Central limit theorems for hyperbolic spaces and Jacobi processes on $[0,\infty[$

0.2 MB·English

by  Michael Voit

Central limit theorems for hyperbolic spaces and Jacobi
processes on [0, [
2 ∞
1
0
2 Michael Voit
Fakult¨at Mathematik, Technische Universit¨at Dortmund
n
a Vogelpothsweg 87, D-44221 Dortmund, Germany
J
e-mail: [email protected]
7
1
January 18, 2012

]
R
P
. Abstract
h
t We present a unified approach to a couple of central limit theorems for radial ran-
a
m dom walks on hyperbolic spaces and time-homogeneous Markov chains on [0, [ whose
∞
transitionprobabilities are defined in terms of the Jacobiconvolutions. The proofs of all
[
results are based on limit results for the associated Jacobi functions. In particular, we

1 consider α , the case ϕ(α,β)(t) for small λ, and ϕ(α,β) (t/n) for n . The proofs
v →∞ iρ−λ iρ−nλ →∞
ofalltheselimitresultsarebasedontheknownLaplaceintegralrepresentationforJacobi
0
9 functions. Partsof the results areknown, other improveknownones,and other arenew.
4
3 KEYWORDS: Laplace integral representation, limits of Jacobi functions, asymptotic
.
1 results, spherical functions, hyperbolic spaces, radial random walks, central limit theorems,
0 normal limits, Rayleigh distributions.
2
Mathematics Subject Classification 2010: 60F05; 60B15; 33C45; 43A90; 43A62; 41A80.
1
:
v
i 1 Introduction
X
r
a We here derive a couple of central limit theorems for the distance of radial random walks
(Sk) from their starting points on the hyperbolic spaces H (F) of dimension k 2 over
thenfine≥ld0sF = R,C,orthequaternionsH. Themainobservationkforproofsisthatthe≥distance
processes are again Markov chains on [0, [ whose transition probabilities are related to to
∞
the product formula for the spherical functions for H (F), i.e., certain Jacobi functions. As
k
all proofs work without additional effort for more general “Jacobi random walks” on [0, [,
∞
i.e., Markov processes on [0, [ whose transition probabilities are related to general Jacobi
∞
functions, we shall derive all results in this context.
To describe the main results, we regard H (F) as symmetric space G/K with
k
F = R : G = SO (1,k), K = SO(k)
o
F = C : G = SU(1,k), K = S(U(1) U(k))
×
F = H : G = Sp(1,k), K = Sp(1) Sp(k)
×
and definethe dimension parameter d := dimRF = 1,2,4. We identify the doublecoset space
1
G//K with the interval [0, [ such that t [0, [ corresponds with the double coset
∞ ∈ ∞
cht 0 ... 0 sht
0 0
 
