths in ${\cal Q}^{k}$. In order for that, we write

$\displaystyle H_{\alpha,N}$ $\displaystyle=$ $\displaystyle
S_{N}+V_{\alpha,N}+S_{N}^{*}$

where $S_{N}$ is the right shift on $\ell^{2}([1,N])$. Expanding
$(S_{N}+V_{\alpha,N}+S_{N}^{*})^{k}$ and taking each terms, we have

$\displaystyle H_{\alpha,N}^{k}$ $\displaystyle=$
$\displaystyle(S_{N}+V_{\alpha,N}+S_{N}^{*})^{k}=\sum_{M\in{\cal M}}M$

where ${\cal M}$ is the set of strings of $S_{N},S_{N}^{*},V_{\alpha,N}$ of
length $k$ :

$\displaystyle{\cal M}$ $\displaystyle:=$
$\displaystyle\left\\{M_{1}M_{2}\cdots
M_{k}\,\middle|\,M_{t}=S_{N},S_{N}^{*},V_{\alpha,N},\;t=1,2,\cdots,k\right\\}.$

For $M=M_{1}M_{2}\cdots M_{k}\in{\cal M}$, we set the corresponding path ${\bf
y}=(y_{0},y_{1},\cdots,y_{k})\in{\cal Q}^{k}$ as

$\displaystyle y_{0}:=0,\quad y_{t}-y_{t-1}$ $\displaystyle:=$
$\displaystyle\begin{cases}1&(M_{t}=S_{N})\\\ -1&(M_{t}=S_{N}^{*})\\\
0&(M_{t}=V_{\alpha,N})\end{cases}$

which gives a bijection between ${\cal M}$ and ${\cal P}^{k}$. We henceforth
write ${\bf y}(M)\in{\cal Q}^{k}$ is a path corresponding to $M\in{\cal M}$
and vice versa.

Lemma 2.1
For $M=M_{1}M_{2}\cdots M_{k}\in{\cal M}$, let ${\bf
y}(M)=(y_{0},y_{1},\cdots,y_{k})\in{\cal Q}^{k}$ be the corresponding path.
For $i,j\in[N]$, we have

$\displaystyle M_{ij}$ $\displaystyle=$ $\displaystyle V^{\beta}\cdot
1\left(\beta,{\bf y}(M)\mbox{ satisfies condition }(C)_{i,j}^{\beta}\right)$

where $1(P)$ is the indicator function associated to a proposition $P$ and we
say a multi-index $\beta$ and ${\bf y}\in{\cal P}^{k}$ satisfies condition
$(C)_{i,j}^{\beta}$ if
($C_{1}$) ${\bf y}+i$ lies in $[1,N]$ :

$\displaystyle 1\leq i+y_{t}\leq N,\quad 1\leq t\leq k.$

($C_{2}$) $\beta^{-i}$ counts the number of flat steps of ${\bf y}$ :

$\displaystyle\sharp\\{t\,|\,y_{t}=y_{t-1}=h\\}=\beta(h+i),\quad\forall
h\in{\bf Z}.$

($C_{3}$) $j-i=y_{k}(M)$.

Note that, the condition $(C_{2})$ determines $\beta$ uniquely, and for each
$i\in[N]$, there is at most one $j$ such that the condition
$(C)_{i,j}^{\beta}$ is satisfied.

Proof. For simplicity of notation, we write $U:=S_{N}$, $D:=S_{N}^{*}$. On the
products in $M=M_{1}M_{2}\cdots M_{k}\in{\cal M}$, we group together the
products of $U$’s which are divided by an appearance of $V$, and denote by
$W_{1},W_{2},\cdots$ the products of $U$’s in each group :

$\displaystyle M$ $\displaystyle=$ $\displaystyle M_{1}M_{2}\cdots
M_{k}=(UD\cdots U)V(DU\cdots D)V\cdots V(DU\cdots U)$ $\displaystyle=:$
$\displaystyle W_{1}VW_{2}V\cdots W_{f}VW_{f+1}$

where the number of flat steps is equal to $f$ in ${\bf y}(M)$. For each
$W_{l}$ ($l=1,2,\cdots,f+1$), let $s_{l}:=\sharp U-\sharp D$ be the differeces
between the number of $U$’s and that of $V$’s. Letting $y_{t}(n_{l}\leq t\leq
m_{l})$ be the part of ${\bf y}(M)$ associated to $W_{l}$, we have

$\displaystyle(W_{l})_{ij}=1\left(1\leq i+y_{t}\leq N,\;n_{l}\leq t\leq
m_{l}\right)\cdot 1\left(j=i+s_{l}\right)$

so that

$\displaystyle M_{ij}$ $\displaystyle=$ $\displaystyle
V(i+s_{1})V(i+s_{1}+s_{2})\cdots V(i+s_{1}+s_{2}+\cdots+s_{f})\times$
$\displaystyle\qquad\times 1\left(1\leq i+y_{t}\leq N,\;1\leq t\leq
k\right)\cdot 1\left(j=i+y_{k}\right)$

Because the set
$\left\\{i+s_{1},i+s_{1}+s_{2},\cdots,i+s_{1}+\cdots+s_{f}\right\\}$ coincides
as a multi-set with that of levels $\\{h\,|\,\exists
t\in\\{0,1,\cdots,k-1\\},\,s.\,t.\,y_{t-1}=y_{t}=h\\}$ of flat steps of ${\bf
y}(M)$, we have

$\displaystyle M_{ij}$ $\displaystyle=$ $\displaystyle V^{\beta}\cdot
1\left(1\leq i+y_{t}\leq N,\;1\leq t\leq k,\;j=i+y_{k}\right)$ $\displaystyle
where\quad\beta(h+i)$ $\displaystyle=$
$\displaystyle\sharp\\{t\,|\,y_{t}=y_{t-1}=h\\},\quad\forall h\in{\bf Z}.$

In what follows, we consider the diagonal element of $H_{\alpha,N}^{k}$ and
set $i=j$, in the corresponding paths of which the number of up steps is equal
to that of down steps. We introduce the subset of ${\cal Q}^{k}$ returning to
the origin satisfying $(C)_{i,i}^{\beta}$, and those satisfying $(C_{2})$ only
:

$\displaystyle{\cal C}^{k}_{i}(\beta)$ $\displaystyle:=$
$\displaystyle\left\\{{\bf y}\in{\cal
Q}^{k}\,\middle|\,y_{0}=y_{k}=0,\;\beta,{\bf y}\mbox{ satisfies
}(C)_{i,i}^{\beta}\right\\}$ $\displaystyle{\cal P}^{k}_{i}(\beta)$
$\displaystyle:=$ $\displaystyle\left\\{{\bf y}\in{\cal
Q}^{k}\,\middle|\,y_{0}=y_{k}=0,\;\sharp\\{t\,|\,y_{t}=y_{t-1}=h\\}=\beta(h+i),\;\forall
h\in{\bf Z}\right\\}$ $\displaystyle{\cal P}^{k}(\beta)$ $\displaystyle:=$
$\displaystyle\bigcup_{i}{\cal P}_{i}^{k}(\beta)$

By definition, ${\cal C}^{k}_{i}(\beta)\subset{\cal P}^{k}_{i}(\beta)$ and
${\cal P}^{k}(\beta)=\bigcup_{i}{\cal P}^{k}_{i}(\beta)$ is a disjoint union.
And let

$p^{k}(\beta):=\sharp{\cal P}^{k}(\beta)$ (2.1)

be the number paths in ${\cal P}^{k}(\beta)$, which is the number of paths
where the number of flat steps on each site is given by a multi-index $\beta$.
For a polynomial $P$, we denote by $[x^{k}]P$ the coefficient of $x^{k}$ in
$P$.

Lemma 2.2

$\displaystyle(1)$
$\displaystyle\quad[V^{\beta}](H^{k}_{\alpha,N})_{ii}=\sharp{\cal
C}^{k}_{i}(\beta),$ $\displaystyle(2)$ $\displaystyle\quad\frac{k}{2}\leq
i\leq
N-\frac{k}{2}\quad\Longrightarrow\quad[V^{\beta}](H^{k}_{\alpha,N})_{ii}=\sharp{\cal
P}^{k}_{i}(\beta).$

Proof. (1) Letting $i=j$ in Lemma 2.1, we have

$\displaystyle M_{ii}=V^{\beta}\cdot 1\left({\bf y}(M)\in{\cal
C}^{k}_{i}(\beta)\right).$

We thus have

$\displaystyle(H^{k}_{\alpha,N})_{ii}$ $\displaystyle=$
$\displaystyle\sum_{{\bf y}\in{\cal P}^{k}}M_{ii}({\bf
y})=\sum_{\beta}\sum_{{\bf y}\in{\cal C}^{k}_{i}(\beta)}M_{ii}({\bf
y})=\sum_{\beta}\sum_{{\bf y}\in{\cal
C}^{k}_{i}(\beta)}V^{\beta}=\sum_{\beta}\sharp{\cal
C}_{i}^{k}(\beta)V^{\beta}.$

(2) If $\frac{k}{2}\leq i\leq N-\frac{k}{2}$ then the condition $(C_{1})$ is
always satisfied and ${\cal C}^{k}_{i}(\beta)={\cal P}^{k}_{i}(\beta)$.

For a multi-index $\beta$, let

$\displaystyle\iota(\beta)$ $\displaystyle:=$
$\displaystyle\min\mathop{\mathrm{supp}}\nolimits\beta$ $\displaystyle
a^{k}_{N}(\beta)$ $\displaystyle:=$
$\displaystyle[V^{\beta}]Tr(H^{k}_{\alpha,N}).$

Lemma 2.3  If $\beta\neq 0$, and $\iota(\beta)\in[k,N-k]$, then
$a^{k}_{N}(\beta)=p^{k}(\beta)$.

