Trace-class Monte Carlo Markov Chains for Bayesian Multivariate Linear Regression with Non-Gaussian Errors

0.16 MB·

by  Qian Qin

Trace-class Monte Carlo Markov Chains for Bayesian
Multivariate Linear Regression with Non-Gaussian Errors
6
1 QianQinand James P. Hobert
0
DepartmentofStatistics
2
n UniversityofFlorida
u
J
January 2016

1

]
T
S
Abstract
.
h
t
a Letπ denotetheintractableposteriordensitythatresultswhenthelikelihoodfromamul-
m
tivariatelinearregressionmodelwitherrorsfromascalemixtureofnormalsiscombinedwith
[ the standard non-informative prior. There is a simple data augmentation algorithm (based

2 on latent data from the mixing density) that can be used to explore π. Let h(·) and d de-
v
notethemixingdensityandthedimensionoftheregressionmodel,respectively. Hobertetal.
6
3 (2016) have recently shown that, if h convergesto 0 at the origin at an appropriate rate, and
1 ∞ d
0 0 u2 h(u)du < ∞, then the Markov chains underlying the DA algorithm and an alterna-
0 RtiveHaarPX-DAalgorithmarebothgeometricallyergodic. Infact,somethingmuchstronger
2. than geometric ergodicity often holds. Indeed, it is shown in this paper that, under simple
0 conditions on h, the Markov operators defined by the DA and Haar PX-DA Markov chains
6
are trace-class, i.e., compact with summable eigenvalues. Many of the mixing densities that
1
: satisfy Hobertetal.’s (2016) conditions also satisfy the new conditions developedin this pa-
v
i per. Thus, for this set of mixing densities, the new results provide a substantial strength-
X
ening of Hobertetal.’s (2016) conclusion without any additional assumptions. For example,
r
a Hobertetal.(2016)showedthattheDAandHaarPX-DAMarkovchainsaregeometricallyer-
godicwheneverthemixingdensityisgeneralizedinverseGaussian,log-normal,Fre´chet(with
shapeparameterlargerthand/2), orinvertedgamma(withshape parameterlargerthand/2).
The results in this papershow that, in each of these cases, the DA and Haar PX-DA Markov
operatorsare,infact,trace-class.
Key words and phrases. Compact operator, Data augmentation algorithm, Haar PX-DA algorithm, Heavy-tailed
distribution,Scalemixture,Markovoperator,Trace-classoperator
1
1 Introduction
Considerthemultivariatelinearregression model
1
Y = Xβ +εΣ2 , (1)
where Y denotes an n×d matrix of responses, X is an n×p matrix of known covariates, β is a
1
p×dmatrix ofunknown regression coefficients, Σ2 isan unknown positive-definite scale matrix,
and ε is an n×d matrix whose rows are iid random vectors from a scale mixture of multivariate
normaldensities. Inparticular, lettingεT denotetheithrowofε,weassumethatε hasdensity
i i
∞ d
f (ε) = u2 exp − uεTε h(u)du ,
h d 2
Z0 (2π)2 n o
where h : (0,∞) → [0,∞) is the so-called mixing density. Error densities of this form are often
usedwhenheavy-tailederrorsarerequired. Forexample,itiswellknownthatifhisaGamma(ν, ν)
2 2
density (with mean 1), then f becomes the multivariate Student’s t density with ν degrees of
h
freedom.
A Bayesian analysis of the data from this regression model requires a prior on (β,Σ). We
consideranimproperdefaultpriorthattakestheformω(β,Σ) ∝ |Σ|−aISd(Σ)whereSd ⊂ Rd(d2+1)
denotesthespaceofd×dpositivedefinitematrices. Takinga = (d+1)/2yieldstheindependence
Jeffreysprior, whichisthestandard non-informative priorformultivariate location scaleproblems.
