Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $β$-ensembles

0.22 MB·English

by  Peter J. Forrester

Large deviation eigenvalue density for the soft edge
Laguerre and Jacobi β-ensembles
Peter J. Forrester
2
1
0
2 Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
email: [email protected]
n
a
J Abstract

5
We analyze the eigenvalue density for the Laguerre and Jacobi β-ensembles in the cases
1
that the corresponding exponents are extensive. In particular, we obtain the asymptotic

] expansion up to terms o(1), in the large deviation regime outside the limiting interval
h
p of support. As found in recent studies of the large deviation density for the Gaussian
- β-ensemble, and Laguerre β-ensemble with fixed exponent, there is a scaling from this
h
t asymptotic expansion to the right tail asymptotics for the distribution of the largest
a
m eigenvalue at the soft edge.
[

1 1 Introduction
v
5
5 Let X be a p N (p N) random matrix with standard real Gaussian entries. It is well known
0 × ≥
in random matrix theory [26, 13] that the eigenvalue probability density function (PDF) of the
3
. corresponding covariance matrix W = XTX is proportional to
1
0
2 N
1 λβa/2e−βλl/2 λ λ β, (1.1)
: l | k − j|
v l=1 1≤j<k≤N
Y Y
i
X
with a = p N + 1 2/β and β = 1. For general β (1.1) is referred to as the Laguerre
r − −
a β-ensemble and is to be denoted ME (λβa/2e−βλ/2).
β,N
In multivariate statistics, thinking of X as a data matrix with successive columns storing
the values of p measurements for each of the N observables, knowledge of (1.1) allows for an
analysisofquestionssuchas“arethemeasuredquantitiesinX correlated?” [26]. Inphysics, the
Laguerre β-ensemble (1.1) with exponent N/2 appears as the eigenvalue PDF of the Wigner-
Smith delay time matrix in mesoscopic transport problems [7] (see also [13, 3.3.1]). The
§
parameter β can then take on one of the three values β = 1,2 or 4 depending on the underlying
time reversal symmetry. We remark too that the case β = 2 of (1.1) corresponds to the
eigenvalue PDF for the so-called Penner model in the theory of matrix models (see e.g. [28]).
It also appears in the analysis of information capacity in the setting of wireless communication
systems [30].
A peculiar feature of the Wigner-Smith time delay matrix example is that the parameter a
in (1.1) is extensive, being proportional to the number of eigenvalues N. This is in fact what is
1
required in the analysis of the Penner model, and this same limit is of interest in the study of
wireless communication systems; see references as cited above. One direction of study relating
to (1.1) with a extensive is the large N asymptotic expansion of the moments
N
λk . (1.2)
l
* +
Xl=1 MEβ,N(λβa/2e−βλ/2)
Recent references on this topic include [22], [24, 25]. These moments probe the non-oscillatory
portion of the large N asymptotic expansion of the eigenvalue density in the domain of its
leading support. Our interest in this paper is also in the large N expansion of the eigenvalue
density with a extensive, but in the domain outside rather than inside its leading support.
Physical motivation to pursue this problem comes from a very recent [16] experimental set
up relating to (1.1) in the case β = 2, which in turn comes about by (1.1) corresponding to
the eigenvalue PDF for matrices X†X, where X is a p N complex Gaussian matrix. In fact,
×
the theoretical discussion in [16] tells us that the so-called round trip propagation matrix M
determining the evolution of the complex electric field for a p N coupled array of high gain
×
lasers can be decomposed in the form X†X for X a complex p N matrix with approximately
×
Gaussian entries. Furthermore, the output power is proportional to the largest eigenvalue of
M. With hundreds of thousands of data points accessible in this way, it is possible to probe
the large deviation portions of the largest eigenvalue PDF from these measurements, something
that was done explicitly for the square case p = N(= 5). For the right tail the large deviation
form of the largest eigenvalue PDF is, to leading order, equal to the density. This happens in
the domain outside the leading order support, where the density takes on exponentially small
values.