Proof. We note that, if $\iota(\beta)\in[k,N-k]$, then the corresponding
$i\in{\bf Z}$, such that $\beta^{-i}$ counts the number of flat steps starting
at the origin, satisfies $i\in\left[\frac{k}{2},N-\frac{k}{2}\right]$. We
decompose $TrH_{\alpha,N}^{k}$ as

$\displaystyle TrH_{\alpha,N}^{k}$ $\displaystyle=$
$\displaystyle\sum_{i}(H^{k}_{\alpha,N})_{ii}=\sum_{i}\sum_{\beta}\sharp{\cal
C}_{i}^{k}(\beta)V^{\beta}$ $\displaystyle=$
$\displaystyle\left(\sum_{\iota(\beta)\in[k,N-k]}+\sum_{\iota(\beta)\in[k,N-k]^{c}}\right)\sum_{i}\sharp{\cal
C}_{i}^{k}(\beta)V^{\beta}$ $\displaystyle=:$ $\displaystyle(1)+(2).$

Since ${\cal P}^{k}(\beta)=\bigcup_{i\in{\bf Z}}{\cal P}_{i}^{k}(\beta)$ is a
disjoint union, $p^{k}(\beta)=\sum_{i}\sharp{\cal P}_{i}^{k}(\beta)$ so that
we have

$\displaystyle(1)$ $\displaystyle=$
$\displaystyle\sum_{\iota(\beta)\in[k,N-k]}\sum_{i}\sharp{\cal
P}_{i}^{k}(\beta)V^{\beta}=\sum_{\iota(\beta)\in[k,N-k]}p^{k}(\beta)V^{\beta}.$

At the end of this subsection, we give some estimates on the number of the
specific paths.

Lemma 2.4

$\displaystyle(1)$
$\displaystyle\quad\sum_{\begin{subarray}{c}\beta\,:\,|\beta|=j\end{subarray}}p^{l}(\beta)\leq\left(\begin{array}[]{c}l\\\
j\end{array}\right)\left(\begin{array}[]{c}l-j\\\
\frac{l-j}{2}\end{array}\right)$ $\displaystyle(2)$
$\displaystyle\quad\sum_{j=0}^{l}\sum_{\begin{subarray}{c}\beta:|\beta|=j\end{subarray}}p^{l}(\beta)C_{X}^{j}\leq(C_{X}+2)^{l}$
$\displaystyle(3)$ $\displaystyle\quad a^{k}(\delta)\leq
p^{k}(\delta)=\frac{1}{\sqrt{2\pi}}\sqrt{k}\cdot 2^{k}(1+o(1)),\quad
k\to\infty$

Proof. (1)

$\displaystyle\sum_{\begin{subarray}{c}\beta\,:\,|\beta|=j\end{subarray}}p^{l}(\beta)$
$\displaystyle\leq$ $\displaystyle\sharp\left\\{{\bf
y}=(y_{0},y_{1},\cdots,y_{l})\in{\cal
P}^{l}\,\middle|\,y_{0}=y_{l}=0,\;\sharp\\{\mbox{flat steps}\\}=j\right\\}$
$\displaystyle\leq$ $\displaystyle\left(\begin{array}[]{c}l\\\
j\end{array}\right)\left(\begin{array}[]{c}l-j\\\
\frac{l-j}{2}\end{array}\right)$

where $\left(\begin{array}[]{c}l\\\ j\end{array}\right)$ is equal to the
number of ways of choosing the flat steps, and $\left(\begin{array}[]{c}l-j\\\
\frac{l-j}{2}\end{array}\right)$ is equal to the number of path of length
$l-j$ starting from and coming back to the origin. (2) We use (1) and the
multinomial theorem to compute

$\displaystyle\sum_{j=0}^{l}\sum_{\begin{subarray}{c}\beta\,:\,|\beta|=j\end{subarray}}p^{l}(\beta)C_{X}^{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{l}\frac{l!}{j!\cdot\left[\left(\frac{l-j}{2}\right)!\right]^{2}}C_{X}^{j}\leq\sum_{\begin{subarray}{c}i,j,k\\\
i+j+k=l\end{subarray}}^{l}\frac{l!}{j!\cdot i!\cdot k!}1^{i}\cdot 1^{k}\cdot
C_{X}^{j}$ $\displaystyle=$ $\displaystyle(1+1+C_{X})^{l}=(C_{X}+2)^{l}.$

(3) The first inequality follows from ${\cal C}^{k}_{i}(\beta)\subset{\cal
P}^{k}_{i}(\beta)$. For the second one, we use the following equations

$\displaystyle p^{k}(\delta)$ $\displaystyle=$
$\displaystyle\frac{k!}{\left[\left(\frac{k-1}{2}\right)!\right]^{2}},\quad
k:odd$ $\displaystyle\left(\begin{array}[]{c}n\\\ n/2\end{array}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\frac{2^{n+1}}{\sqrt{n}}(1+o(1)),\quad
n:even,\;n\to\infty.$

###  2.2 Some Lemmas for the proof of Theorem 1

We prepare some lemmas which are used to show the convergence of the
fluctuation.

Lemma 2.5
For given $j,\beta$ we can find $a^{j}(\beta)$ such that

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}(a_{N}^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle\stackrel{{\scriptstyle
N\to\infty}}{{\to}}\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,j/2]\\\
|\beta|=\ell\end{subarray}}(a^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right),\quad a.s.$

Proof. We first estimate the absolute value of the quantities in question, to
apply the dominated convergence theorem. Note that
$a_{N}^{j}(\beta)=p^{j}(\beta)$ if $\iota(\beta)\in[j/2,N-j/2]$ which yields

$\displaystyle c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}(a_{N}^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle=$ $\displaystyle
c_{j}\left(\sum_{\begin{subarray}{c}\iota(\beta)\in[1,\frac{j}{2}]\\\
|\beta|=\ell\end{subarray}}+\sum_{\begin{subarray}{c}\iota(\beta)\in[N-\frac{j}{2},N]\\\
|\beta|=\ell\end{subarray}}\right)(a_{N}^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right).$

Then by Lemma 2.4, we have

$\displaystyle|c_{j}|\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta}-{\bf
E}[V^{\beta}]\right|$ $\displaystyle=$
$\displaystyle|c_{j}|\left(\sum_{i=1}^{j/2}+\sum_{i=N-\frac{j}{2}}^{N}\right)\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right|$ $\displaystyle\leq$
$\displaystyle|c_{j}|j\binom{j}{\ell}\dbinom{j-\ell}{\frac{j-\ell}{2}}2C_{X}^{\ell}\leq
2|c_{j}|j(2+C_{X})^{j}.$

Here we have $2\sum_{j=0}^{\infty}|c_{j}|j(2+C_{X})^{j}<\infty$ by assumption.
On the other hand, if $j<N$, $\iota(\beta)\in[1,j/2]$, $a_{N}^{j}(\beta)$ does
not depend on $N$ : $a_{N}^{j}(\beta)=:a^{j}(\beta)$. Moreover,

$\displaystyle|c_{j}|\sum_{\begin{subarray}{c}\iota(\beta)\in[N-\frac{j}{2},N]\\\
|\beta|=\ell\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta}-{\bf
E}[V^{\beta}]\right|$ $\displaystyle\leq$
$\displaystyle|c_{j}|\sum_{i=N-\frac{j}{2}}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right|$ $\displaystyle\leq$
$\displaystyle|c_{j}|j\binom{j}{\ell}\dbinom{j-\ell}{\frac{j-\ell}{2}}\frac{2C_{X}^{\ell}}{\left(N-\frac{j}{2}\right)^{\alpha\ell}}\stackrel{{\scriptstyle
N\to\infty}}{{\to}}0.$

Therefore the conclusion follows from the dominated convergence theorem.

Lemma 2.6

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq\ell_{0}\end{subarray}}(a_{N}^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle\stackrel{{\scriptstyle
N\to\infty}}{{\to}}\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,j/2]\\\
|\beta|\geq\ell_{0}\end{subarray}}(a^{j}(\beta)-p^{j}(\beta))\left(V^{\beta}-{\bf
E}[V^{\beta}]\right),\quad a.s.$

Proof. It suffices to make sure that the condition to apply the dominated
convergence theorem is satisfied : then we sum up the result of Lemma 2.5 with
respect to $\ell\geq\ell_{0}$. The point is that, for fixed $j$,
$\sum_{\ell\geq\ell_{0}}$ becomes a finite sum. In fact,

$\displaystyle|c_{j}|\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq\ell_{0}\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta}-{\bf
E}[V^{\beta}]\right|$ $\displaystyle=$
$\displaystyle|c_{j}|\sum_{\ell=\ell_{0}}^{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}|a_{N}^{j}(\beta)-p^{j}(\beta)|\left|V^{\beta}-{\bf
E}[V^{\beta}]\right|$ $\displaystyle\leq$
$\displaystyle|c_{j}|\sum_{\ell=\ell_{0}}^{j}j\binom{j}{\ell}\dbinom{j-\ell}{\frac{j-\ell}{2}}2C_{X}^{\ell}\leq
2|c_{j}|j^{2}(2+C_{X})^{j}.$

Lemma 2.7
Suppose $\alpha\ell_{0}>1$. Then

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$

converges as $N\to\infty$ almost surely.

Proof. Since

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\sum_{i=1}^{N}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right),$

it suffices to show that RHS converges absolutely. In fact,

$\displaystyle\sum_{j=0}^{\infty}\sum_{i=1}^{\infty}|c_{j}|\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left|V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right|$ $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}\sum_{i=1}^{\infty}|c_{j}|\sum_{\ell=\ell_{0}}^{j}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}p^{j}(\beta)\frac{2C_{X}^{\ell}}{i^{\alpha\ell}}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}\sum_{i=1}^{\infty}|c_{j}|\sum_{\ell=\ell_{0}}^{j}\binom{j}{\ell}\dbinom{j-\ell}{\frac{j-\ell}{2}}\frac{2C_{X}^{\ell}}{i^{\alpha\ell_{0}}}$
$\displaystyle\leq$ $\displaystyle
2\sum_{j=0}^{\infty}|c_{j}|(2+C_{X})^{j}\sum_{i=1}^{\infty}\frac{1}{i^{\alpha\ell_{0}}}<\infty.$

Lemma 2.8
Suppose $\alpha\ell>\frac{1}{2}$. Then

$\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}\sum_{j=0}^{\infty}c_{j}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$

converges as $N\to\infty$ in probability.

Proof. We show that the quantity in question is Cauchy in probability.