Of course, whenever an improper prior is used, one must check that the corresponding posterior
distribution is proper. Letting y denote the observed value of Y, the joint density of the data from
model(1)canbeexpressed as
n ∞ d
f(y|β,Σ)= u2 exp − u y −βTx TΣ−1 y −βTx h(u)du .
Yi=1"Z0 (2π)d2|Σ|21 (cid:26) 2(cid:16) i i(cid:17) (cid:16) i i(cid:17)(cid:27) #
Define
m(y) = f(y|β,Σ)ω(β,Σ)dβdΣ .
S Rp×d
Z dZ
The posterior distribution is proper precisely when m(y) < ∞. Let Λ stand for the n×(p +d)
matrix (X : y). Straightforward arguments (using ideas from Ferna´ndezandSteel (1999)) show
that,together, thefollowingfourconditions aresufficientforposterior propriety:
(S1) rank(Λ) = p+d;
(S2) n > p+2d−2a ;
∞ d
(S3) u2 h(u)du < ∞ ;
0
R∞ −n−p+2a−2d−1
(S4) u 2 h(u)du < ∞ .
0
R
2
Thesefourconditions areassumedtoholdthroughout thispaper.
Remark1. Conditions(S1)&(S2)areknowntobenecessaryforposteriorpropriety(Ferna´ndez andSteel,
1999;Hobertetal.,2016).
Remark 2. Condition (S3) clearly concerns the tail behavior ofh. Similarly, condition (S4)con-
cernsthebehaviorofhneartheorigin,unlessn−p+2a−2d−1isnegative,whichispossible. Note,
however,that(S2)impliesthat−(n−p+2a−2d−1)/2 < 1/2. Consequently,ifn−p+2a−2d−1
isnegative, then(S4)isimpliedby(S3).
Ofcourse,theposterior density of(β,Σ)giventhedatatakestheform
f(y|β,Σ)ω(β,Σ)
π(β,Σ|y) = .
m(y)
Thereisawell-knowndataaugmentation(DA)algorithmthatcanbeusedtoexplorethisintractable
density (Liu, 1996). Hobertetal. (2016) (hereafter HJK&Q)performed convergence rate analyses
of the Markov chains underlying this DA algorithm and an alternative Haar PX-DA algorithm. In
this paper, we provide a substantial improvement of HJK&Q’s main result. A formal statement
of the DA algorithm requires some buildup. Let z = (z ,...,z ) have strictly positive elements,
1 n
and let Q = Q(z) be the n×n diagonal matrix whose ith diagonal element is z−1. Also, define
i
Ω = (XTQ−1X)−1 andµ = (XTQ−1X)−1XTQ−1y. Foreachs ≥ 0,defineaunivariate density
asfollows
d −su
ψ(u;s) = b(s)u2 e 2 h(u) , (2)
where b(s) is the normalizing constant. The DA algorithm uses draws from the inverse Wishart
(IW ) and matrix normal (N ) distributions. These densities are defined in the Appendix. If
d p,d
the current state of the DA Markov chain is (β ,Σ ) = (β,Σ), then we simulate the new state,
m m
(β ,Σ ),usingthefollowingthree-step procedure.
m+1 m+1
Iteration m+1oftheDAalgorithm:
1. Draw {Z }n independently with Z ∼ ψ · ; βTx −y TΣ−1 βTx −y , and call the
i i=1 i i i i i
resultz =(z1,...,zn). (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
2. Draw
−1
Σ ∼ IW n−p+2a−d−1, yTQ−1y−µTΩ−1µ .
m+1 d
(cid:18) (cid:19)
(cid:16) (cid:17)
3. Drawβ ∼ N µ,Ω,Σ
m+1 p,d m+1
(cid:0) (cid:1)
3
DenotetheDAMarkovchainbyΦ = {(β ,Σ )}∞ ,anditsstatespacebyX := Rp×d×S . For
m m m=0 d
positiveintegerm,letkm :X×X → (0,∞)denotethem-stepMarkovtransition density(Mtd)of
Φ,sothatifAisameasurable setinX,
P (β ,Σ )∈ A (β ,Σ )= (β,Σ) = km (β′,Σ′) (β,Σ) dβ′dΣ′ .