There is another random matrix ensemble that allows for extensive exponents. This is the
Jacobi β-ensemble, specified by an eigenvalue PDF proportional to
N
λβa1/2(1 λ )βa2/2 λ λ β, 0 < λ < 1 (1.3)
l − l | k − j| l
l=1 1≤j<k≤N
Y Y
andreferred toas ME (λβa1/2(1 λβa2/2)). Inthe special case a = 0,a = p N+1 2/β and
β,N 2 1
− − −
β = 1,2 and 4 (1.3) gives the PDF of the non-zero squared singular values of an N p sub-block
×
of a (N +p) (N +p) matrix from the circular orthogonal ensemble (β = 1), circular unitary
×
ensemble (β = 2) and circular symplectic ensemble (β = 4) (see [13, 3.8.2]). The propagation
§
of electron fluxes in a disordered mesoscopic wire is well described by scattering matrices from
the circular ensembles. And an N p sub-block then corresponds to the transmission matrix
×
relating a state of p channels entering the left of the wire and a state of N channels exiting the
right. The corresponding singular values then determine the conductance (see e.g. [3]). We
will compute the asymptotic form of the spectral density for (1.3) with a and a extensive,
1 2
in the asymptotically small region outside the leading support. With (1.3) in the case β = 1
occurring in mathematical statistics, and the largest eigenvalue having special significance in
various statistical tests [19], the fact that our results imply the right tail asymptotic form of the
distribution of the largest eigenvalue thus provides us with motivation in an applied setting.
Our task then is to compute the eigenvalue density for both the Laguerre and Jacobi β-
ensembles with extensive exponents, in the exponentially small regime outside the leading
2
support. In addition to the interest from the viewpoint of the problems discussed above, this
complements therecent computation of the largedeviation density for the Gaussian β-ensemble
[14, 6], and the Laguerre β-ensemble with a fixed [14]. Here we will use the method of [14]
based on writing the density as an average of the β moment of the characteristic polynomial,
and then using a Gaussian fluctuation theorem to deduce the asymptotics. This is distinct from
the method of [6] which is based on the loop equations for correlators. In Section 2 we review
results on this problem already documented in the literature, and then present our general
strategy. The details of our calculations, leading to the sought large deviation formulas, are
given in Section 3. In the case of the Laguerre ensemble with weight λβαN/2+β/2−1e−βλ/2, where
uL is given by (2.9) below and x (0,(√α+1 1)2) ((√α+1+1)2, ), we obtain
∈ − ∪ ∞
Nβ uL +x 2 α uL x α
NρL (Nx) exp uL +2log − − 2αlog − −
(1),N N→∼∞ ( 2 − 2(1+α)1/2 − 2x1/2(1+α)1/2 !)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) x(uL +(cid:12)x 2 α) (cid:12)
exp (1 3β/2)log u(cid:12)L (1 β/2)lo(cid:12)g (cid:12) − − (cid:12)
× − | |− − 2
( )
(cid:12) (cid:12)
(cid:12) (cid:12)
N(1+α)1/2 2 β/2 (cid:12) (cid:12)
Γ(1+β/2) (cid:12) (cid:12) (1.4)
× 2π βN
(cid:18) (cid:19)
while for the Jacobi ensemble with weight λβα1N/2+β/2−1(1 λ)βα2N/2+β/2−1, where uJ is given
−
by (3.24) below and x (0,c ) (c ,1) with c ,c given by (2.3), (2.4), our final result reads
1 2 1 2
∈ ∪
x (c +c )/2+uJ
ρJ (x) exp Nβ log − 1 2
(1),N N→∼∞ ( (cid:12) (c2 c1)/2 (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
√c c(cid:12)+x uJ (cid:12) (1 c )(1 c ) x+1+uJ
1 2 1 2
α log | (cid:12) − | α log | (cid:12) − − − |
− 1 (√c1 +√c2)x1/2 − 2 p(√1 c1 +√1 c2)(1 x)1/2 !)