$\displaystyle{\bf
E}\left[\left|\sum_{\begin{subarray}{c}\iota(\beta)\in[N+1,M]\\\
|\beta|=\ell\end{subarray}}\sum_{j=0}^{\infty}c_{j}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)\right|^{2}\right]$ $\displaystyle=$ $\displaystyle{\bf
E}\left[\left|\sum_{i=N+1}^{M}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{j=0}^{\infty}c_{j}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)\right|^{2}\right]$ $\displaystyle=$
$\displaystyle{\bf
E}\Biggl{[}\sum_{i_{1}=N+1}^{M}\sum_{i_{2}=N+1}^{M}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\left(\sum_{j\geq
0}c_{j}p^{j}(\beta)\right)\left(\sum_{j^{\prime}\geq
0}c_{j^{\prime}}p^{j^{\prime}}(\gamma)\right)$ $\displaystyle\hskip
100.00015pt\times\left(V^{\beta^{i_{1}}}-{\bf
E}[V^{\beta^{i_{1}}}]\right)\left(V^{\gamma^{i_{2}}}-{\bf
E}[V^{\gamma^{i_{2}}}]\right)\Biggl{]}$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=N+1}^{M}\sum_{i_{2}=N+1}^{M}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\left(\sum_{j\geq
0}c_{j}p^{j}(\beta)\right)\left(\sum_{j^{\prime}\geq
0}c_{j^{\prime}}p^{j^{\prime}}(\gamma)\right)Cov\left(V^{\beta^{i_{1}}},V^{\gamma^{i_{2}}}\right).$

We remark that, the estimate

$\displaystyle
p^{j}(\beta)\leq\binom{j}{\ell}\binom{j-\ell}{\frac{j-\ell}{2}}\leq\binom{j}{\ell}\cdot
2^{j-\ell}$

guarantee the absolute convergence of $\sum_{j\geq 0}c_{j}p^{j}(\beta)$. We
further compute

$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\left(\sum_{j_{1}\geq
0}c_{j_{1}}p^{j_{1}}(\beta)\right)\left(\sum_{j_{2}\geq
0}c_{j_{2}}p^{j_{2}}(\gamma)\right)\sum_{i_{1}=N+1}^{M}\sum_{i_{2}=N+1}^{M}Cov\left(V^{\beta^{i_{1}}},V^{\gamma^{i_{2}}}\right)$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\left(\sum_{j_{1}\geq
0}c_{j_{1}}p^{j_{1}}(\beta)\right)\left(\sum_{j_{2}\geq
0}c_{j_{2}}p^{j_{2}}(\gamma)\right)$ $\displaystyle\hskip
50.00008pt\times\left(\sum_{i_{1}=N+1}^{M}\sum_{\begin{subarray}{c}i_{2}\in[N+1,M]\\\
\cap[i_{1},i_{1}+j_{1}]\end{subarray}}+\sum_{i_{2}=N+1}^{M}\sum_{\begin{subarray}{c}i_{1}\in[N+1,M]\\\
\cap[i_{2}+1,i_{2}+j_{2}]\end{subarray}}\right)Cov\left(V^{\beta^{i_{1}}},V^{\gamma^{i_{2}}}\right)$
$\displaystyle\leq$ $\displaystyle 2\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\left(\sum_{j_{1}\geq
0}|c_{j_{1}}|p^{j_{1}}(\beta)\right)\left(\sum_{j_{2}\geq
0}|c_{j_{2}}|p^{j_{2}}(\gamma)\right)\sum_{i_{1}=N+1}^{M}\sum_{\begin{subarray}{c}i_{2}\in[N+1,M]\\\
\cap[i_{1},i_{1}+j_{1}]\end{subarray}}\frac{\left|Cov\left(X^{\beta^{i_{1}}},X^{\gamma^{i_{2}}}\right)\right|}{\prod_{h}h^{\alpha(\beta_{h}^{i_{1}}+\gamma_{h}^{i_{2}})}}.$

By using

$\displaystyle\prod_{h}h^{\alpha(\beta_{h}^{i_{1}}+\gamma_{h}^{i_{2}})}$
$\displaystyle\geq$ $\displaystyle
i_{1}^{\alpha(|\beta|+|\gamma|)}=i_{1}^{2\alpha\ell},\qquad\left|Cov\left(X^{\beta^{i_{1}}},X^{\gamma^{i_{2}}}\right)\right|\leq(2C_{X})^{2\ell}$

we have

$\displaystyle\leq$ $\displaystyle 2\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}\sum_{j_{1}\geq
0}|c_{j_{1}}|p^{j_{1}}(\beta)\sum_{j_{2}\geq
0}|c_{j_{2}}|p^{j_{2}}(\gamma)\sum_{i_{1}=N+1}^{M}\sum_{\begin{subarray}{c}i_{2}\in[N+1,M]\\\
\cap[i_{1},i_{1}+j_{1}]\end{subarray}}\frac{4C_{X}^{2\ell}}{i_{1}^{2\alpha\ell}}$
$\displaystyle\leq$ $\displaystyle 8\left(\sum_{j_{1}\geq
0}|c_{j_{1}}|(j_{1}+1)\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}p^{j_{1}}(\beta)\right)\left(\sum_{j_{2}\geq
0}|c_{j_{2}}|\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|=\ell\end{subarray}}p^{j_{2}}(\gamma)\right)C_{X}^{2\ell}\sum_{i_{1}=N+1}^{M}\frac{1}{i_{1}^{2\alpha\ell}}.$

Here we notice that

$\displaystyle\sum_{j_{1}\geq
0}|c_{j_{1}}|(j_{1}+1)\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=\ell\end{subarray}}p^{j_{1}}(\beta)\leq\sum_{j_{1}\geq
0}|c_{j_{1}}|(j_{1}+1)\binom{j}{\ell}2^{j_{1}-\ell}<\infty.$

Thus by Chebyshev’s inequality,

$\displaystyle{\bf
P}\left(\left|\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)-\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,M]\\\
|\beta|=\ell\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)\right|\geq\epsilon\right)$ $\displaystyle\leq$
$\displaystyle\epsilon^{-2}{\bf
E}\left[\left|\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=\ell\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)-\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,M]\\\
|\beta|=\ell\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)\right|^{2}\right]$ $\displaystyle\to$ $\displaystyle
0,\quad N,M\to\infty$

Before ending this section, we show below that the “limit” of the random
variable in Lemma 2.8 really converges as a random series. This fact is not
used in this paper, but we include here for completeness.

Lemma 2.9 Suppose $\alpha\ell_{0}>\frac{1}{2}$. Then the random series

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,\infty)\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$

converges and define a random variable.

Remark  Lemma 2.9 says that, the random series

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)$

converges almost surely, if one takes $\sum_{i=1}^{\infty}$ first and then
$\sum_{j=0}^{\infty}$. Since we are not able to show the absolute convergence,
we do not know whether a.s. convergence holds if we change the order of
summation.

Proof. We first note that, by Lemma 4.2 in [2], The random series

$\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)=\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)$

converges a.s. and define a random variable.

$\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)\in[1,\infty)\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)=\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)$

with finite variance of which we estimate below.

$\displaystyle
Var\left(\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)\right)$ $\displaystyle\leq$
$\displaystyle\liminf_{N\to\infty}{\bf
E}\left[\left(\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)\right)^{2}\right]$ $\displaystyle\leq$
$\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq\ell_{0}\end{subarray}}p^{j}(\gamma)\limsup_{N\to\infty}\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N}\left|Cov\left(\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right),\left(V^{\gamma^{i^{\prime}}}-{\bf
E}[V^{\gamma^{i^{\prime}}}]\right)\right)\right|$

Since the covariance vanishes if $|i-i^{\prime}|>\frac{j}{2}$,

$\displaystyle\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N}\left|Cov\left(\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right),\left(V^{\gamma^{i^{\prime}}}-{\bf
E}[V^{\gamma^{i^{\prime}}}]\right)\right)\right|\leq\sum_{i=1}^{N}j\cdot\frac{4C_{X}^{|\beta|+|\gamma|}}{i^{2\alpha\ell_{0}}}$

so that by Lemma 2.4,

$\displaystyle\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq\ell_{0}\end{subarray}}p^{j}(\gamma)\limsup_{N\to\infty}\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N}\left|Cov\left(\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right),\left(V^{\gamma^{i^{\prime}}}-{\bf
E}[V^{\gamma^{i^{\prime}}}]\right)\right)\right|$ $\displaystyle\leq$
$\displaystyle 4\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq\ell_{0}\end{subarray}}p^{j}(\gamma)jC_{X}^{|\beta|+|\gamma|}\sum_{i=1}^{\infty}\frac{1}{i^{2\alpha\ell_{0}}}\leq
4j(C_{X}+2)^{2j}\sum_{i=1}^{\infty}\frac{1}{i^{2\alpha\ell_{0}}}.$

By Chebyshev’s inequality,

$\displaystyle{\bf
P}\left(\left|\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)\right|>j^{\delta}(C_{X}+2)^{j}\right)\leq
4j^{1-2\delta}\sum_{i=1}^{\infty}\frac{1}{i^{2\alpha\ell_{0}}}$

which is summable with respect to $j$ if $\delta>1$. Thus by Borel-Cantelli
lemma,

$\displaystyle\left|\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)\right|\leq j^{\delta}(C_{X}+2)^{j}$

for sufficiently large $j$, almost surely. Therefore

$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{i=1}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq\ell_{0}\end{subarray}}p^{j}(\beta)\left(V^{\beta^{i}}-{\bf
E}[V^{\beta^{i}}]\right)$

converges almost surely.

##  3 Proof of Theorem 1

We prove A(1), A(2), $\cdots$, C(2) in Theorem 1 respectively in each
subsection below.

###  3.1 Proof of A(1)

Since $\sigma(H_{\alpha,N})\subset[-C_{X}-2,C_{X}+2]$, and since
$r(f)>C_{X}+2$, the power series $f(H)=\sum_{l\geq 0}a_{l}H^{l}$ converges in
the operator norm topology. Hence $Trf(H)=\sum_{l\geq 0}c_{l}Tr(H^{l})$ is
absolutely convergent so that by Fubini theorem,

$\displaystyle{\bf E}[Trf(H)]$ $\displaystyle=$ $\displaystyle\sum_{l\geq
0}c_{l}{\bf E}[Tr(H^{l})].$

Letting $a_{\beta}^{j}:=[V^{\beta}]\left(Tr(H_{\alpha,N}^{j})\right)$, we
compute

$\displaystyle Trf(H_{\alpha,N})-{\bf E}\left[Trf(H_{\alpha,N})\right]$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}c_{j}\left\\{Tr\left(H_{\alpha,N}^{j}\right)-{\bf
E}\left[Tr\left(H_{\alpha,N}^{j}\right)\right]\right\\}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}c_{j}\left(\sum_{|\beta|=1}+\sum_{|\beta|\geq
2}\right)a_{\beta}^{j}\left(V_{\alpha}^{\beta}-{\bf
E}\left[V_{\alpha}^{\beta}\right]\right)$ $\displaystyle=:$ $\displaystyle
I+II.$

Since $|\beta|=1$ implies $\beta=\delta^{n}$, for some $n=1,2,\cdots,N$, we
have
$I=\sum_{n=1}^{N}\sum_{j=0}^{\infty}c_{j}a_{\delta^{n}}^{j}\dfrac{X_{n}}{n^{\alpha}}$.
Here we use

Lemma 3.1 ([5] Theorem 2.5.6)
Suppose $\\{Y_{n}\\}_{n}$ are independent with ${\bf E}[Y_{n}]=0$. If
$\sum_{n}Var(Y_{n})<\infty$, then $\sum_{n}Y_{n}$ is convergent a.s.