m m 0 0
(cid:16) (cid:12) (cid:17) ZA (cid:0) (cid:12) (cid:1)
The1-stepMtd,k ≡ k1,canb(cid:12)eexpressed as (cid:12)
′ ′ ′ ′ ′
k (β ,Σ) (β,Σ) = π(β |Σ,z,y)π(Σ |z,y)π(z|β,Σ,y)dz ,
Rn
Z +
(cid:0) (cid:12) (cid:1)
where the precise forms o(cid:12)f the conditional densities π(z|β,Σ,y), π(Σ|z,y), and π(β|Σ,z,y) can
begleanedfromsteps1.,2.,and3. oftheDAalgorithm,respectively. IfthereexistM : X→ [0,∞)
andλ ∈ [0,1)suchthat,forallm,
km β,Σ β˜,Σ˜ −π(β,Σ y) dβdΣ ≤ M(β˜,Σ˜)λm ,
S Rp×d
Z dZ (cid:12) (cid:12)
(cid:12) (cid:0) (cid:12) (cid:1) (cid:12) (cid:12)
then the chain Φ is geomet(cid:12)rically erg(cid:12)odic. The bene(cid:12)fits(cid:12)of using a geometrically ergodic Monte
Carlo Markov chain have been well documented (see, e.g. Flegaletal., 2008; JonesandHobert,
2001; RobertsandRosenthal, 1998). HJK&Q showed that, if h converges to zero at the origin at
an appropriate rate, then Φ is geometrically ergodic. In order to state HJK&Q’s result, we must
introduce three classes of mixing densities. Leth : (0,∞) → [0,∞) be a mixing density. Ifthere
is an η > 0 such that h(u) = 0 for all u ∈ (0,η), then wesay that h is zero near the origin. Now
assumethathisstrictlypositiveinaneighborhood of0. Ifthereexistsac> −1suchthat
h(u)
lim ∈ (0,∞) ,
u→0 uc
then we say that h is polynomial near the origin with power c. Finally, if for every c > 0, there
exists anη > 0such thatthe ratio h(u) isstrictly increasing in(0,η ), then wesay thathisfaster
c uc c
than polynomial near the origin. HJK&Q showed that every mixing density that is a member of a
standard parametricfamilyisinoneofthesethreeclasses, andtheyprovedthefollowingresult.
Theorem 1 (HJK&Q). If the mixing density, h, is zero near the origin, or faster than polynomial
near the origin, or polynomial near the origin with power c > n−p+2a−d−1, then the DA Markov
2
chainisgeometrically ergodic.
Remark3. Itisnotnecessarytocheckthat(S4)holdsbeforeapplyingTheorem1because,together
with(S3),thehypothesis ofTheorem1impliesthat(S4)issatisfied.
Inthispaper,weshowthatsomethingmuchstronger thangeometricergodicityoftenholds. We
begin withsome requisite background material onMarkov operators. Theposterior density can be
usedtodefineaninnerproduct
hf ,f i= f (β,Σ)f (β,Σ)π(β,Σ|y)dβdΣ ,
1 2 1 2
X
Z
4
andnormkfk= hf,fiontheHilbertspace
p
L2 = f :X → R: f2(β,Σ)π(β,Σ|y)dβdΣ < ∞ and f(β,Σ)π(β,Σ|y)dβdΣ = 0 .
0
X X
(cid:26) Z Z (cid:27)
NowdefinetheDAMarkovoperatorK : L2 → L2 asthatwhichtakesf ∈ L2 into
0 0 0
′ ′ ′ ′ ′ ′
(Kf)(β,Σ) = f(β ,Σ)k (β ,Σ) (β,Σ) dβ dΣ .