− − −
x(1 x)(uJ +x (c +c )/2)
exp (1 3β/2)log uJ (1 β/2)log − − 1 2
× − | |− − 2+α +α
( (cid:12) 1 2 (cid:12))
(cid:12) (cid:12)
N(c c ) 1 β/2 (cid:12) (cid:12)
2 − 1 Γ(1+β/2). (cid:12) ((cid:12)1.5)
× 4π βN
(cid:18) (cid:19)
The notation in (1.4) and (1.5) means that LHS/RHS 1 in the appropriate limit. Both
∼ →
these asymptotic formulas are shown to exhibit a scaling to the asymptotic form of the right
tail for the distribution of the largest eigenvalue at the soft edge. In the Appendix we critique a
recent [20] claimed large deviation-type asymptotic expansion for the smallest eigenvalue PDF
in the Laguerre β-ensemble in the case that a is fixed.
3
2 Previous results and strategy
2.1 Previous results
For the Laguerre β-ensemble with a = αN+O(1) the leading support of the eigenvalue density,
upon scaling the eigenvalues λ Nλ , is the interval (a2,a2) with
l 7→ l 1 2
a = √α+1 1, a = √α+1+1. (2.1)
1 2
−
Moreover, the corresponding leading eigenvalue density is given by the so-called Marˇcenko-
Pastur law
N
Nρ (Nx) (a2 x)(x a2), a2 < x < a2 (2.2)
(1) ∼ 2πx 2 − − 1 1 2
q
(see e.g. [13, eq. (3.61) with x2 x]). Thus for α > 0 the smallest eigenvalue is bounded away
7→
from the origin with a density profile that vanishes like a square root (a soft edge), while for
α = 0 the smallest eigenvalue is part of an accumulation of eigenvalues about the origin with
a density profile diverging like a reciprocal of a square root (a hard edge).
A similar effect is exhibited by the Jacobi β-ensemble (1.3) with a = α N + O(1),a =
1 1 2
α N +O(1), except now there is no need to scale the eigenvalues. Thus the leading eigenvalue
2
support is (c ,c ) [0,1] with
1 2
⊂
2(α2 α2)
2(c +c 1) = 1 − 2 (2.3)
1 2 − (α +α +2)2
1 2
2(α2 +α2)
(2c 1)(2c 1) = 1 2 1. (2.4)
1 − 2 − (α +α +2)2 −
1 2
Notice that (2.3) and (2.4) are unchanged by
(c ,c ,α ,α ) (1 c ,1 c ,α ,α ). (2.5)
1 2 1 2 1 2 2 1
7→ − −
The corresponding eigenvalue density is given by
2+α +α (x c )(c x)
1 2 1 2
ρ (x) N − − (2.6)
(1)
∼ 2π x(1 x)
p −
(see e.g. [13, eq. (3.77) with y 2y 1]).
7→ −
We are interested in the eigenvalue density outside of [a2,a2] for the Laguerre β-ensemble,
1 2
and outside of [c ,c ] for the Jacobi β-ensemble. In the former case there are some prior results
1 2
from the existing literature, which we will now summarize. But before doing so, we make note
of the work [23]. This relates to the case α = 0 in the Laguerre ensemble, and so strictly
speaking is not relevant to our setting of extensive exponents. However, the leading term of the
asymptotic expansion of the density does not contain any information on the exponent a for a
of order unity, and so is expected to correspond to the case α = 0. Actually the concern of [23]
is not the density in the exponentially small region as such, but the large deviation asymptotic
form of the distribution of the largest eigenvalue pL (s). However, it is a simple result that for
N,β
x in this regime, the two are identical [13, eq. (14.136)],
pL (Nx) ρL (Nx). (2.7)
N,β ∼ (1)
4
The leading large N form of (2.7) for x > 4 in the case α = 0 as calculated in [23] is given by
setting α = 0 in (2.8) below.