Put
$Y_{n}:=\sum_{j=0}^{\infty}c_{j}a_{\delta^{n}}^{j}\dfrac{X_{n}}{n^{\alpha}}$.
Then by Lemma 2.4(3),

$\displaystyle Var(Y_{n})$ $\displaystyle\leq$
$\displaystyle\left(\sum_{j=0}^{\infty}c_{j}a_{\delta^{n}}^{j}\right)^{2}\frac{\eta^{2}}{n^{2\alpha}}\leq(Const.)\left(\sum_{j=0}^{\infty}c_{j}(2+C_{X})^{j}\right)^{2}\frac{\eta^{2}}{n^{2\alpha}}$

so that $\sum_{n}Var(Y_{n})<\infty$ implying $I=\sum_{n}Y_{n}$ is convergent
a.s.
For $II$, the relevant paths have more than one flat steps which produces the
convergent factor. In fact, by Lemma 2.4(2),

$\displaystyle|II|$ $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}|c_{j}|\sum_{i=1}^{N}\sum_{|\beta|\geq
2}a_{\beta^{i}}^{j}\left|\frac{X^{\beta}-{\bf E}[X^{\beta}]}{\prod_{n\in{\bf
N}}n^{\alpha\beta_{n}}}\right|$ $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}|c_{j}|\sum_{i=1}^{N}\sum_{|\beta|\geq
2}p^{j}(\beta)\frac{2(C_{X})^{|\beta|}}{i^{2\alpha}}$ $\displaystyle\leq$
$\displaystyle(Const.)\sum_{j=0}^{\infty}|c_{j}|(C_{X}+2)^{j}\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}<\infty$

Therefore $II$ converges a.s.

###  3.2 Proof of A(2)

The key lemma is :

Lemma 3.2 ([1] Theorem 25.5)  Suppose that

$\\{Y_{N}\\}_{N}$, $\\{X_{N,k}\\}$, $N,k\in{\bf N}$ are random variables
satisfying
(i) $X_{N,k}\stackrel{{\scriptstyle d}}{{\to}}X_{k}$, $N\to\infty$,  for any
fixed  k
(ii) $X_{k}\stackrel{{\scriptstyle d}}{{\to}}X$, $k\to\infty$
(iii) For any $\epsilon>0$, $\lim_{k\to\infty}\limsup_{N\to\infty}{\bf
P}\left(|Y_{N}-X_{N,k}|>\epsilon\right)=0.$
Then $Y_{N}\stackrel{{\scriptstyle d}}{{\to}}X$.

We set

$\displaystyle X_{N,k}$ $\displaystyle:=$
$\displaystyle\frac{Trf_{k}(H_{\alpha,N})-{\bf
E}[Trf_{k}(H_{\alpha,N})]}{g_{2\alpha}(N)},\quad
X_{k}:=N(0,\sigma_{A}(P_{k})^{2})$ $\displaystyle Y_{N}$ $\displaystyle:=$
$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{2\alpha}(N)},\quad X:=N(0,\sigma_{A}(f)^{2}).$

to apply Lemma 3.2, where $f_{k}(x):=\sum_{j=0}^{k}c_{j}x^{j}$ is the
polynomial by truncating the Taylor expansion of $f$. To finish the proof, it
suffices to check the three conditions in Lemma 3.2
(i) By Theorem 1.1 in [2], we have $X_{N,k}\stackrel{{\scriptstyle
d}}{{\to}}X$.
(ii) $X_{k}\stackrel{{\scriptstyle d}}{{\to}}X$ is obvious. In fact, the
characteristic function of $X_{k}$ converges to that of $X$ :
$\exp\left(-\frac{1}{2}\sigma_{A}(f_{k})^{2}t^{2}\right)\stackrel{{\scriptstyle
k\to\infty}}{{\to}}\exp\left(-\frac{1}{2}\sigma_{A}(f)^{2}t^{2}\right)$.
(iii) As in the proof for Theorem 1(1), we can show

$Var\left(Y_{N}-X_{N,k}\right)={\bf E}\left[\left(\sum_{j\geq
k+1}c_{j}\frac{\sum_{|\beta|\geq 1}a_{\beta}^{j}(V_{\alpha}^{\beta}-{\bf
E}[V_{\alpha}^{\beta}])}{g_{2\alpha}(N)}\right)^{2}\right]$ (3.1)

We further compute

${\bf E}\left[\left\\{\sum_{j\geq k+1}c_{j}\sum_{|\beta|\geq
1}a_{\beta}^{j}(V_{\alpha}^{\beta}-{\bf
E}[V_{\alpha}^{\beta}])\right\\}^{2}\right]=\sum_{j_{1}\geq
k+1}\sum_{j_{2}\geq k+1}c_{j_{1}}c_{j_{2}}\sum_{|\beta|\geq
1}\sum_{|\gamma|\geq
1}a_{\beta}^{j_{1}}a_{\gamma}^{j_{2}}\,\cdot\,Cov(V_{\alpha}^{\beta},V_{\alpha}^{\gamma})$
(3.2)

and moreover

$\displaystyle\sum_{|\beta|,|\gamma|\geq
1}a_{\beta}^{j_{1}}a_{\gamma}^{j_{2}}\,\cdot\,|Cov(V_{\alpha}^{\beta},V_{\alpha}^{\gamma})|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N}\sum_{\iota(\beta)=0}\sum_{\iota(\gamma)=0}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)|Cov(V_{\alpha}^{\beta^{i}},V_{\alpha}^{\gamma^{i^{\prime}}})|$
$\displaystyle=$
$\displaystyle\sum_{\iota(\beta)=0}\sum_{\iota(\gamma)=0}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)\sum_{\ell=-\infty}^{\infty}|Cov(X^{\beta},X^{\gamma^{\ell}})|\sum_{i\in
I_{N,\ell}}\prod_{n\in{\bf
N}}\frac{1}{n^{\alpha(\beta_{n-i}+\gamma_{n-i-\ell})}}$

where we set $I_{N,\ell}:=\left\\{i\,\middle|\,1\leq i,i+\ell\leq N\right\\}$.
The covariance and the sum of the products are bounded by

$\displaystyle|Cov(X^{\beta},X^{\gamma^{\ell}})|\leq\sqrt{V(X^{\beta})\cdot
V(X^{\gamma})}\leq C_{X}^{|\beta|+|\gamma|}\leq C_{X}^{j_{1}+j_{2}}$
$\displaystyle\sum_{i\in I_{N,\ell}}\prod_{n\in{\bf
N}}\frac{1}{n^{\alpha(\beta_{n-i}+\gamma_{n-i-\ell})}}\leq\sum_{i\in
I_{N,\ell}}\frac{1}{i^{\alpha|\beta|}\dot{(}i+\ell)^{\alpha|\gamma|}}\leq\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}.$

On the other hand, since $supp\;\beta\subset\left[0,\frac{j_{1}}{2}\right]$,
$supp\;\gamma\subset\left[0,\frac{j_{2}}{2}\right]$, the number of $\ell$’s
such that $Cov(X^{\beta},X^{\gamma^{\ell}})\neq 0$ is at most
$(j_{1}+j_{2})(\leq j_{1}\cdot j_{2})$. Using all these estimates and Lemma
2.4 yields

$\displaystyle\sum_{|\beta|,|\gamma|\geq
1}\alpha_{\beta}^{j_{1}}\alpha_{\gamma}^{j_{2}}\,\cdot\,|Cov(V_{\alpha}^{\beta},V_{\alpha}^{\gamma})|$
$\displaystyle\leq$
$\displaystyle\sum_{\iota(\beta)=0}\sum_{\iota(\gamma)=0}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)j_{1}\cdot
j_{2}\cdot C_{X}^{|\beta|+|\gamma|}\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}$
$\displaystyle\leq$ $\displaystyle j_{1}(C_{X}+2)^{j_{1}}\cdot
j_{2}(C_{X}+2)^{j_{2}}\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}.$

Plugging this one to (3.2) and then (3.1), we have

$\displaystyle{\bf E}\left[\left\\{\sum_{j\geq k+1}c_{j}\sum_{|\beta|\geq
1}\alpha_{\beta}^{j}(V_{\alpha}^{\beta}-{\bf
E}[V_{\alpha}^{\beta}])\right\\}^{2}\right]$ $\displaystyle\leq$
$\displaystyle\left(\sum_{j=k+1}^{\infty}|c_{j}|j(C_{X}+2)^{j}\right)^{2}\sum_{i=1}^{N}\frac{1}{i^{2\alpha}},$
$\displaystyle Var\left(Y_{N}-X_{N,k}\right)$ $\displaystyle\leq$
$\displaystyle
D_{k}^{2}\frac{\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}}{g_{2\alpha}(N)}.$

where we set $D_{k}:=\sum_{j=k+1}^{\infty}|c_{j}|j(C_{X}+2)^{j}$. Therefore,

$\displaystyle\lim_{k\to\infty}\limsup_{N\to\infty}P(|X_{N,k}-Y_{N}|\geq\epsilon)\leq\lim_{k\to\infty}\limsup_{N\to\infty}D_{k}^{2}\frac{\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}}{g_{2\alpha}(N)}=0.$

###  3.3 Proof of B(1)

By assumption $c_{j}=0$ for $j$ : odd so that if $|\beta|\notin 2{\bf N}$,
${\cal C}_{i,j}(\beta)=\emptyset$, $\forall i$. We then compute

$\displaystyle\sum_{j=0}^{\infty}c_{j}\left\\{Tr(H_{\alpha,N}^{j})-{\bf
E}[Tr(H_{\alpha,N}^{j})]\right\\}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=2\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)+\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq 4\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$
$\displaystyle+\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq
2\end{subarray}}\left(a_{\beta}^{j,N}-p^{j}(\beta)\right)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle=:$ $\displaystyle I+II+III.$

Here we apply Lemma 2.8 to $I$, Lemma 2.7 to $II$, and Lemma 2.6 to $III$, to
have the desired result.

