On spectral measures of random Jacobi matrices

0.21 MB·

by  Trinh Khanh Duy

On spectral measures of random Jacobi matrices
∗
Trinh Khanh Duy
6
1
0
2 Abstract

n The paper studies the limiting behaviour of spectral measures of random Jacobi
u
matricesofGaussian,WishartandMANOVAbetaensembles. Weshowthatthespec-
J
tral measures converge weakly to a limit distribution which is the semicircle distri-
2
bution, Marchenko-Pastur distributions or Kesten-Mckey distributions, respectively.
2
The Gaussian fluctuation around the limit is then investigated.

] Keywords: spectralmeasure;randomJacobimatrix;Gaussianbetaensemble;Wishart
R
beta ensemble; MANOVA beta ensemble.
P AMS MSC 2010: Primary 60B20; Secondary 60F05, 47B36,47B80
.
h
t
a 1 Introduction
m
[
Three classical random matrix ensembles on the real line, Gaussian beta ensembles,

2 Wishart beta ensembles and MANOVA beta ensembles, are now realized as eigenvalues
v
of certain random Jacobi matrices. For instance, the following random Jacobi matrices
6
4 whose components are independent and distributed as
1
1 a b (0,2) χ
1 1 (n 1)β
0 N −
01. Hn,β = b1 .a.2. .b.2. ... ∼ √1nβ χ(n−1)β N(.0..,2) χ(n.−..2)β ... 
   
6  b a   χ (0,2)
1  n−1 n  β N 
   
:
v are matrix models of (scaled) Gaussian beta ensembles for any β > 0. Here (µ,σ2)
i N
X denotes the normal (or Gaussian) distribution with mean µ and variance σ2, and χ
k
r denotes the chi distribution with k degrees of freedom. Namely, the eigenvalues of H
a n,β
are distributed as Gaussian beta ensembles,
n
nβ
(λ ,...,λ ) ∆(λ)βexp λ2 ,
1 n ∝ | | − 4 i
!
i=1
X
where ∆(λ) = (λ λ ) denotes the Vandermonde determinant.
i<j j − i
Threespecialvaluesofbeta,β = 1,2and4,correspondtoGaussianorthogonal,unitary
Q
and symplectic ensembles (GOE, GUE and GSE) in which the above formula describes
thejoint distributionof eigenvalues of randommatrices with real, complex andquaternion
entries, respectively. As a generalization, Gaussian beta ensembles were originally defined
as ensembles of pointson thereallinewhosejointdensity function isgiven as above. They
can be also viewed as the equilibrium measure of a one dimensional Coulomb log-gas at
the inverse temperature β. Using the idea of tridiagonalizing a GOE matrix, Dumintriu
and Eldelman [6] introduced the model H for Gaussian beta ensembles. In the same
n,β
∗This work is supported by JSPS Grant-in-Aid for YoungScientists (B) no. 16K17616
1
paper, they also gave a matrix model for Wishart beta ensembles. A model for MANOVA
beta ensembles was discovered later by Killip and Nenciu [11].
One of main objects in random matrix theory is to study the limiting behaviour of the
empirical distribution of eigenvalues
n
1
L = δ ,
n n λi
i=1
X
where δ denotes the Dirac measure. For Gaussian beta ensembles, as n tends to infinity,
the empirical distributions converge weakly, almost surely, to the semicircle distribution,
which is well known as Wigner’s semicircle law. The convergence means that for any
bounded continuous function f on R,
n
1
L ,f = f(λ ) sc,f almost surely as n ,
n i
h i n → h i → ∞
i=1
X
with sc denoting the semicircle distribution, a probability measure supported on [ 2,2]
−
withdensity sc(x) = (2π) 1√4 x2. Afluctuation aroundthelimit was alsoinvestigated.
−
−
To be more precise, it was shown that for a ‘nice’ test function f,
n
n( L ,f µ ,f ) = (f(λ ) µ ,f ) d (0,a2),
h n i−h ∞ i i −h ∞ i →N f
i=1
X
where a2 can be written as a quadratic functional of f. There are several ways to prove
f
thoseresults. SeeJohansson[10]foranapproachbasedonjointdensityfunction,Dumitriu
andEdelman[7]andDumitriuandPaquette [8]foracombinatorial approachbasedonthe
random Jacobi matrix models. Since GOE and GUE have their original matrix models,
we can see more approaches in books [1, 14].
ThespectralmeasuresofrandomJacobimatrices associatedwiththosebetaensembles
have been investigated recently. The weak convergence to a limit distribution, a central
limit theorem for moments and large deviations have been established [5, 9, 12]. The
spectral measure of a finite Jacobi matrix, a symmetric tridiagonal matrix of the form,
a b
1 1
b a b
1 2 2
J =  ... ... ...,(ai ∈ R,bi > 0),
 