X
Z
(cid:0) (cid:12) (cid:1)
Because K isbased onaDAalgorithm, itisself-adjoint and(cid:12) positive (Liuetal.,1994). If, inaddi-
tion,K isalsoacompactoperator,thenK hasapureeigenvalue spectrum,allofitseigenvalues re-
sidein[0,1), andthecorresponding Markovchainisgeometrically ergodic (see,e.g.,Hobertetal.,
2011;MiraandGeyer,1999). WenotethatthesetofMonteCarloMarkovchains whoseoperators
are compact is a small subset of those that are geometrically ergodic (see, e.g., ChanandGeyer,
1994, p. 1755). Taking this a step further, K is said to be trace-class if it is compact and its
eigenvalues are summable (see, e.g. Conway, 1990, p. 267). In this paper, we provide sufficient
conditions (on h) for K to be trace-class. The benefits of using trace-class Markov operators are
spelledoutinKhareandHobert(2011),andweexploittheirresults inSection4.
A statement of our main result requires substantial build-up, so here in the Introduction we
present only one simple, but powerful, corollary. Let R denote the set (0,∞), and define a para-
+
metricfamilyoffunctions g : R → [0,∞)asfollows. Forρ ∈R andτ ∈ R,let
ρ,τ + +
g (u) = exp −ρ(logu)2+τ logu .
ρ,τ
(cid:8) (cid:9)
Thefollowingresultisacorollary ofTheorem2inSection2.
Corollary 1. Let h be a mixing density. If there exist ρ ∈ R , τ ∈ R and η > 0 such that
+
h(u)/g (u)isnon-decreasing in(0,η),thentheDAMarkovoperator, K,istrace-class.
ρ,τ
An immediate consequence of Corollary 1 is that, if h is zero near the origin, then K is trace-
class. Indeed, for any (ρ,τ) ∈ R ×R, h(u)/g (u) is constant (and equal to zero) in a neigh-
+ ρ,τ
borhood of the origin. Corollary 1 also implies that if h is a member of one of the standard
parametric families that are faster than polynomial near the origin (inverted gamma, log-normal,
generalized inverse Gaussian, andFre´chet), then the corresponding Markov operator istrace-class.
For example, consider the case where the mixing density is inverted gamma. In particular, let
h(u) = bu−α−1e−γ/uIR (u),whereα > d/2, γ > 0andb = b(α,γ)isthenormalizing constant.
+
(Werequireα > d/2sothatcondition(S3)issatisfied.) Takingρ = 1andτ = −(α+1),wehave
d h(u) d γ b γ γ
= b exp − +(logu)2 = +2logu exp − +(logu)2 ,
dug (u) du u u u u
ρ,τ
n o h i n o
which is clearly positive in a neighborhood of 0. Hence, Corollary 1 implies that K is trace-class.
(This result was established by JungandHobert (2014) in the special case where d = 1.) Similar
5
arguments can be used for the other three families (log-normal, generalized inverse Gaussian and
Fre´chet), and these are given in Section 2. Indeed, for a large class of mixing densities (including
the ones just mentioned) our results provide a substantial strengthening of Hobertetal.’s (2016)
conclusion without any additional assumptions. On the other hand, as we now explain, there are
still many mixing densities that satisfy the hypotheses of Theorem 1, but to which our results are
notapplicable.
Thefollowinglemma,whichisproveninSection2,provides asufficient condition forK tobe
trace-class, andisoneofthekeypiecesoftheproofofTheorem2(andhenceofCorollary1).