###  3.4 Proof of B(2)

Proof of B(2) and C(2) is similar to that of A(1) : let

$\displaystyle X_{N,k}$ $\displaystyle:=$
$\displaystyle\frac{Trf_{k}(H_{\alpha,N})-{\bf
E}[Trf_{k}(H_{\alpha,N})]}{g_{4\alpha}(N)},\quad
X_{k}:=N(0,\sigma_{B}(f_{k})^{2})$ $\displaystyle Y_{N}$ $\displaystyle:=$
$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{4\alpha}(N)},\quad X:=N(0,\sigma_{B}(f)^{2})$

and show that the conditions to apply Lemma 3.2 are satisfied.

(i) $X_{N,k}\stackrel{{\scriptstyle d}}{{\to}}X_{k}$, $N\to\infty$ has been
proved in [2].

(ii) To show $X_{k}\stackrel{{\scriptstyle d}}{{\to}}X$, let $\varphi_{k}(t)$
be the characteristic function of $X_{k}$. Then

$\displaystyle\varphi_{k}(t)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\sigma_{B}(f_{k})^{2}t^{2}\right)$
$\displaystyle=$
$\displaystyle\exp\left[-\frac{1}{2}\left\\{\left(\sum_{j=2}^{k}c_{j}p^{j}(2\delta)\right)^{2}\left({\bf
E}[X_{1}^{4}]-\eta^{4}\right)+\sum_{s=1}^{\infty}\left(\sum_{j=2}^{k}c_{j}p^{j}(\delta+\delta^{s})\right)^{2}\eta^{4}\right\\}t^{2}\right].$

To estimate that the second term in the exponential factor in RHS, we use

$\displaystyle p^{j}(2\delta)$ $\displaystyle=$ $\displaystyle j2^{j-3},$
$\displaystyle p^{j}(\delta+\delta^{s})$ $\displaystyle\leq$
$\displaystyle\frac{j(j-1)}{2}\dbinom{j-2}{(j-2)/2}\leq\frac{j^{2}}{2}\frac{1}{\sqrt{j-2}}2^{j-1}(1+o(1))\leq(Const.)j^{3/2}2^{j-1},$

and thus

$\displaystyle\sum_{s=1}^{\infty}\left(\sum_{j=2}^{k}c_{j}p^{j}(\delta+\delta^{s})\right)^{2}$
$\displaystyle=$
$\displaystyle\sum_{s=1}^{\infty}\sum_{j=2}^{k}\sum_{j^{\prime}=2}^{k}c_{j}\cdot
c_{j^{\prime}}p^{j}(\delta+\delta^{s})p^{j^{\prime}}(\delta+\delta^{s})$
$\displaystyle=$ $\displaystyle\sum_{j=2}^{k}\sum_{j^{\prime}=2}^{k}c_{j}\cdot
c_{j^{\prime}}\left(\sum_{s=1}^{j\vee
j^{\prime}}p^{j}(\delta+\delta^{s})p^{j^{\prime}}(\delta+\delta^{s})\right)$
$\displaystyle\leq$
$\displaystyle\left(\sum_{j=2}^{k}c_{j}j^{5/2}2^{j-1}\right)^{2}.$

Therefore the RHS converges as $k\to\infty$ so that

$\displaystyle\varphi_{k}(t)\stackrel{{\scriptstyle
k\to\infty}}{{\to}}\varphi(t)=\exp\left[-\frac{1}{2}\sigma_{B}(f)^{2}t^{2}\right].$

(iii) We note that if $|\beta|\notin 2{\bf N}$ ${\cal
C}_{i,j}(\beta)=\emptyset$, $\forall i$. In particular $a_{N}^{j}(\beta)=0$
for $|\beta|=1$. Hence

$\displaystyle Var\left(\sum_{j\geq k+1}c_{j}\frac{Tr(H^{j}_{\alpha,N})-{\bf
E}[Tr(H^{j}_{\alpha,N})]}{g_{4\alpha}(N)}\right)$ $\displaystyle=$
$\displaystyle{\bf E}\left[\left\\{\sum_{j\geq k+1}c_{j}\sum_{|\beta|\geq
2}\frac{a_{N}^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)}{g_{4\alpha}(N)}\right\\}^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{g_{4\alpha}(N)^{2}}\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq
k+1}c_{j_{1}}\cdot c_{j_{2}}\sum_{|\beta|\geq 2}\sum_{|\gamma|\geq
2}a_{N}^{j_{1}}(\beta)a_{N}^{j_{2}}(\gamma)Cov(V^{\beta},V^{\gamma}).$

Then its absolute value is bounded from above by

$\displaystyle\frac{1}{g_{4\alpha}(N)^{2}}\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|\sum_{|\beta|\geq 2}\sum_{|\gamma|\geq
2}a_{N}^{j_{1}}(\beta)a_{N}^{j_{2}}(\gamma)|Cov(V^{\beta},V^{\gamma})|$
$\displaystyle\leq$ $\displaystyle\frac{1}{g_{4\alpha}(N)^{2}}\sum_{j_{1}\geq
k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq 2\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0,\\\
|\gamma|\geq 2\end{subarray}}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)$
$\displaystyle\qquad\times\sum_{\ell\in{\bf
Z}}|Cov(X^{\beta},X^{\gamma^{\ell}})|\sum_{i\in I_{N,\ell}}\prod_{n\in{\bf
N}}\frac{1}{n^{\alpha(\beta_{n-i}+\gamma_{n-i-\ell})}}$ $\displaystyle\leq$
$\displaystyle\frac{2}{g_{4\alpha}(N)^{2}}\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|\left(j_{1}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq
2\end{subarray}}p^{j_{1}}(\beta)C_{X}^{|\beta|}\right)\left(j_{2}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq
2\end{subarray}}p^{j_{2}}(\gamma)C_{X}^{|\gamma|}\right)\sum_{i=1}^{N}\frac{1}{i^{4\alpha}}$
$\displaystyle\leq$
$\displaystyle\frac{2}{g_{4\alpha}(N)^{2}}\left(\sum_{j\geq
k+1}|c_{j}|j(C_{X}+2)^{j}\right)^{2}\sum_{i=1}^{N}\frac{1}{i^{4\alpha}}.$

Take $\epsilon>0$ arbitrary small. Then by Chebyshev’s inequality,

$\displaystyle{\bf P}\left(\left|\sum_{j\geq
k+1}c_{j}\frac{Tr(H^{j}_{\alpha,N})-{\bf
E}[Tr(H^{j}_{\alpha,N})]}{g_{4\alpha}(N)}\right|\geq\epsilon\right)$
$\displaystyle\leq$
$\displaystyle\frac{2}{g_{4\alpha}(N)^{2}}\left(\sum_{j\geq
k+1}|c_{j}|j(C_{X}+2)^{j}\right)^{2}\sum_{i=1}^{N}\frac{1}{i^{4\alpha}}\frac{1}{\epsilon^{2}}.$

Therefore by the definition of $g_{4\alpha}(N)$, we have

$\displaystyle\limsup_{N}{\bf P}\left(\left|\sum_{j\geq
k+1}c_{j}\frac{Tr(H^{j}_{\alpha,N})-{\bf
E}[Tr(H^{j}_{\alpha,N})]}{g_{4\alpha}(N)}\right|\geq\epsilon\right)\leq\frac{2}{\epsilon^{2}}\left(\sum_{j\geq
k+1}|c_{j}|j(C_{X}+2)^{j}\right)^{2}$ $\displaystyle\lim_{k}\limsup_{N}{\bf
P}\left(\left|\sum_{j\geq k+1}c_{j}\frac{Tr(H^{j}_{\alpha,N})-{\bf
E}[Tr(H^{j}_{\alpha,N})]}{g_{4\alpha}(N)}\right|\geq\epsilon\right)=0.$

###  3.5 Proof of C(1)

The idea is the same as that of the proof of B(1). In fact,

$\displaystyle Trf(H_{\alpha,N})-{\bf E}[Trf(H_{\alpha,N})]$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}c_{j}\Biggl{\\{}\Bigl{(}Tr(H_{\alpha,N}^{j})-{\bf
E}\left[Tr(H_{\alpha,N}^{j})\right]\Bigr{)}-\Bigl{(}p^{j}(\delta)Tr(H_{\alpha,N})-{\bf
E}\left[p^{j}(\delta)Tr(H_{\alpha,N})\right]\Bigr{)}\Biggr{\\}}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=3\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)+\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|=5\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf E}[V^{\beta}]\right)$
$\displaystyle+\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq 7\end{subarray}}p^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)+\sum_{j=0}^{\infty}c_{j}\sum_{\begin{subarray}{c}\iota(\beta)\in[1,N]\\\
|\beta|\geq
1\end{subarray}}\left(a_{\beta}^{j,N}-p^{j}(\beta)\right)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)$ $\displaystyle=:$ $\displaystyle I+II+III+IV.$

We apply Lemma 2.8 to $I,II$, Lemma 2.7 to $III$, and Lemma 2.6 to $IV$ to
obtain the desired conclusion.

###  3.6 Proof of C(2)

As in the proof of B(2), we set

$\displaystyle X_{N,k}$ $\displaystyle:=$
$\displaystyle\frac{Trf_{k}(H_{\alpha,N})-{\bf
E}[Trf_{k}(H_{\alpha,N})]}{g_{6\alpha}(N)},\quad
X_{k}:=N(0,\sigma_{C}(f_{k})^{2})$ $\displaystyle Y_{N}$ $\displaystyle:=$
$\displaystyle\frac{Trf(H_{\alpha,N})-{\bf
E}[Trf(H_{\alpha,N})]}{g_{6\alpha}(N)},\quad X:=N(0,\sigma_{C}(f)^{2})$

and we show that the three conditions (i), (ii), (iii) to apply Lemma 3.2 are
satisfied. (i) has been done in [2].