There are at least three existing works containing explicit asymptotic formulas for ρL (Nx)
(1)
(or equivalently, according to (2.7), pL (Nx)) outside of [a2,a2] for general α. In order of
N,β 1 2
publication date the first [13, eq. (4.83)] is the leading large N form
ρL (Nx)
(1)
βN α(α+uL x) 2x uL +x 2 α
= exp uL +αlog − − +(2+α)log − − +o(N) ,
2 − 2(α+1)1/2x 2(α+1)1/2
(cid:26) (cid:18) (cid:12) (cid:12) (cid:12) (cid:12)(cid:19) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12) (2.8)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where
(x a2)(x a2), x (a2, )
uL := − 1 − 2 ∈ 2 ∞ (2.9)
 q (x a2)(x a2), x (0,a2).
 − − 1 − 2 ∈ 1
q
The second [20] is presented as the leading asymptotic form for pL (Nx) in the region x
N,β ∈
[0,a2]. In terms of our notation the result of [20] reads
1
ρL (Nx) = exp βNΦmin (1 √α+1)2 x +o(N) , (2.10)
(1) − − − −
(cid:16) (cid:16) (cid:17) (cid:17)
where
1 α x
Φmin(x) = x x+4√α+1 log 1
− −2 − 2 − (1 √α+1)2
r (cid:16) (cid:17) (cid:18) − (cid:19)
x+4√α+1 √x
+2log −
p 4√α+1 !
p2√x x+4√α+1 √x
+αlog 1+ − . (2.11)
√α+1 1 p 4√α+1 !!
−
Elementary algebraic manipulation involving the simple identities
α(α+uL x) 2x uL+x 2 α
αlog − − +αlog − −
2(α+1)1/2 2(α+1)1/2x
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) α+uL +(cid:12)x (cid:12) α uL +(cid:12)x
(cid:12)= 2αlog (cid:12) = 2(cid:12)αlog − (cid:12) (2.12)
(cid:12)2√α+1√x(cid:12) − (cid:12)2√α+1√x(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
shows that (2.8) and (2.10) are in(cid:12)fact identical(cid:12)expressions.(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
The third and most recent result [1] was derived for the case β = 2 only, but in addition
to the leading exponential term also gives the algebraic correction. Thus, after making the
correspondences λ = α,ξ = a2/(α + 1),ξ = a2/(α + 1),x x/(α + 1) we read off from [1]
0 1 1 2 7→
that for exponent a = αN
(α+1)1/2 1
ρL Nx e−Nκ(x), (2.13)
(1) ∼ 2πN (x a2)(x a2)
− 1 − 2
(cid:0) (cid:1)
5
where
κ(x) = uL +a a arcosh x1 − 21 a121 + a122 a21 +a22 arcosh a21+2a22 −x . (2.14)
− 1 2 (cid:12)(cid:12) 1 1(cid:16) 1 (cid:17)(cid:12)(cid:12)− 2 (cid:12) a21−a22 (cid:12)
(cid:12) 2 a21 − a22 (cid:12) (cid:12)(cid:12) 2 (cid:12)(cid:12)
(cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
Recalling that for x > 1 (cid:12) (cid:12) (cid:12) (cid:12)
arcoshx = log(x+√x2 1)
−
shows that the leading (exponential) term of (2.13) coincides with the β = 2 case of (2.8).
2.2 Strategy
We seek to extend (2.8) to the same order as the β = 2 result (2.13), and thus give the
asymptotic form of ρL (Nx) for x [0,a2) (a2, ) up to terms o(1) as N . We seek
(1) ∈ 1 ∪ 2 ∞ → ∞
too an analogous expansion in the Jacobi case. Our method relies on the fact that the density
can be written in terms of the β-moment of the corresponding characteristic polynomial, and
as such is an application of the strategy used recently in the case of the Gaussian β-ensemble,
and the Laguerre β-ensemble with a fixed [14].