Lemma 2. Let h be a mixing density that is strictly positive in a neighborhood of the origin. If
thereexistζ ∈ (1,2)andη > 0suchthat
η ud2h(u)
du < ∞ , (3)
ζu d
Z0 0 v2h(v)dv
thenK istrace-class. R
Notethat(3)cannotholdif,foreachη >0,
η ud2h(u)
du = ∞ . (4)
2u d
Z0 0 v2h(v)dv
Consequently, when h isstrictly positivRe in aneighborhood of the origin and (4)holds, our results
cannot be applied to h. For example, assume that h is polynomial near the origin with power c so
that h(u) → l ∈ R as u → 0. Then for all u is some small neighborhood of the origin, we have
uc +
l/2 < h(u) < 2l. So,ifη > 0issmallenough, thenforallu∈ (0,η),
uc
ud2h(u) ≥ ud22luc = b ,
2uvd2h(v)dv 2uvd22lvcdv u
0 0
where b = b(d,l) is a positivRe constant. Thus, sRince η 1 du diverges for every η > 0, (4) holds.
0 u
Consequently, ourresultsarenotapplicabletomixingdensitiesthatarepolynomial neartheorigin.
R
Furthermore, in Section 2, we give an example of a mixing density that is faster than polynomial
neartheorigin, butforwhich(4)holds.
The remainder of this paper is organized as follows. The main result is stated and proven in
Section 2. In Section 3, weexamine the consequences of the main result when the mixing density
is faster than polynomial near the origin. In Section 4, we show that Theorem 2 has important
implicationsforaHaarPX-DAvariantoftheDAalgorithmthatwasintroducedbyRoyandHobert
(2010)andextendedbyHJK&Q.Finally,theAppendixcontainsthedefinitionsoftheIW andN
d p,d
families,aswellassometechnical details.
6
2 Main Result
In this section, we will be dealing with functions g : R → [0,∞) that are strictly positive and
+
differentiable inaneighborhood oftheorigin. LetAdenote thesetofallsuch functions, andletK
denotethesubsetofAconsisting offunctions whosereciprocals areintegrable neartheorigin,i.e.,
η 1
K = κ ∈ A: du < ∞ forsomeη > 0 .
κ(u)
n Z0 o
Thefunction κ(u) = u(logu)2 isamemberofK, andwewillusethis factinthesequel. Now,for
fixedκ ∈ K andfixedζ ∈ (1,2), letC(κ,ζ)denote thesubset ofAcontaining thefunctions g that
satisfythefollowingthreeconditions:
d
1. u2g(u)isbounded inaneighborhood oftheorigin,
d
2. limu→0κ(u)u2g(u) = 0,
3. Thereexistl ,l ∈ Rsuchthat
1 2
′
dκ(u) g(u) κ(u)g (u)
′
lim κ(u)+ = l and lim = l . (5)
1 2
u→0 2 u g(ζu) u→0 g(ζu)
(cid:16) (cid:17)
ThefollowingresultisprovenintheAppendix.
Proposition1. Fix(ρ,τ) ∈ R ×R. Theng ∈C(κ,3/2), withκ(u) = u(logu)2. Furthermore,
+ ρ,τ
d
u2gρ,τ(u)isnon-decreasing inaneighborhood oftheorigin.
Hereisourmainresult.
Theorem 2. Let h be a mixing density. Each of the following three conditions is sufficient for the
corresponding DAMarkovoperator, K,tobetrace-class.
1. Themixingdensityhiszeroneartheorigin.
2. Thereexistκ ∈ K,ζ ∈(1,2)andg ∈ C(κ,ζ)suchthatlimu→0 hg((uu)) ∈ R+.
3. There exist κ ∈ K, ζ ∈ (1,2) and g ∈ C(κ,ζ) such that both ud2g(u) and h(u) are non-
g(u)
decreasing inaneighborhood oftheorigin.
Remark 4. Suppose that h ∈ C(κ,ζ). Then, by taking g = h, the second condition of Theorem 2
is satisfied, so K is trace-class. However, this argument requires that h be differentiable in a
neighborhoodoftheorigin. Thesurrogatefunction,g,allowsustohandlenon-differentiable mixing
densities.