 b a 
n 1 n
 − 
 
is defined to be a unique probability measure µ on R satisfying
µ,xk = Jke ,e = Jk(1,1),k = 0,1,...,
1 1
h i h i
where e = (1,0,...,0)t Rn. Let λ ,...,λ be the eigenvalues of J and v ,...,v
1 1 n 1 n
∈ { } { }
be the corresponding eigenvectors which are chosen to be an orthonormal basis of Rn.
Then the spectral measure µ can be written as
n
µ = q2δ , q = v (1).
i λi i | i |
i=1
X
Notethattheeigenvalues λ aredistinctandtheweights q2 areallpositive. Moreover,
{ i} { i}
a finite Jacobi matrix of size n is one-to-one correspondence with a probability measure
supported on n real points.
2
Let µ be the spectral measure of H ,
n n,β
n
µ = q2δ .
n i λi
i=1
X
Inthiscase, andinallthreebetaensemblesinthepaper,theweights q2 areindependent
{ i}
ofeigenvaluesandhaveDirichletdistributionwithparameterβ/2. Thedistributionof q2
{ i}
is the same as that of the vector
χ2 χ2
β,1 β,n
,..., ,
n χ2 n χ2
i=1 β,i i=1 β,i!
where χ2 n is an i.i.d. sequePnce of randomPvariables having chi-squared distributions
{ β,i}i=1
with β degrees of freedom. In connection with empirical distributions, Nagel [13] showed
that as n tends to infinity, the Kolmogorov distance between L and µ converges almost
n n
surely to 0. Thus the spectral measures and the empirical distributions converge to the
same limit. Moreover, the limiting behaviour of spectral measures of Jacobi matrices can
be read off from the convergence of their entries. For Gaussian beta ensembles, it is clear
that
0 1
H 1 0 1 =:J , almost surely as n .
n,β   free
→ .. .. .. → ∞
. . .
 