(ii) We shall show that, if $0<\alpha\leq 1/6$, the characteristic function
$\varphi_{k}(t)$ of $N(0,\sigma_{C}(f_{k})^{2})$ satisfies

$\displaystyle\varphi_{k}(t)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\sigma_{C}(f_{k})^{2}t^{2}\right)\to\exp\left(-\frac{1}{2}\sigma_{C}(f)^{2}t^{2}\right).$

In fact, we have

$\displaystyle\sigma_{C}(f_{k})^{2}$ $\displaystyle=$ $\displaystyle\langle
f_{k},C_{C}^{(k)}f_{k}\rangle=\sum_{\begin{subarray}{c}3\leq j\leq k\\\
j:odd\end{subarray}}\sum_{\begin{subarray}{c}3\leq j^{\prime}\leq k\\\
j^{\prime}:odd\end{subarray}}c_{j}c_{j^{\prime}}M_{j,j^{\prime}}$

where

$\displaystyle M_{j,j^{\prime}}$ $\displaystyle:=$
$\displaystyle\lim_{N\to\infty}Cov\left(\frac{TrP_{j}(H_{\alpha,N})-{\bf
E}[TrP_{j}(H_{\alpha,N})]}{g_{6\alpha}(N)},\frac{TrP_{j^{\prime}}(H_{\alpha,N})-{\bf
E}[TrP_{j}(H_{\alpha,N})]}{g_{6\alpha}(N)}\right).$

Since

$\displaystyle|M_{j,j^{\prime}}|$ $\displaystyle\leq$
$\displaystyle\lim_{N\to\infty}\left|Cov\left(\frac{TrP_{j}(H_{\alpha,N})-{\bf
E}[TrP_{j}(H_{\alpha,N})]}{g_{6\alpha}(N)},\frac{TrP_{j^{\prime}}(H_{\alpha,N})-{\bf
E}[TrP_{j}(H_{\alpha,N})]}{g_{6\alpha}(N)}\right)\right|$ $\displaystyle\leq$
$\displaystyle\sum_{\begin{subarray}{c}|\beta|\geq 3\\\
\iota(\beta)=0\end{subarray}}\sum_{\begin{subarray}{c}|\gamma|\geq 3\\\
\iota(\gamma)=0\end{subarray}}p^{j}(\beta)p^{j^{\prime}}(\gamma)\sum_{\ell\in{\bf
Z}}|Cov(X^{\beta},X^{\gamma^{\ell}})|\leq\sum_{\begin{subarray}{c}|\beta|\geq
3\\\ \iota(\beta)=0\end{subarray}}\sum_{\begin{subarray}{c}|\gamma|\geq 3\\\
\iota(\gamma)=0\end{subarray}}p^{j}(\beta)p^{j^{\prime}}(\gamma)j\cdot
j^{\prime}\cdot 4C_{X}^{|\beta|+|\gamma|}$ $\displaystyle\leq$ $\displaystyle
j(C_{X}+2)^{j}j^{\prime}(C_{X}+2)^{j^{\prime}}$

we have

$\displaystyle\sigma_{C}(f_{k})^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{\begin{subarray}{c}3\leq j\leq k\\\
j:odd\end{subarray}}\sum_{\begin{subarray}{c}3\leq j^{\prime}\leq k\\\
j^{\prime}:odd\end{subarray}}|c_{j}|\cdot|c_{j^{\prime}}|\cdot j\cdot
j^{\prime}\cdot(C_{X}+2)^{j}(C_{X}+2)^{j^{\prime}}$

which ensures the convergence of $\lim_{k\to\infty}\sigma_{C}(f_{k})^{2}$.
(iii) By assumption, $c_{j}=0$ for $j$ : even so that ${\cal
C}_{i,j}(\beta)=\emptyset$, $\forall i$ for $|\beta|\notin 2{\bf N}-1$. We
compute

$\displaystyle Var\left(\sum_{j\geq
k+1}c_{j}\left\\{\frac{Tr(H_{\alpha,N}^{j})-{\bf
E}[Tr(H_{\alpha,N}^{j})]}{g_{6\alpha}(N)}-p^{j}(\delta)\frac{Tr(H_{\alpha,N})-{\bf
E}[Tr(H_{\alpha,N})]}{g_{6\alpha}(N)}\right\\}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}Var\Biggl{(}\sum_{j\geq
k+1}c_{j}\Biggl{\\{}\sum_{|\beta|=1}a_{N}^{j}(\beta)(V^{\beta}-{\bf
E}[V^{\beta}])+\sum_{|\beta|\geq 3}a_{N}^{j}(\beta)(V^{\beta}-{\bf
E}[V^{\beta}])$ $\displaystyle\hskip
80.00012pt-p^{j}(\delta)\sum_{i=1}^{N}\left(V^{\delta^{i}}-{\bf
E}[V^{\delta^{i}}]\right)\Biggr{\\}}\Biggr{)}$ $\displaystyle=$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}Var\Biggl{(}\sum_{j\geq
k+1}c_{j}\Biggl{\\{}\sum_{i=1}^{N}\left(a_{N}^{j}(\delta^{i})-p^{j}(\delta)\right)\left(V^{\delta^{i}}-{\bf
E}[V^{\delta^{i}}]\right)+\sum_{|\beta|\geq 3}a_{N}^{j}(\beta)(V^{\beta}-{\bf
E}[V^{\beta}])\Biggr{\\}}\Biggr{)}$ $\displaystyle\leq$
$\displaystyle\frac{2}{g_{6\alpha}(N)^{2}}\Biggl{\\{}Var\Biggl{(}\sum_{j\geq
k+1}c_{j}\sum_{|\beta|\geq 3}a_{N}^{j}(\beta)(V^{\beta}-{\bf
E}[V^{\beta}])\Biggr{)}$ $\displaystyle\hskip
50.00008pt+Var\Biggl{(}\sum_{j\geq
k+1}c_{j}\Biggl{\\{}\sum_{i=1}^{N}\left(a_{N}^{j}(\delta^{i})-p^{j}(\delta)\right)\left(V^{\delta^{i}}-{\bf
E}[V^{\delta^{i}}]\right)\Biggr{)}\Biggr{\\}}$ $\displaystyle=:$
$\displaystyle 2(I+II).$

For $I$,

$\displaystyle I$ $\displaystyle=$ $\displaystyle{\bf
E}\left[\left\\{\sum_{j\geq k+1}c_{j}\sum_{|\beta|\geq
3}\frac{a_{N}^{j}(\beta)\left(V^{\beta}-{\bf
E}[V^{\beta}]\right)}{g_{6\alpha}(N)}\right\\}^{2}\right]$ $\displaystyle\leq$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|a_{N}^{j_{1}}(\beta)a_{N}^{j_{2}}(\gamma)|Cov(V^{\beta},V^{\gamma})|$
$\displaystyle\leq$ $\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}\sum_{j_{1}\geq
k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq 3\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq
3\end{subarray}}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)\sum_{\ell\in{\bf
Z}}|Cov(X^{\beta}X^{\gamma^{\ell}})|$ $\displaystyle\times\sum_{i\in
I_{N,\ell}}\prod_{n\in{\bf
N}}\frac{1}{n^{\alpha(\beta_{n-i}+\gamma_{n-i-\ell})}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq
k+1}|c_{j_{1}}|\cdot|c_{j_{2}}|\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|\geq 3\end{subarray}}\sum_{\begin{subarray}{c}\iota(\gamma)=0\\\
|\gamma|\geq 3\end{subarray}}p^{j_{1}}(\beta)p^{j_{2}}(\gamma)\cdot j_{1}\cdot
j_{2}\cdot 4C_{X}^{|\beta|+|\gamma|}\cdot\sum_{i=1}^{N}\frac{1}{i^{6\alpha}}$
$\displaystyle\leq$ $\displaystyle
4\frac{\sum_{i=1}^{N}i^{-6\alpha}}{g_{6\alpha}(N)^{2}}\left(\sum_{j\geq
k+1}|c_{j}|\cdot j\cdot(C_{X}+2)^{j}\right)^{2}.$

For $II$,

$\displaystyle II$ $\displaystyle=$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}Var\left(\sum_{j\geq
k+1}c_{j}\sum_{i=1}^{N}\left(a_{N}^{j}(\delta^{i})-p^{j}(\delta)\right)(V^{\delta^{i}}-{\bf
E}[V^{\delta^{i}}])\right)$ $\displaystyle=$
$\displaystyle\frac{1}{g_{6\alpha}(N)^{2}}\sum_{i=1}^{N}\left\\{\sum_{j\geq
k+1}c_{j}\left(a_{N}^{j}(\delta^{i})-p^{j}(\delta)\right)\right\\}^{2}\frac{{\bf
E}[X_{i}^{2}]}{i^{2\alpha}}$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\sum_{j_{1}\geq
k+1}\sum_{j_{2}\geq
k+1}c_{j_{1}}c_{j_{2}}\sum_{i=1}^{N}\left(a_{N}^{j_{1}}(\delta^{i})-p^{j_{1}}(\delta)\right)\left(a_{N}^{j_{2}}(\delta^{i})-p^{j_{2}}(\delta)\right)\frac{1}{i^{2\alpha}}.$

Here we decompose

$\displaystyle\sum_{j_{1}\geq k+1}\sum_{j_{2}\geq k+1}$ $\displaystyle=$
$\displaystyle\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=k+1}^{N}+\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=N+1}^{\infty}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=k+1}^{N}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=N+1}^{\infty}$

Since $a_{N}^{j}(\delta^{i})=p^{j}(\delta)$ for $\frac{j}{2}\leq i\leq
N-\frac{j}{2}$,

$\displaystyle II$ $\displaystyle=$
$\displaystyle\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=k+1}^{N}c_{j_{1}}c_{j_{2}}\left(\sum_{i=1}^{\min\left(\frac{j_{1}-1}{2},\frac{j_{2}-1}{2}\right)}+\sum_{i=N-\min\left(\frac{j_{1}-1}{2},\frac{j_{2}-1}{2}\right)}^{N}\right)$
$\displaystyle\hskip
30.00005pt\cdot\left(a_{N}^{j_{1}}(\delta^{i})-p^{j_{1}}(\delta)\right)\left(a_{N}^{j_{2}}(\delta^{i})-p^{j_{2}}(\delta)\right)\frac{1}{i^{2\alpha}}$
$\displaystyle+\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\left(\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=N+1}^{\infty}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=k+1}^{N}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=N+1}^{\infty}\right)c_{j_{1}}c_{j_{2}}$
$\displaystyle\hskip
30.00005pt\cdot\sum_{i=1}^{N}\left(a_{N}^{j_{1}}(\delta^{i})-p^{j_{1}}(\delta)\right)\left(a_{N}^{j_{2}}(\delta^{i})-p^{j_{2}}(\delta)\right)\frac{1}{i^{2\alpha}}$
$\displaystyle\leq$
$\displaystyle\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=k+1}^{N}|c_{j_{1}}||c_{j_{2}}|\min(j_{1},j_{2})\cdot
j_{1}2^{j_{1}}\cdot j_{2}2^{j_{2}}$
$\displaystyle+\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\left(\sum_{j_{1}=k+1}^{N}\sum_{j_{2}=N+1}^{\infty}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=k+1}^{N}+\sum_{j_{1}=N+1}^{\infty}\sum_{j_{2}=N+1}^{\infty}\right)|c_{j_{1}}||c_{j_{2}}|\cdot
N\cdot j_{1}2^{j_{1}}\cdot j_{2}2^{j_{2}}.$

Here we used the following estimates.