Remark5. NotethatCorollary1(fromtheIntroduction) followsimmediatelyfromTheorem2and
Proposition 1.
7
OurproofofTheorem2isbasedonthreelemmas,whichwenowstateandprove.
Lemma 1. Let h be a mixing density, and let ψ(u;s) be as in (2). Suppose there exist ζ < 2 and
ν :R → [0,∞)with ν(u)du < ∞suchthat
+ R
+
R
(ζ −1)us
ψ(u;s) ≤ exp ν(u) (6)
2
(cid:26) (cid:27)
forallu∈ R andalls ∈ [0,∞). ThenK istrace-class.
+
Proof. For i = 1,2,...,n, define r = r (β,Σ) = βTx − y TΣ−1 βTx − y . Of course,
i i i i i i
r ≥ 0. First,itsufficestoshowthat
i (cid:0) (cid:1) (cid:0) (cid:1)
k (β,Σ) (β,Σ) dβdΣ < ∞ ,
S Rp×d
Z dZ
(cid:0) (cid:12) (cid:1)
(cid:12)
(see,e.g.,KhareandHobert,2011). Routinecalculations showthat
π(β,Σ z,y)π(z β,Σ,y) = |Σ|−n+22a exp − 12 ni=1rizi n ψ(z ;r )
(cid:12) (cid:12) Sd Rp×d|Σ|−n+22a exp(cid:8) − 1P2 ni=1riz(cid:9)i dβdΣ Yi=1 i i
(cid:12) (cid:12) ≤ R R |Σ|−n+22a exp −(cid:8)(2−2ζ)Pni=1rizi(cid:9) n ν(z ).
Sd Rp×d|Σ|−n+22a e(cid:8)xp − 12 Pni=1rizi(cid:9)dβdΣ Yi=1 i
′ R R (cid:8) P (cid:9)
Thetransformation Σ = Σ/(2−ζ),yields
n
(2−ζ)
−n+2a
|Σ| 2 exp − rizi dβdΣ
S Rp×d 2
Z dZ n Xi=1 o
n
1 1
−n+2a
= (2−ζ)(n+2a−2d−1)d ZSdZRp×d|Σ| 2 expn− 2Xi=1riziodβdΣ .
Itfollowsthat,
n
1
π(β,Σ z,y)π(z β,Σ,y)dβdΣ ≤ ν(z ) .
(n+2a−d−1)d i
ZSdZRp×d (2−ζ) 2 i=1
(cid:12) (cid:12) Y
(cid:12) (cid:12)
Therefore,
k (β,Σ) (β,Σ) dβdΣ = π(β,Σ|z,y)π(z|β,Σ,y)dβdΣdz
S Rp×d Rn S Rp×d
Z dZ Z +Z dZ
(cid:0) (cid:12) (cid:1) n
(cid:12) 1
≤ ν(u)du < ∞ .
(n+2a−d−1)d
(2−ζ) 2 (cid:18)ZR+ (cid:19)
ThefollowinglemmawasgivenintheIntroduction, andisrestated hereforconvenience.
8
Lemma2. Lethbeamixingdensitythatisstrictlypositiveinaneighborhoodoftheorigin. Ifthere
existζ ∈ (1,2)andη > 0suchthat
η ud2h(u)
du < ∞ , (3)
ζu d
Z0 0 v2h(v)dv
thenK istrace-class. R
Proof. First,notethat
(ζ−1)su d
ud2 e−s2u h(u) ud2 e−s2u h(u) exp 2 u2 h(u)
ψ(u;s) = ≤ ≤ .
d −sv ζu d −sv nζu d o
R+v2 e 2 h(v)dv 0 v2 e 2 h(v)dv 0 v2 h(v)dv
R R R
ByLemma1,itsufficestoshowthat
d
u2 h(u)
du < ∞ .