 
Here the convergence means the piecewise convergence of entries. The non-random Ja-
cobi matrix J is called the free Jacobi matrix whose spectral measure is nothing but
free
the semicircle distribution [15, Section 1.10]. Consequently the spectral measures µ ,
n
and hence, the empirical distributions L , converge to the semicircle distribution. Thus
n
studying spectral measures may provide us a new look on classical results.
A natural problem now is to study the fluctuation of spectral measures around the
limit, or a central limit theorem for µ ,f with a ‘nice’ function f. Ina work which is not
n
h i
so related to random matrix theory, Dette and Nagel [5] derived a central limit theorem
for moments µ ,xk of spectral measures. The result covers Gaussian, Wishart beta
n
{h i}
ensembles and MANOVA beta ensembles with fixed parameters. The aim of this paper
is to reconsider the central limit theorem. We propose a universal approach which can be
easily applied to all three beta ensembles. The idea is that for a polynomial test function,
when n is large enough, µ ,p = p(H )(1,1) is a polynomial of finite variables. Then
n n,β
h i
the central limit theorem follows from the limiting behaviour of entries of Jacobi matrices.
Furthermore, by a relation between spectral measures and empirical measures, we obtain
an explicit formula for the limit variance and can extend the central limit theorem to a
large class of test functions. Our main result for Gaussian beta ensembles can be stated
as follows.
Theorem 1.1. (i) The spectral measures µ converge weakly, almost surely, to the
n
semicircle distribution as n , that is, for any bounded continuous function
→ ∞
f,
n
µ ,f = q2f(λ ) sc,f a.s. as n .
h n i i i → h i → ∞
i=1
X
(ii) For a function f with continuous derivative of polynomial growth,
√nβ
( µ ,f E[ µ ,f ]) d (0,σ2(f)) as n ,
n n
√2 h i− h i →N → ∞
3
where σ2(f) = sc,f2 sc,f 2 = Var [f]. Here ‘ d ’ denotes convergence in
sc
h i − h i →
distribution or weak convergence of random variables.
The results for Wishart and MANOVA beta ensembles are analogous where the semicircle
distribution is replaced by Marchenko-Pastur distributions and Kesten-Mckey distribu-
tions, respectively.
The paper is organized as follows. In the next section, we consider general random
Jacobi matrices and derive the weak convergence of spectral measures as well as the
central limit theorem for polynomial test functions. Applications to Gaussian, Wishart
and MANOVA beta ensembles are then investigated in turn. The last section is devoted
to extend the central limit theorem to a larger class of test functions.
2 Limiting behaviour of spectral measures of random Jacobi
matrices
Let us begin by introducing some spectral properties of (non random) Jacobi matrices. A
semi-infinite Jacobi matrix is a symmetric tridiagonal matrix of the form
a b
1 1
J = b1 a2 b2 , where ai R,bi > 0.
... ... ... ∈
 