$\displaystyle\quad\sum_{i=1}^{N}\frac{1}{i^{2\alpha}}\leq
N,\qquad|a_{N}^{j}(\delta^{i})-p^{j}(\delta)|\leq p^{j}(\delta)\leq j\cdot
2^{j}.$

We note that $N\leq j_{2}$ in the 2nd term, $N\leq j_{1}$ in the 3rd term, and
$N\leq j_{1}$ in the 4th term. Therefore

$\displaystyle
II\leq\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\left(\sum_{j=k+1}^{\infty}|c_{j}|\cdot
j^{2}\cdot 2^{j}\right)^{2}.$

Plugging these estimates for $I,II$ above into Chebyshev’s inequality, we have

$\displaystyle{\bf P}\left(\left|\sum_{j\geq
k+1}c_{j}\left(\frac{TrH_{\alpha,N}^{j}-{\bf
E}[TrH_{\alpha,N}^{j}]}{g_{6\alpha}(N)}-p^{j}(\delta)\frac{TrH_{\alpha,N}-{\bf
E}[TrH_{\alpha,N}]}{g_{6\alpha}(N)}\right)\right|>\epsilon\right)$
$\displaystyle\leq$ $\displaystyle\frac{2(I+II)}{\epsilon^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{2}{\epsilon^{2}}\left[\frac{4\sum_{i=1}^{N}i^{-6\alpha}}{g_{6\alpha}(N)^{2}}\left(\sum_{j\geq
k+1}|c_{j}|j(C_{X}+2)^{j}\right)^{2}+\frac{\eta^{2}}{g_{6\alpha}(N)^{2}}\left(\sum_{j=k+1}^{\infty}|c_{j}|\cdot
j^{2}\cdot 2^{j}\right)^{2}\right]$ $\displaystyle\stackrel{{\scriptstyle
N\to\infty}}{{\to}}$ $\displaystyle\frac{2\cdot
4}{\epsilon^{2}}\left(\sum_{j\geq
k+1}|c_{j}|j(C_{X}+2)^{j}\right)^{2}\stackrel{{\scriptstyle
k\to\infty}}{{\to}}0$

which yields the conclusion.

##  4 Proof of Theorem 2

Take any ${\bf\theta}:=(\theta_{1},\cdots,\theta_{d})\in{\bf R}^{d}$ and let
$\widetilde{F}_{\theta,N}:={\bf\theta}\cdot{\bf F}_{N}$,
$f_{\theta}:=\sum_{t=1}^{d}\theta_{t}f_{t}$, $\widetilde{F}_{\theta}\sim
N(0,\sigma(f_{\theta})^{2})$. By Theorem 1, we have

$\displaystyle\widetilde{F}_{\theta,N}$ $\displaystyle:=$
$\displaystyle\sum_{t=1}^{d}\theta_{t}F_{t,N}=\frac{1}{g_{\alpha/\alpha_{c}}(N)}\left(Tr(f_{\theta}(H_{\alpha,N}))-{\bf
E}[Tr(f_{\theta}(H_{\alpha,N}))]\right)\stackrel{{\scriptstyle
d}}{{\to}}N(0,\sigma(f_{\theta})^{2})$

which shows $\widetilde{F}_{\theta,N}\stackrel{{\scriptstyle
d}}{{\to}}\widetilde{F}_{\theta}$. We denote by $\varphi_{X}(t)$ the
characteristic function of the random variable $X$. Since
$\widetilde{F}_{\theta}\sim N(0,\sigma(f_{\theta})^{2})$ and since
$r(f)>C_{X}+2$, we have

$\displaystyle\varphi_{\widetilde{F}_{\theta}}(\xi)$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{1}{2}\sigma(f_{\theta})^{2}\xi^{2}\right]=\exp\left[-\frac{1}{2}\xi^{2}\left(\sum_{t=1}^{d}\theta_{t}\sigma(f_{t})\right)^{2}\right]=\varphi_{{\bf\theta}\cdot{\bf
F}}(\xi)$

thus $\widetilde{F}_{\theta}\stackrel{{\scriptstyle d}}{{=}}\theta\cdot{\bf
F}$. By the method of Cramer-Wald, this implies ${\bf
F}_{N}\stackrel{{\scriptstyle d}}{{\to}}{\bf F}$, completing the proof of
Theorem 2.

##  5 Proof of Theorem 3 : Polynomial case

In this section we prove Theorem 3 when $f$ is a polynomial of degree $k$.
General case will be considered in the next section. Our goal in this section
is :

Theorem 5.1

${\bf
E}[Tr(H_{\alpha,N}^{k})]=A_{k,1}N+A_{k,0}+B^{k}(N)+\sum_{j=1}^{k}C_{j,k}\sum_{i=1}^{N}\frac{1}{i^{\alpha
j}}+D^{k}(N)$ (5.1)

where

$\displaystyle A_{k,1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}k\\\
k/2\end{array}\right)1\left(k:even\right),\quad A_{k,0}=-\sum_{{\bf y}\in{\cal
R}^{k}}(\max{\bf y}-\min{\bf y})$ $\displaystyle{\cal R}^{k}$
$\displaystyle:=$ $\displaystyle\left\\{{\bf y}=(y_{0},\cdots,y_{k})\in{\cal
P}^{k}\,\middle|\,y_{0}=y_{k}=0,\;{\bf y}\mbox{ has no flat steps }\right\\}$
$\displaystyle C_{j,k}$ $\displaystyle:=$
$\displaystyle\sum_{\begin{subarray}{c}\beta\,:\,\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf E}[X^{\beta}]$ $\displaystyle
B^{k}(N)$ $\displaystyle:=$
$\displaystyle\sum_{\iota(\beta)\in[k,N-k]^{c}}(a_{\beta}-p^{k}(\beta)){\bf
E}[V_{\alpha}^{\beta}]$ $\displaystyle B^{k}(N)$
$\displaystyle\stackrel{{\scriptstyle N\to\infty}}{{\to}}$ $\displaystyle
B^{k},\quad B^{k}:=\sum_{\iota(\beta)\in[1,k]}(a_{\beta}-p^{k}(\beta)){\bf
E}[V_{\alpha}^{\beta}]$ $\displaystyle D^{k}(N)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[X^{\beta}]\sum_{i=1}^{N}\left(\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}-\frac{1}{i^{\alpha|\beta|}}\right)$
$\displaystyle D^{k}(N)$ $\displaystyle\stackrel{{\scriptstyle
N\to\infty}}{{\to}}$ $\displaystyle D^{k},\quad
D^{k}:=\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[X^{\beta}]\sum_{i=1}^{\infty}\left(\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}-\frac{1}{i^{\alpha|\beta|}}\right)$
$\displaystyle|D^{k}|$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[|X|^{\beta}]\sum_{i=1}^{k}\frac{1}{i^{\alpha|\beta|}}$

(i) $A^{k}=A_{k,1}N+A_{k,0}$ is the contribution from the paths with no flat
steps, which can be derived explicitly. (ii) $B^{k}(N)$ is a boundary effect.
(iii) $\sum_{j=1}^{k}C_{j,k}\sum_{i=1}^{N}\dfrac{1}{i^{\alpha j}}={\cal
O}(N^{1-j\alpha})$ is the main term. The asymptotic behavior for
$Tr(P(H_{\alpha,N}))$ for a polynomial $P$ can be derived by summing up eq.
(5.1).

Proof. As in the proof of Theorem 1, let
$a_{\beta}:=[V_{\alpha}^{\beta}]\left(Tr(H_{\alpha,N}^{k})\right)$ be the
coefficient of $V^{\beta}$, $\beta\neq 0$ in $Tr(H_{\alpha,N}^{k})$. Then by
Lemma 2.3,

$\displaystyle{\bf E}[TrH_{\alpha,N}^{k}]$ $\displaystyle=$
$\displaystyle\sum_{|\beta|\geq 1}a_{\beta}{\bf
E}[V_{\alpha}^{\beta}]+A^{k}(N)=:A^{k}(N)+B^{k}(N)+C^{k}(N)$

where $A^{k}(N)$ is the contribution of the paths with no flat steps which we
write

$\displaystyle A^{k}(N)$ $\displaystyle=$ $\displaystyle A_{k,1}N+A_{k,0}$

and

$\displaystyle B^{k}(N)$ $\displaystyle:=$
$\displaystyle\sum_{\iota(\beta)\in[k,N-k]^{c}}(a_{\beta}-p^{k}(\beta)){\bf
E}[V_{\alpha}^{\beta}]$ $\displaystyle C^{k}(N)$ $\displaystyle:=$
$\displaystyle\sum_{\iota(\beta)\in[1,N]}p^{k}(\beta){\bf
E}[V_{\alpha}^{\beta}]$

We compute these terms separately below.