ζu d
ZR+ 0 v2 h(v)dv
But,foranyη > 0,wehave R
∞ ud2 h(u) η∞ud2 h(u)du
du ≤ < ∞ ,
ζu d ζη d
Zη 0 v2 h(v)dv R0 v2 h(v)dv
andtheresultfollows. R R
Lemma3. Letg ∈ C(κ,ζ)forsomeκ ∈ Kandsomeζ ∈ (1,2). Byassumption,thereexistsη > 0
0
suchthatg isstrictlypositive anddifferentiable on(0,η ). Thenforanyη ∈ (0,η ),wehave
0 0
η ud2g(u)
du < ∞ .
ζu d
Z0 0 v2g(v)dv
d R
Proof. Sinceu2g(u)isbounded inaneighborhood oftheorigin,wehave
ζu
d
lim v2g(v)dv = 0 .
u→0Z0
Hence,anapplication ofL’Hoˆpital’sruleyields
lim κ(u)ud2g(u) = lim κ′(u)ud2 +κ(u)d2ud2−1 g(u)+κ(u)ud2g′(u)
u→0 ζuvd2g(v)dv u→0(cid:2) ζ(ζu)d2g(cid:3)(ζu)
0
′
R 1 ′ dκ(u) g(u) κ(u)g (u)
= lim κ(u)+ + lim
ζd2+1(u→0(cid:18) 2 u (cid:19)g(ζu) u→0 g(ζu) )
l +l
1 2
= ≥ 0 . (7)
ζd2+1
9
Putl3 = (l1 +l2)/ζd2+1. Itfollows from(7)thatforanyη ∈ (0,η0),there exists0 < η1 < η such
that
d
u2g(u) l3+1
≤
ζuvd2g(v)dv κ(u)
0
wheneveru ∈ (0,η1). Then,sinceκ ∈R K,thereexistsη2 ∈ (0,η1)suchthat
η2 1
du < ∞ .
κ(u)
Z0
η d
Furthermore, since g is continuous on [η2,η], η2u2g(u)du < ∞. Putting all of this together, we
haveforanyη < η ,
0 R
η ud2g(u) η2 ud2g(u) η ud2g(u)
du = du+ du
ζu d ζu d ζu d
Z0 0 v2g(v)dv Z0 0 v2g(v)dv Zη2 0 v2g(v)dv
η d
R η2 lR +1 u2g(u)dRu
≤ 3 du+ η2
Z0 κ(u) R0ζη2vd2g(v)dv
< ∞ . R
ProofofTheorem2. Assume that h is zero near the origin, and define η = sup η ∈ R :
0 +
η d 3η0 d
0 u2h(u)du = 0 . Clearly,J := 0 2 u2h(u)du > 0. Now,fors ∈ [0,∞),wehav(cid:8)e
R (cid:9) R 3η0
d −sv 2 d −sv −3η0s
v2e 2 h(v)dv ≥ v2e 2 h(v)dv ≥ Je 4 .
ZR+ Z0
Therefore, foru∈ R ands ∈ [0,∞),wehave
+
d −su
ψ(u;s) = u2 e 2 h(u) ≤ J−1ud2h(u)e−s2u+3η40s .
d −sv
R v2 e 2 h(v)dv
+
R
Now,byconsidering u≥ η andu < η separately, wecanseethat
0 0
ψ(u;s) ≤ J−1ud2h(u)es4u
for all u ∈ R and all s ∈ [0,∞). Hence, (6) of Lemma 1 holds with ζ = 3/2 and ν(u) =
+
J−1ud2h(u),sotheresultfollows.
Wenowprovethatthesecondconditionissufficient. Assumethatthereexistsg ∈ C(κ,ζ)such
thatlimu→0 hg((uu)) = l ∈ R+. ThenbyLemma3,thereexistsη > 0suchthat
η ud2 g(u)
du < ∞ ,
ζu d
Z0 0 v2 g(v)dv
R
10