 
To a Jacobi matrix J, there exists a probability measure µ such that
µ,xk = xkdµ = Jke ,e ,k = 0,1,...,
1 1
h i R h i
Z
where e = (1,0,...,)t ℓ2. Then µ is unique, or µ is determined by its moments, if and
1
∈
only if, J is an essentially self-adjoint operator on ℓ2. If the parameters a and b are
i i
{ } { }
bounded, or more generally, if b 1 = , then J is essentially self-adjoint [15, Corollary
−i ∞
3.8.9]. In case of uniqueness, we call µ the spectral measure of J, or of (J,e ). See [4,
1
P
Chapter 2] or [15, Section 3.8] for more details on Jacobi matrices.
When J is a finite Jacobi matrix of order n, then the spectral measure µ is supported
on the eigenvalues λ of J with weights q2 = v (1)2 ,
{ i} { i} { i }
n
µ = q2δ .
i λi
i=1
X
Here v ,...,v are the correspondingeigenvectors which arechosen to bean orthogonal
1 n
{ }
basis of Rn. The eigenvalues λ n are distinct and the weights q2 n are all positive
{ i}i=1 { i}i=1
[4, Proposition 2.40].
We are now in a position to study the convergence of spectral measures of Jacobi ma-
trices. Recall that spectral measures are defined by their moments. Does the convergence
of moments imply the weak convergence of probability measures? The following lemma
gives us the answer. It is a classical result which can be found in some textbooks in
probability theory.
Lemma 2.1. Assume that µ and µ are probability measures on R such that for all
{ n}∞n=1
k = 0,1,...,
µ ,xk µ,xk as n .
n
h i→ h i → ∞
4
Assume further that the measure µ is determined by its moments. Then µ converges
n
weakly to µ as n . Moreover, if f is a continuous function of polynomial growth, that
→ ∞
is, there is a polynomial P such that f(x) P(x) for all x R, then we also have
| | ≤ ∈
µ ,f µ,f as n .
n
h i→ h i → ∞
Proof. The first part of this lemma is a well-known moment problem [2, Theorem 30.2].
The second part follows by a truncated argument. For the sake of completeness, we give
proof here. Assume that the sequence µ converges weakly to µ and that µ ,P con-
n n
{ } h i
verges to µ,P for all polynomials P. Let f be a continuous function which is dominated
h i
by a polynomial P, f(x) P(x) for all x R. For M > 0, write f for the truncated
M
| | ≤ ∈
function
M, if f(x) M,
− ≤ −
f (x) = f(x), if f(x) M,
M 
| |≤
M, if f(x) M.
≥
Then it is clear that f fM P PM, where PM is the truncated function of P. Thus
| − |≤ −
by the triangle inequality,
µ ,f µ,f µ ,f f + µ ,f µ,f + µ,f f
n n M n M M M
|h i−h i|≤ |h − i| |h i−h i| |h − i|
µ ,P P + µ ,f µ,f + µ,P P
n M n M M M
≤ h − i |h i−h i| h − i
= µ ,P µ ,P + µ ,f µ,f + µ,P P .
n n M n M M M
h i−h i |h i−h i| h − i
Asn ,thefirsttermconvergesto µ,P bytheassumption,thesecondtermconverges
→ ∞ h i
to µ,P and the third term converges to 0 because P and f are bounded continuous
M M M
h i
functions. Therefore
limsup µ ,f µ,f 2 µ,P P .
n M
|h i−h i| ≤ h − i
n
→∞
Finally, by letting M , µ,P P 0 by the monotone convergence theorem. The
M
→ ∞ h − i→
lemma is proved.
To random probability measures, we deal with two types of convergence, almost sure
convergence and convergence in probability. A result for almost sure convergence is a
direct consequence of the above deterministic result. However, it is not the case for
convergence in probability. When the limit measure has compact support, the following
result on convergence in probability of random measures can be derived by a method of
polynomials approximation, see subsection 2.1.2 in [1], for instance.
Lemma 2.2. Let µ be a sequence of random probability measures and µ be a non-
{ n}∞n=1
random probability measure which isdetermined byitsmoments. Assume that anymoment
of µ converges almost surely to that of µ, that is, for any k = 0,1,...,
n
µ ,xk µ,xk a.s. as n .
n
h i→ h i → ∞
Then as n , the sequence of measures µ converges weakly, almost surely, to µ,
n
→ ∞ { }
namely, for any bounded continuous function f,
µ ,f µ,f a.s. as n .
n
h i→ h i → ∞
The convergencestillholds foracontinuousfunctionf ofpolynomial growth. Ananalogous
result holds for convergence in probability.
5
Proof. The case of almost sure convergence is a direct consequence of Lemma 2.1. Indeed,
for k 1, let
≥
A = ω : µ (ω),xk µ,xk as n .
k n
{ h i → h i → ∞}
Then P(Ak) = 1 by the assumption. Therefore P(A := ∞k=1Ak) = 1. Applying
Lemma 2.1 to the sequence of probability measures µ (ω) , for ω A, yields the desired
n
{ } T ∈
result.
Next we consider the case of convergence in probability. The idea here is to use the
following criterion for convergence in probability [2, Theorem 20.5]: a sequence X
n
{ }
converges to X in probability if and only if for every subsequence X , there is a
n(m)
{ }
further subsequence X that converges almost surely to X. Let f be a continuous
{ n(mk)}
function which is dominated by some polynomial. Given a subsequence n(m) , the aim
{ }
now is to find a subsequence n(m ) such that µ ,f converges almost surely to
k {h n(mk) i}
µ,f . Let n(0,m) = n(m) . For k 1, using the necessary condition in the criterion,
h i { } ≥
we can find a subsequence n(k,m) of n(k 1,m) such that
{ } { − }
µ ,xk µ,xk a.s. as n .
n(k,m)
h i→ h i → ∞
By selecting the diagonal, we get a subsequence n(m ) = n(k,k) for which all moments
k
{ }
of µ converge almost surely to the corresponding moments of µ. Consequently, the
{ n(mk)}
sequence µ converges weakly, almost surely, to µ by the first part of this lemma,
{ n(mk)}
which implies that µ ,f µ,f almost surely. The proof is complete.
{h n(mk) i}→ h i
Let us now explain the main idea of this paper. Consider the sequence of random
Jacobi matrices
(n) (n)
a b
1 1
(n) (n) (n)
b a b
 1 2 2 
J = ,
n ... ... ...
 