(A) We compute $A_{k}(N)=A_{k,1}N+A_{k,0}$. On the expansion of $H^{k}$,

$\displaystyle H^{k}$ $\displaystyle=$
$\displaystyle(V+U+D)^{k}=\sum_{M\in{\cal M}}M$

we consider the terms $M=\prod_{t}M_{t}\in{\cal M}$ such that $M_{t}\neq V$
and $TrM\neq 0$. Since we must have $\sharp U=\sharp D$, in $\\{M_{t}\\}$, $k$
must be even. In that case, such terms are expressed as

$\displaystyle U^{u_{1}}D^{d_{1}}\cdots U^{u_{j}}D^{d_{j}},\quad
u_{1}+\cdots+u_{j}=d_{1}+\cdots+d_{j}=k/2$

while the set of corresponding paths is

$\displaystyle{\cal R}^{k}$ $\displaystyle:=$ $\displaystyle\left\\{{\bf
y}=(y_{0},\cdots,y_{k})\in{\cal P}^{k}\,\middle|\,\;{\bf y}\mbox{ has no flat
steps }\right\\}$

By an explicit computation, we see that, in the corresponding $M\in{\cal M}$,
$\max{\bf y}(M)$ components in the leftmost diagpnal part and $-\min{\bf
y}(M)$ components in the rightleast diagonal part are all zero. Thus for even
$k$,

$\displaystyle Tr\left((U+D)^{k}\right)$ $\displaystyle=$
$\displaystyle\sum_{{\bf y}(M)}\left\\{N-(\max{\bf y}(M)-\min{\bf
y}(M))\right\\}$ $\displaystyle=$ $\displaystyle N\left(\begin{array}[]{c}k\\\
k/2\end{array}\right)-\sum_{{\bf y}\in{\cal R}^{k}}(\max{\bf y}-\min{\bf y}).$

(B) Since $V_{\alpha}$ is decaying, $B^{k}(N)$ is bounded and convergent :

$\displaystyle B^{k}(N)\stackrel{{\scriptstyle N\to\infty}}{{\to}}B^{k},\quad
B^{k}:=\sum_{\iota(\beta)\in[1,k]}(a_{\beta}-p^{k}(\beta)){\bf
E}[V_{\alpha}^{\beta}]$

(C) Using ${\bf
E}[V_{\alpha}^{\beta}]=\prod_{l}\left[\dfrac{X^{\beta_{l}}}{l^{\alpha\beta_{l}}}\right]={\bf
E}[X^{\beta}]\prod_{l\in{\bf N}}\dfrac{1}{l^{\alpha\beta_{l}}}$, we have

$C^{k}(N)=\sum_{j=1}^{k}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=i\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf E}[X^{\beta}]\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l}}}.$ (5.4)

Since $\iota(\beta)=i$, $|\beta|=j$, $p^{k}(\beta)\neq 0$, $\beta_{l}>0$
implies $l\in\left[i,i+\frac{k}{2}\right]\subset[i,i+k]$, and thus

$\displaystyle\frac{1}{(i+k)^{\alpha|\beta|}}=\prod_{l\in{\bf
N}}\frac{1}{(i+k)^{\alpha\beta_{l}}}\leq\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l}}}\leq\prod_{l\in{\bf
N}}\frac{1}{i^{\alpha\beta_{l}}}=\frac{1}{i^{\alpha|\beta|}}$

We replace $\prod_{l\in{\bf N}}\dfrac{1}{l^{\alpha\beta_{l}}}$ by
$\dfrac{1}{i^{\alpha|\beta|}}$ in (5.4), and denote the error by $D^{k}(N)$ :

$\displaystyle C^{k}(N)$ $\displaystyle=:$
$\displaystyle\sum_{j=1}^{k}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=i\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[X^{\beta}]\frac{1}{i^{\alpha|\beta|}}+D^{k}(N)$

By some change of variables, we have

$\displaystyle D^{k}(N)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf E}[X^{\beta}]\left(\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}-\frac{1}{i^{\alpha|\beta|}}\right)=\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf E}[X^{\beta}](-b_{N}(\beta))$
$\displaystyle where\quad
b_{N}(\beta):=\sum_{i=1}^{N}\left(\frac{1}{i^{\alpha|\beta|}}-\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}\right)$

Since $b_{N}(\beta)$ is monotonically increasing and bounded,

$\displaystyle\lim_{N\to\infty}b_{N}(\beta)=b(\beta):=\sum_{i=1}^{\infty}\left(\frac{1}{i^{\alpha|\beta|}}-\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}\right).$

Therefore

$\displaystyle D^{k}(N)\stackrel{{\scriptstyle N\to\infty}}{{\to}}D^{k},\quad
D^{k}$ $\displaystyle:=$
$\displaystyle\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[X^{\beta}]\sum_{i=1}^{\infty}\left(\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}-\frac{1}{i^{\alpha|\beta|}}\right)$
$\displaystyle|D^{k}|$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[|X|^{\beta}]\sum_{i=1}^{k}\frac{1}{i^{\alpha|\beta|}}$

On the other hand,

$\displaystyle C^{k}(N)-D^{k}(N)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k}C_{j,k}\sum_{i=1}^{N}\frac{1}{i^{\alpha j}},\quad
C_{j,k}:=\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf E}[X^{\beta}].$

Here we note that $C_{1,k}=0$, since
$\\{\beta\,|\,\iota(\beta)=0,|\beta|=1\\}=\\{\delta\\}$. The proof of Theorem
5.1 is now complete.

##  6  Proof of Theorem 3 : general case

In this section, we use Theorem 5.1 to finish the proof of Theorem 3. To be
more concrete, we show

Theorem 6.1

$\displaystyle{\bf E}[Trf(H)]$ $\displaystyle=$
$\displaystyle\left(\sum_{l}c_{l}C_{0,l}\right)N+\left(\sum_{l}c_{l}C_{2,l}\right)S_{2}(N)+\cdots+\left(\sum_{l}c_{l}C_{m,l}\right)S_{m}(N)+C_{N}(f)$
$\displaystyle C_{N}(f):=\left(\sum_{l}c_{l}A_{l,0}\right)+\sum_{l\geq
m+1}c_{l}\sum_{j=m+1}^{l}C_{j,l}S_{j}(N)+\sum_{l}c_{l}B^{l}(N)+\sum_{l}c_{l}D^{l}(N)$

where

$\displaystyle\lim_{N\to\infty}C_{N}(f)$ $\displaystyle=$
$\displaystyle\left(\sum_{l}c_{l}A_{l,0}\right)+\sum_{l\geq
m+1}c_{l}\sum_{j=m+1}^{l}C_{j,l}S_{j}+\sum_{l}c_{l}B^{l}+\sum_{l}c_{l}D^{l}$
$\displaystyle S_{j}$ $\displaystyle:=$
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{j\alpha}},\quad j\geq m+1$
$\displaystyle B^{k}$ $\displaystyle:=$
$\displaystyle\sum_{\iota(\beta)\in[1,k]}(a_{\beta}-p^{k}(\beta)){\bf
E}[V_{\alpha}^{\beta}]$ $\displaystyle D^{k}$ $\displaystyle:=$
$\displaystyle\sum_{j=1}^{k}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{k}(\beta){\bf
E}[X^{\beta}]\sum_{i=1}^{\infty}\left(\prod_{l\in{\bf
N}}\frac{1}{l^{\alpha\beta_{l-i}}}-\frac{1}{i^{\alpha|\beta|}}\right)$

Proof. By Theorem 5.1,

$\displaystyle{\bf E}[TrH^{k}]$ $\displaystyle=$ $\displaystyle
C_{0,k}N+C_{2,k}S_{2}(N)+\cdots+C_{m,k}S_{m}(N)+C_{m+1,k}S_{m+1}(N)+\cdots+C_{k,k}S_{k}(N)$
(6.1) $\displaystyle+B^{k}(N)+D^{k}(N)+A_{k,0}$

where $C_{0,k}=A_{k,1}$, $S_{m+1}(N),\cdots,S_{k}(N)$ are bounded w.r.t. $N$.
If $k\leq m$, we do not have terms of the form
$C_{m+1,k}S_{m+1}+\cdots+C_{k,k}S_{k}$. As we mentioned in the proof of
Theorem 1, $f(H)=\sum_{l\geq 0}a_{l}H^{l}$ is norm convergent, so that by
Fubini theorem,

$\displaystyle{\bf E}[Trf(H)]$ $\displaystyle=$ $\displaystyle\sum_{l\geq
0}c_{l}{\bf E}[Tr(H^{l})].$

Plugging it into (6.1) yields

$\displaystyle{\bf E}[Trf(H)]$ $\displaystyle=$
$\displaystyle\sum_{l}c_{l}{\bf E}[Tr(H^{l})]$ $\displaystyle=$
$\displaystyle\sum_{l}c_{l}\Bigl{(}C_{0,l}N+A_{l,0}+C_{2,l}S_{2}(N)+\cdots+C_{m,l}S_{m}(N)$
$\displaystyle\qquad+C_{m+1,l}S_{m+1}(N)+\cdots+C_{l,l}S_{l}(N)+B^{l}(N)+D^{l}(N)\Bigr{)}$

Here we would like to change the order of summation. In order for that, it
suffices to show that,

$\displaystyle(i)$ $\displaystyle\quad\sum_{l\geq
j}c_{l}C_{j,l},\quad(ii)\quad\sum_{l\geq
m+1}c_{l}\sum_{j=m+1}^{l}C_{j,l}S_{j}(N),\quad(iii)\quad\sum_{l}c_{l}B^{l}(N)$
$\displaystyle(iv)$
$\displaystyle\quad\sum_{l}c_{l}D^{l}(N),\quad(v)\quad\sum_{l}c_{l}A_{l,0}$

are absolutely convergent, and the quantities in (ii), (iii), (iv) converge as
$N\to\infty$.

(i) By Lemma 2.4 (2), we have $|C_{j,l}|\leq(C_{X}+2)^{l}$ so that since
$r(f)>C_{X}+2$, (i) is absolutely convergent.
(ii) That (ii) is absolutely convergent is shown similarly as (i). Since

$\displaystyle S(N)$ $\displaystyle:=$ $\displaystyle\max_{j\geq
m+1}S_{j}(N)<\infty,\quad\lim_{N\to\infty}S_{j}(N)=S_{j}:=\sum_{n=1}^{\infty}\frac{1}{n^{j\alpha}},\quad
j\geq m+1$

we have

$\displaystyle\lim_{N\to\infty}\sum_{l\geq
m+1}c_{l}\sum_{j=m+1}^{l}C_{j,l}S_{j}(N)=\sum_{l\geq
m+1}c_{l}\sum_{j=m+1}^{l}C_{j,l}S_{j}.$

(iii) Absolute convergence is similar as (i), (ii), and it is easy to see

$\displaystyle\lim_{N\to\infty}\sum_{l}c_{l}B^{l}(N)=\sum_{l}c_{l}B^{l}.$

(iv) We have only to use the following estimate :

$\displaystyle|D^{l}(N)|$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{l}\sum_{\begin{subarray}{c}\iota(\beta)=0\\\
|\beta|=j\end{subarray}}p^{l}(\beta)C_{X}^{j}(Const.)\max\\{1,l^{1-\alpha|\beta|}\\}$
$\displaystyle\leq$
$\displaystyle(C_{X}+2)^{l}(Const.)\max\\{1,l^{1-\alpha|\beta|}\\}.$

(v) By Stirling’s formula, we have

$\displaystyle|A_{k,0}|$ $\displaystyle\leq$ $\displaystyle\sum_{{\bf
y}\in{\cal R}_{k}}\left(\max{\bf y}-\min{\bf y}\right)\leq
k\left(\begin{array}[]{c}k\\\
\frac{k}{2}\end{array}\right)\leq(Const.)\sqrt{k}2^{k}$

so that $\sum_{l}c_{l}\cdot A_{l,0}$ is absolutely convergent. The proof of
Theorem 6.1 is now complete.

This work is partially supported by JSPS KAKENHI Grant Number 20K03659(F.N.).

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