We have referred to the PDFs (1.1) and (1.3) by ME (w) with w(λ) = λaβ/2e−βλ/2 and
β,N
w(λ) = λa1β/2(1 λ)a2β/2 respectively. Let us denote the corresponding normalizations by
−
C [w]. Then it follows from the definitions (see e.g. [14, eq. (2.11)]) that
β,N
N
(N +1)C [w]
ρ (x) = β,N w(x) x λ β . (2.15)
(1),N+1 l
C [w] | − |
β,N+1 * +
Yl=1 MEβ,N(w)
The key observation is that the average in (2.15) can be interpreted as a particular case of the
characteristic function for the linear statistic V(x) = N log x λ . But for a large class of
l=1 | − l|
weights w the latter is expected to be a Gaussian (see e.g. [13, 14.4]), and this implies, with
P §
x outside the interval of support,
N
x λ β eβµN(v)e(βσ(v))2/2, (2.16)
l
| − | ∼
* +
Yl=1 MEβ,N(w)
where, with v(t) := log x t and (d ,d ) the limiting interval of support,
1 2
| − |
d2
µ (v) = ρ (t)v(t)dt (2.17)
N (1),N
Zd1
1 d2 v(t ) d2 v′(t ) (d t )(t d ) 1/2
(σ(v))2 = dt 1 dt 2 2 − 2 2 − 1
βπ2 1 1/2 2 t t
Zd1 (d2 t1)(t1 d1) Zd1 (cid:0) 2 − 1 (cid:1)
− −
1 ∞ 2 π d +d d d
= ka2, (cid:0) a = v (cid:1) 1 2 + 2 − 1 cosθ coskθ dθ. (2.18)
2β k k π 2 2
k=1 Z0 (cid:18) (cid:19)
X
6
Theorems relating to the asymptotic form of the characteristic function for linear statistics
in random matrix theory were first proved in [17]. However, the technical assumptions therein
did not permit the linear statistic corresponding to v(t) = log x t due to its singularity
| − |
at x = t. But for x outside the eigenvalue support (2.18) is well defined and on physical
grounds it is expected that (2.16) will still apply [13, 14.4]. Fortunately, very recently [5],
§
under the conditions that the eigenvalue support is a single interval, and with both endpoints
soft edges (or a hard edge with parameter a = 0) the validity of (2.16) for any v analytic in the
neighbourhood of [d ,d ] has been rigorously established. In our setting of the Laguerre and
1 2
Jacobi ensembles with extensive parameters both end points of the eigenvalue support are soft
edges, so (2.16) thus follows from [5].
Our immediate task then is to compute the explicit form of (2.17) and (2.18). We will
perform this task in the next section. Substituting (2.16) in (2.15) then gives the sought
asymptotic expansions of the densities.
At a technical level, with the parameters being extensive, and the occurrence of both N +1
and N in (2.15), there is some advantage in first manipulating (2.15) before applying (2.16).
Thus in the Laguerre case the weight required in (2.15) is
w(x) = xα(N+1)β/2+β/2−1e−(N+1)βx/2. (2.19)
With this w we observe
N N
C [w˜]
x λ β = β,N λαβ/2e−βλl/2 x λ β , (2.20)
| − l| C [w] l | − l|
* + β,N * +
Yl=1 MEβ,N(w) Yl=1 MEβ,N(w˜)
where w˜(x) = xαNβ/2+β/2−1e−Nβx/2, and we observe too that
C [w˜] 1
β,N
= . (2.21)
Cβ,N[w] h Nl=1λαlβ/2e−βλl/2iMEβ,N(w˜)
Similarly, in the Jacobi case, the weiQght required in (2.15) is
w(x) = xα1(N+1)β/2+β/2−1(1 x)α2(N+1)β/2+β/2−1 (2.22)
−
and we have
N N
C [w˜]
x λ β = β,N λα1β/2(1 λ )α2β/2 x λ β , (2.23)
| − l| C [w] l − l | − l|
* + β,N * +
Yl=1 MEβ,N(w) Yl=1 MEβ,N(w˜)
where w˜(x) = xα1Nβ/2+β/2−1(1 x)α2Nβ/2+β/2−1, together with
−
C [w˜] 1
β,N
= . (2.24)
Cβ,N[w] h Nl=1λlα1β/2(1−λl)α2β/2iMEβ,N(w˜)
Ourimmediate taskthenistocomQputetheexplicit formof(2.17)and(2.18)fortheaverages
on the RHS’s of (2.20) and (2.23) with
log x t + α logt 1t, Laguerre
v(t) = | − | 2 − 2 (2.25)
log x t + α1 logt+ α2 log 1 t , Jacobi,
(cid:26) | − | 2 2 | − |
7
and with
α logt 1t, Laguerre
v(t) = 2 − 2 (2.26)
α1 logt+ α2 log 1 t , Jacobi.