 (n) (n)
 b a 
 n 1 n 
 − 
and let µ be the spectral measure of (J ,e ). Assume that each entry of J converges
n n 1 n
almost surely to a non random limit as n , that is, for any fixed i, as n ,
→ ∞ → ∞
a(n) a¯ ; b(n) ¯b a.s. (1)
i → i i → i
Here we require that a¯ and ¯b are non random and ¯b > 0. Assume further that the
i i i
spectral measure of (J ,e ), denoted by µ , is unique, where J is the infinite Jacobi
1
matrix consisting of a¯∞ and ¯b , ∞ ∞
i i
{ } { }
a¯ ¯b
1 1
J = ¯b1 a¯2 ¯b2 .
 
∞ ... ... ...
 
 
Thenthemeasureµ isdeterminedbyitsmoments, andhencewegetthefollowingresult.
∞
Theorem 2.3. The spectral measures µ converge weakly, almost surely, to the limit
n
measure µ as n .
∞ → ∞
Remark 2.4. If in the assumption (1), convergence in probability is assumed instead of
almost sure convergence, then the spectral measures µ converge weakly, in probability,
n
to µ as n .
∞ → ∞
6
Proof. Let p be a polynomial of degree m. When n is large enough, µ ,p = p(J )(1,1)
n n
h i
(n) (n)
is a polynomial of {ai ,bi }i=1,...,⌈m2⌉. Therefore, as n → ∞,
µ ,p µ ,p a.s.,
n
h i → h ∞ i
which implies the weak convergence of µ by Lemma 2.2.
n
Next, we consider the second order of the convergence of spectral measures, or a type
of central limit theorem. It turns out that the central limit theorem for polynomial test
functions is a direct consequence of a joint central limit theorem for entries of Jacobi
matrices. Indeed, assume that there are random variables η and ζ defined on the
i i
{ } { }
same probability space such that for some fixed r > 0, for any i, as n ,
→ ∞
a˜(n) = nr(a(n) a¯ ) d η ,
i i − i → i (2)
˜b(n) = nr(b(n) ¯b ) d ζ .
i i − i → i
Moreover, we assume that the joint weak convergence holds. This means that any finite
linear combination of a˜(n) and˜b(n) converges weakly to the corresponding linear combina-
i i
tion of η and ζ as n , namely, for any real numbers c and d ,
i i i i
→ ∞
(c a˜(n)+d˜b(n)) d (c η +d ζ ).
i i i i → i i i i
finite finite
X X
For now on, both conditions (1) and (2) will be written in a compact form
a¯ ¯b η ζ
1 1 1 1
Jn ≈ ¯b1 .a¯.2 .¯b.2 .. + n1r ζ1 .η.2 .ζ.2 .. .
. . . . . .
   
   
Let f be a polynomial of 2k variables (a ,...,a ,b ,...,b ). For simplicity, we write
1 k 1 k
f(a ,b ) instead of f(a ,...,a ,b ,...,b ).
i i 1 k 1 k
Lemma 2.5. (i) As n ,
→ ∞
k
∂f ∂f
nr f(a(n),b(n)) f(a¯ ,¯b ) (a¯ ,¯b )a˜(n)+ (a¯ ,¯b )˜b(n) P 0.
i i − i i − ∂a i i i ∂b i i i →
(cid:16) (cid:17) Xi=1(cid:18) i i (cid:19)
P
Here ‘ ’ denotes the convergence in probability.
→
(ii) As n ,
→ ∞
k
∂f ∂f
nr f(a(n),b(n)) f(a¯ ,¯b ) d (a¯ ,¯b )η + (a¯ ,¯b )ζ .
i i − i i → ∂a i i i ∂b i i i
(cid:16) (cid:17) Xi=1(cid:18) i i (cid:19)
Proof. Write
1 1
a(n) = a¯ + a˜(n);b(n) =¯b + ˜b(n).
i i nr i i i nr i
Then use the Taylor expansion of f(a(n),b(n)) at (a¯ ,¯b ) with noting that the Taylor ex-
i i i i
pansion of a polynomial consists of finitely many terms,
k
1 ∂f ∂f
f(a(n),b(n)) = f(a¯ ,¯b )+ (a¯ ,¯b )a˜(n)+ (a¯ ,¯b )˜b(n) + .
i i i i nr ∂a i i i ∂b i i i ∗
i=1(cid:18) i i (cid:19)
X X
7
Each term in the finite sum has the following form,
∗
P k
c(α,β) (a(n) a¯ )αi(b(n) ¯b )βi,
i − i i − i
i=1
Y
where α and β are non negative integers and k (α +β ) 2. Therefore, when
{ i} { i} i=1 i i ≥
that term is multiplied by nr, it converges to 0 in distribution, and hence, in probability
P
by Slutsky’s theorem.
By using Slutsky’s theorem again, we see that (ii) is a consequence of (i). The proof
is complete.
Let p be a polynomial of degree m > 0. Then there is a polynomial of 2 m variables
⌈2⌉
such that for n > m/2,
(n) (n) (n) (n)
µ ,p = p(J )(1,1) = f(a ,...,a ,b ,...,b ).
h n i n 1 ⌈m2⌉ i ⌈m2⌉
Therefore, by Lemma 2.5, we obtain the central limit theorem for polynomial test func-
tions.
Theorem 2.6. For any polynomial p, nr( µ ,p µ ,p ) converges weakly to a limit
n
h i−h ∞ i
as n .
→ ∞
(n) (n)
Since we do not assume that all moments of a and b are finite, even the
{ i } { i }
expectation of µ ,p , E[ µ ,p ] may not exist. Thus we need further assumptions to
n n
h i h i
ensure the convergence of mean and variance in the central limit theorem above. Our
assumptions are based on the following basic result in probability theory (the corollary
following Theorem 25.12 in [2]).
Lemma 2.7. Assume that the sequence X converges weakly to a random variable X.
n
{ }
If for some δ > 0,
supE[X 2+δ]< ,
n
| | ∞
n
then E[X ] E[X] and Var[X ] Var[X] as n . In general, if X converges to X
n n n
→ → → ∞
in probability and for q 1 and δ > 0,
≥
supE[X q+δ] < ,
n
| | ∞
n
then X converges to X in L .
n q
We make the following assumptions
(n) (n)
(i) all moments of a and b are finite and in addition, the convergences in (1)
{ i } { i }
also hold in L for all q < , which is equivalent to the following conditions
q
∞
supE[a(n) k]< , supE[b(n) k] < , for all k = 1,2,...; (3)
| i | ∞ | i | ∞
n n
(ii) E[η ]= 0,E[ζ ]= 0, and for some δ > 0,
i i
supE[a˜(n) 2+δ]< , supE[˜b(n) 2+δ]< . (4)
| i | ∞ | i | ∞
n n
8
Lemma 2.8. As n ,
→ ∞
k
2 ∂f ∂f
E f(a(n),b(n)) f(a¯ ,¯b ) Var (a¯ ,¯b )η + (a¯ ,¯b )ζ , (5)
i i − i i → ∂a i i i ∂b i i i
(cid:20)(cid:16) (cid:17) (cid:21) "Xi=1(cid:18) i i (cid:19)#
nr E[f(a(n),b(n))] f(a¯ ,¯b ) 0, (6)
i i − i i →
(cid:16) (cid:17) k
∂f ∂f
Var f(a(n),b(n)) Var (a¯ ,¯b )η + (a¯ ,¯b )ζ . (7)
i i → ∂a i i i ∂b i i i
h i "Xi=1(cid:18) i i (cid:19)#
Proof. It is just a direct consequence of Lemma 2.7.
We state now a slightly different form of the central limit theorem for polynomial test
functions.
Theorem 2.9. Under assumptions (1)–(4), for any polynomial p, as n ,
→ ∞
µ ,p µ ,p a.s. and in Lq for all q 1;
n
h i → h ∞ i ≥
nr( µ ,p E[ µ ,p ]) d ξ (p).
n n
h i− h i → ∞
Here ξ (p) denotes the limit distribution. Moreover, E[ξ (p)]= 0 and
∞ ∞
n2rVar[ µ ,p ] Var[ξ (p)] as n .
n
h i → ∞ → ∞
3 Gaussian beta ensembles or β-Hermite ensembles
LetH bearandomJacobimatrixwhoseelementsareindependent(uptothesymmetric
n,β
constraint) and are distributed as
(0,2) χ
(n 1)β
N −
Hn,β = √1nβ χ(n−1)β N(.0..,2) χ(n.−..2)β ... .
 