(cid:26) 2 2 | − |
For future purposes, especially in labellings, it will be convenient to refer to (2.25) and (2.26)
as choices (1) and (2) respectively. We will proceed with required computations in the next
section.
3 The asymptotic forms
3.1 The Laguerre β-ensemble
For the weight w˜ in (2.20) it is known from [11] that
N
ρ (t) = (a2 t)(t a2)
(1),N 2πt 2 − − 1
q
1 1 1 1 1 1 1
+ δ(t a2)+ δ(t a2) +O . (3.1)
β − 2 2 − 1 2 − 2 − π (t a2)(a2 t)! N
(cid:18) (cid:19) − 1 2 − (cid:16) (cid:17)
We remark that the precise form of the O(1) term in (3.1p) relies crucially on the O(1) portion
of the exponent in w˜ equalling β/2 1. Note that the latter is precisely the O(1) portion of
−
the exponent appearing in (1.1) in relation to covariance matrices. We remark too that the
expansion (3.1) ignores possible oscillatory terms which appear at next order [9], as these do
not contribute to the next correction in (2.17).
As is consistent with the labelling noted below (2.26), let (2.17) in the case ρ (t) is
(1),N
L,(1)
given by (3.1), and v(t) by the Laguerre cases of (2.25) and (2.26), be denoted µ and
N
L,(2) L,(1) L,(2)
µ respectively. Then µ µ is equal to (2.17) with ρ (t) given by (3.1) and
N N − N (1),N
v(t) = log x t . It can be evaluated by making use of the following integral evaluations.
| − |
Proposition 3.1 Let uL be given by (2.9). For x (0,a2) (a2, ) we have
∈ 1 ∪ 2 ∞
a22 1
(t a2)(a2 t) log x t dt =
2tπ − 1 2 − | − |
Za21 q
1 α(α+uL x) 2x uL +x 2 α
x α uL 2+αlog − − +(2+α)log − − (3.2)
2 − − − 2x2 2
(cid:18) (cid:12) (cid:12) (cid:12) (cid:12)(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
and (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1 a22 log x t a2 a2 1
| − | dt = log 2 − 1 +log x˜ √x˜2 1
π Za21 (a22 −t)(t−a21) 2 2 (cid:12) ± − (cid:12)
uL +x 2 α(cid:12) (cid:12)
p = log − − (cid:12) , (cid:12) (3.3)
2
(cid:12) (cid:12)
(cid:12) (cid:12)
where in the first equality of (3.3) we take + ( )(cid:12)according to x (cid:12) (a2, ) x (0,a2) and
− (cid:12) ∈(cid:12) 2 ∞ ∈ 1
2 a2 +a2 (cid:0) (cid:1)
x˜ = x 1 2 . (3.4)
a2 a2 − 2
2 − 1 (cid:18) (cid:19)
8
Proof: The result (3.2) is derived in [13, Exercises 14.4 q.6(i)] for Re(x) > a2; its value for
2
x (0,a2) follows by analytic continuation. The first equality in (3.3) is a simple corollary of
∈ 1
the integral evaluation
1 1 log x t 1
| − |dt = log (x+√x2 1) , x > 1,
π √1 t2 2 −
Z−1 − (cid:18) (cid:19)
the derivation of which can be found, for example, in [14, eq. (3.3)]. The second equality in
(3.3) now follows by noting
a2 a2
2 − 1√x˜2 1 = uL. (3.5)
± 2 −
(cid:3)
Proposition 3.1 implies that (2.17) in the case ρ (t) is given by (3.1) and v(t) by the
(1),N
Laguerre case of (2.25) has the large N evaluation
N α(α+uL x) 2x
µL,(1) µL,(2) = x α uL 2+αlog − −
N − N 2 − − − 2x2
(cid:18) (cid:12) (cid:12)
uL +x 2 α 1(cid:12) 1 (cid:12) uL +x 2 α 1
+(2+α)log − − + (cid:12) log uL lo(cid:12)g − − +O .