 χ (0,2)
 β N 
 
Then the eigenvalues λ of H have Gaussian beta ensembles [6], that is,
i n,β
{ }
n
nβ
(λ ,λ ,...,λ ) ∆(λ)βexp λ2 .
1 2 n ∝ | | − 4 j
!
i=1
X
The weights w = q2 are independent of λ and have Dirichlet distribution with
{ i} { i} { i}
parameters (β/2,...,β/2), that is,
n
β 1
(w1,...,wn−1)∝ wi2− 1{w1+···+wn−1<1,wi>0}, wn =1−(w1+···+wn−1).
i=1
Y
Recall that the distribution of w is the same as that of the vector
i
{ }
χ2 χ2
β,1 β,n
,..., ,
n χ2 n χ2
i=1 β,i i=1 β,i!
with χ2 n being i.i.d. sequePnce of randomPvariables having chi-squared distributions
{ β,i}i=1
with β degrees of freedom.
9
Lemma 3.1. (i) As k ,
→ ∞
χ
k 1 in probability and in L for all q < .
q
√k → ∞
A sequence {χkn/√kn} converges almost sure to 1, if ∞n=1e−εkn < ∞ for anyε > 0.
(ii) As k , P
→ ∞
χ 1
√k k 1 = (χ √k) d (0, ).
k
√k − − →N 2
(cid:18) (cid:19)
Proof. The convergence in probability and a central limit theorem are standard results.
Let us prove the almost sure convergence part. For almost sure convergence, usually
there is some relation between random variables in the sequence. If there is no relation,
the following criterion, a direct consequence of the Borel-Cantelli lemma, becomes useful.
The sequence X converges almost surely to x as n , if for any ε> 0,
n
→ ∞
∞
P(X x > ε) < .
n
| − | ∞
n=1
X
For the proof here and later for Beta distributions, we use the following bounds (see
Lemma 4.1 in [13])
P χ2k 1 > ε 2e−k8ε2,
k − ≤
(cid:18)(cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
P(Beta(a,b) (cid:12)(cid:12) E[Beta(cid:12)(cid:12)(a,b)] > ε) 4e−1ε228a3a+bb3.
| − | ≤
Here Beta(a,b) denotes the beta distribution with parameters a and b. Then the proof
follows immediately from the criterion above.
Since the Jacobi parameters a ,b of H (β) are independent, it follows that joint
i i n
{ }
convergence in distribution holds. Thus we can write
0 1 (0,2) (0, 1)
H 1 0 1 + 1 N(0, 1) N(0,22) (0, 1) .
n,β ≈  ... ... ... √βn N 2 N... N ...2 ...
   
   
ThenonrandomJacobimatrixintheaboveexpressioniscalledfreeJacobimatrix,denoted
by J , whose spectral measure is the semicircle distribution
free
1
sc(x) = 4 x2,( 2 x 2).
2π − − ≤ ≤
p
There are several ways to derive that fact. For example, in the theory of orthogonal
polynomials on the real line, it is derived from the relation of Chebyshev polynomials of
the second kind [15, Section 1.10]. One can also calculate moments and show that they
match moments of the semicircle distribution. However, we introduce here a method to
find the spectral measure by calculating its Stieltjes transform, which is applicable to the
next two cases.
Let J be a Jacobi matrix with bounded coefficients,
a b
1 1
J = b1 a2 b2 .
 
... ... ...
 
 
10