(cid:12) (cid:12)
2 β − 2 | |− 2 N
(cid:12) (cid:12)(cid:19) (cid:18) (cid:19)(cid:18) (cid:12) (cid:12)(cid:19)
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12) (3.6)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
It remains to evaluate (2.18) with v(t) given by the Laguerre cases of (2.25) and (2.26).
Consider the former case first. Making use of the second equality in (2.18) shows we must
evaluate
(1) (2) (3)
a = a +a +a , k = 1,2,... (3.7)
k k k k
where
2 π cosθ
(1)
a = log 1 coskθ dθ, (3.8)
k π − x˜
Z0 (cid:18) (cid:19)
with x˜ given by (3.4), and
α
(2) (1)
a = a , (3.9)
k 2 k
x=0
a2 (cid:12) a2 π
a(3) = 2 −(cid:12)(cid:12) 1 cosθcoskθ dθ. (3.10)
k − 2π
Z0
From [14, Lemma 2] we know that
cosθ
log 1 = log(1 ν eiθ)(1 ν e−iθ) log(1+ν2), (3.11)
− x˜ − x − x − x
(cid:18) (cid:19)
where ν is such that ν < 1 and has the explicit value
x x
| |
x˜ (x˜2 1)1/2, x (a2, ),
ν = − − ∈ 2 ∞ (3.12)
x x˜+(x˜2 1)1/2, x (0,a2).
(cid:26) − ∈ 1
9
We see immediately from (3.11) that
2νk
a(1) = x. (3.13)
k − k
According to (3.9), we see from this, (3.12), (3.4) and (2.1) that
α 1
a(2) = νk, ν = . (3.14)
k −k 0 0 −√α+1
And in relation to (3.10), an elementary calculation shows
√α+1, k = 1,
(3)
a = − (3.15)
k 0, otherwise.
(cid:26)
Substituting these explicit forms in (3.7) we see
∞ ∞
1
ka2 =α+1+2√α+1(2ν +αν )+ 2νk +ανk 2
k x 0 k x 0
k=1 k=1
X X (cid:0) (cid:1)
= α+1+2(x (α+2) uL) 4log(1 ν2)
− − − − − x
4αlog(1 ν ν ) α2log(1 ν2). (3.16)
− − x 0 − − 0
With v(t) given by the Laguerre case of (2.26), in the notation of (3.7) we have a =
k
(2) (3)
a +a . The first line of (3.16) then remains valid but with ν = 0, so in this case we have
k k x
∞
ka2 = α+1 α2log(1 ν2). (3.17)
k − − − 0
k=1
X
Recalling now (2.18) and using a superscript notation analogous to that used in (3.6) we
read off from (3.16) and (3.17) that
1
(σ2)L,(1) (σ2)L,(2) = 2(x (α+2) uL) 4log(1 ν2) 4αlog(1 ν ν ) . (3.18)
− 2β − − − − x − − x 0
(cid:16) (cid:17)
Use of (3.12) and (3.14) allows us to compute that
x (α+2)+uL
log(1 ν2) = log uL log | − |,
− x | |− 2
x+α uL
log(1 ν ν ) = log | − |. (3.19)
x 0
− 2(α+1)
Substituting in (3.18) then gives the explicit evaluation of the variance
1 2 2 x (α+2)+uL
(σ2)L,(1) (σ2)L,(2) = (x (α+2) uL) log uL + log | − |
− β − − − β | | β 2
2α x+α uL
log | − |. (3.20)
− β 2(α+1)
10