. .
KatK with at = .. Ik 1 .. ;
 − 
 0 0 
 
sht 0 ... 0 cht
 
 
see e.g. [F]or[Hel]. Usingthis homeomorphismϕ : G//K [0, [, wedefinethehyperbolic
→ ∞
distance on G/K by d(xK,yK) = ϕ(Ky 1xK). In this way, G acts on G/K = H (F)
− k
isometrically in a two-point homogeneous way, i.e., for x ,x ,y ,y H (F) with d(x ,x ) =
1 2 1 2 k 1 2
∈
d(y ,y ) there exists g G with g(x )= y and g(x )= y .
1 2 1 1 2 2
∈
Now consider atime-homogeneous Markov chain (Sk) on H (F)with transition kernel
n n 0 k
K starting at time 0 in eK G/K = H (F). This Mark≥ov chain is called a radial random
k k
∈
walkonH (F),ifK isG-invariant,i.e.,ifforallg G,x H (F),andBorelsetsA H (F),
k k k k
∈ ∈ ⊂
K (g(x),g(A)) = K (g,A). It is well-known (see e.g. Lemma 4.4 of [RV1]) that then for the
k k
canonical projection π : G/H = H (F) G//H = [0, [, the image process (π(Sk)) is a
k → ∞ n n≥0
time-homogeneous Markov chain on [0, [ with kernel
∞
K˜ (x,A) = (ν δ )(A) for x [0, [ and Borel sets A [0, [,
k x
∗ ∈ ∞ ⊂ ∞
where ν M1([0, [) is the distribution of d(Sk,Sk ) (which is independent of n), δ is a
∈ ∞ n n+1 x
point measure, and denotes the the double coset convolution on G//H = [0, [. We point
∗ ∞
out that the Markov kernel K and thus the finite-dimensional distributions of (Sk) are
k n n 0
determineduniquelybyν M1([0, [), andthatthedistributionsofthedistancesd(S≥k,eK)
∈ ∞ n
from the starting point eK are the n-th convolution powers ν(n) of ν w.r.t. .
∗
We are now interested in central limit theorems (CLTs) for d(Sk,eK) for n .
n → ∞
For the first type of results, we fix ν M1([0, [) and H (F), and introduce for each
k
∈ ∞
c ]0,1] the compressing map D :x cx on [0, [ as well as the compressed measure ν :=
c c
∈ 7→ ∞
D (ν) M1([0, [). Now consider the radial random walk (S(k,c)) on H (F) associated
c n n 0 k
∈ ∞ (k,n−r) ≥
with thecompressedmeasureν . We nowlook forCLTsfor (S ) dependingonr 0
c n n 0
≥ ≥
which include or supplement the results in [KTS], [Tu], [Gr], Section 3.2 of [Te], [Tr], and in
[Z1], [Z2], and Section 7.4 of the monograph [BH] in our setting.
The most classical case appears for r = 1/2 and is well-known; see Theorem 7.4.1 of [BH]
and references cited there. Here, the distance processes 1 d(S(k,n−r),eK) from the starting
√n n
point tend in distribution to some known limit depending on H (F) and the second moment
k
of ν. These limit distributions are known as (radial parts) of Gaussians on H (F). As this
k
case is studied precisely in the literature, we omit this case here.
Thecaser = 0withoutinitial compressionis dueto [Z1]; seeSection 7.4of [BH]. Wehere
improve this CLT by a rate of convergence. The CLT needs the modified moment functions
1 π
m (t) := m (d,k;t) := ln cht+r eiϕsht j dm (r,ϕ) (j N)
j j dk/2 1,d/2 1
Z0 Z0 | · | − − ∈
(cid:0) (cid:0) (cid:1)(cid:1)
with the probability measure m M1([0,1] [0,π]) defined below in (1.4). It
dk/2 1,d/2 1
− − ∈ ×
will turnout in (2.4) below that this definition agrees with that in [Z1] and [BH]. It is known
that m 0 and m (x)2 m (x) x2 for x 0 with equality precisely for x = 0.
1 1 2
≥ ≤ ≤ ≥
2
1.1 Theorem. Let ν M1([0, [) with ν = δ and finite second moment. For j N let
0
∈ ∞ 6 ∈
Mj := 0∞mj(t)dν(t) be the modified moments of ν with M1,M2 < ∞ by our assumption.
Then
R (k,1)
d(S ,eK) nM
n 1
−
√n
tends in distribution to N(0,M M2) with M M2 > 0.
2 − 1 2− 1
Moreover, if ν M1([0, [) in addition has a finite third moment, then the distribution
∈ ∞
functions of the random variables above tend uniformly on R to the distribution function of
N(0,M M2) of order n 1/3.
2 − 1 −
The case r > 1/2 was studied in [V2]. We reprove the result here in a different way:
1.2 Theorem. Let ν ∈ M1([0,∞[) with ν 6= δ0 and finite second moment m2 := 0∞x2dµ(x)
(which is automatically positive). Then
R
1/2
dk nr 1/2 d(S(k,n−r),eK)
m · − · n
(cid:18) 2(cid:19)
tends in distribution to the Rayleigh distribution ρ , where ρ has the Lebesgue density
dk/2 1 α
−
1
x2α+1e x2/2 (x 0).
−
2αΓ(α+1) ≥
Notice that in the preceding result dimRHk(F) = dk and that ρdk/2 1 is precisely the
radial part of a dk-dimensional standard normal distribution on Rdk. The−preceding theorem
therefore means that for r > 1/2, the CLT forgets the curvature and admits the behavior of
classical sums of i.i.d. random variables on the tangent space.
The case r ]0,1/2[ was not considered before. We here obtain:
∈
1.3 Theorem. Let ν M1([0, [) with ν = δ , with compact support, and thus with finite
0
∈ ∞ 6
moments ml := 0∞xldµ(x) ∈]0,∞[ (l ∈ N). Then:
(1) For r ]1/R6,1/2[,
∈ d(Sn(k,n−r),eK)− (d(k+1)/2−dk1)m2n1−2r
n1/2 r
−
tends in distribution to N(0, m2).
dk
(2) If r = 1/6, then the random variables of part (1) tend in distribution to N( M,m2)
− dk
with the mean
(d(k+1)/2 1)d(k+3)/2 2) m
4
M := − − · .
− 6dk(dk/2+1)
(3) If r ]0,1/6[, then
∈
d(Snk,n−r),eK)− (d(k+1)d/k2−1)m2 ·n1−2r M
n1 4r −→
−
in probability with M as in (2).
Besides the preceding limit theorems for a fixed hyperbolic space, we also derive the
following CLT for a fixed field F, where the dimension k and the number n of steps tend to
infinity. It generalizes a result in [V3]:
3
1.4 Theorem. Let (k ) N be increasing with lim n/k = 0, and fix F as above. Let
n n 1 n n
ν M1([0, [) with finit≥e s⊂econd moment ∞x2 dν(x→),∞and consider the associated radial
∈ ∞ 0
random walks (Snk)n≥0 on Hk(F) for k ∈ N.RThen, mj := 0∞(ln(chx))dν(x) < ∞ exist for
j = 1,2, and
R
d(Skn,Skn) nm
n 0 − 1
√n
tends in distribution for n to N(0,m m2).
→ ∞ 2− 1
An extension of this CLT without the restriction lim n/k = 0 was recently derived
n n
→∞
by Grundmann [G] by using completely different methods.
We now brieflydescribethecommon roots of theproofof theprecedingCLTs. We regard
the spherical functions of the Gelfand pair (G,H) above as continuous functions on [0, [
∞
whicharemultiplicative w.r.t. ∗, i.e., f(x)f(y)= 0∞fd(δx∗δy)forx,y ≥ 0. Itiswell-known
(see [Ko2]) that in our case all spherical functions are given by Jacobi functions
R
ϕ(α,β)(t) := F ((α+β+1 iλ)/2,(α+β+1+iλ)/2;α+1; sh2t) (λ C) (1.1)
λ 2 1 − − ∈
with the parameters
α = dk/2 1, β = d/2 1 with d:= dimRF = 1,2,4. (1.2)
− −
Moreover, the double coset convolutions on [0, [ for the hyperbolic spaces above can be
∗ ∞
regardedasspecialcasesofJacobiconvolution on[0, [whichwereinvestigated mainly
(α,β)
∗ ∞
by Flensted-Jensen and Koornwinder. In the following we refer to the survey [Ko2] on the
subject. For α > β 1/2 with α> 1/2, this convolution is given by
≥ − −
1 π
δ δ (f):= f arch chs cht+reiϕshs sht dm (r,ϕ) (1.3)
s (α,β) t α,β
∗ · ·
Z0 Z0
(cid:0) (cid:12) (cid:12)(cid:1)
(cid:12) (cid:12)
for f C ([0, [) and for the probability measure dm with
b α,β
∈ ∞
2Γ(α+1)
dm (r,ϕ) = (1 r2)α β 1(rsinϕ)2β rdrdϕ (1.4)
α,β − −
Γ(1/2)Γ(α β)Γ(β +1/2) · − ·
−
for α> β > 1/2. For α> β = 1/2, the measure degenerates into
− −
2Γ(α+1) 1
dm (r,ϕ) = (1 r2)α 1/2dr d(δ +δ )(ϕ), (1.5)
α,−1/2 Γ(1/2)Γ(α+1/2) − − · 2 0 π
and for α = β > 1/2 into
−
2Γ(α+1)
dm (r,ϕ) = sin2αϕdϕ dδ (r). (1.6)
α,α 0
Γ(1/2)Γ(α+1/2) ·
Nowfixα β 1/2withα > 1/2. Itiswell-knownthattheJacobiconvolutionabove
≥ ≥ − −
can be extended uniquely in a weakly continuous, bilinear way to a probability-preserving
convolution on M ([0, [), and that one obtains the so-called Jacobi-type hypergroups
(α,β) b
∗ ∞
on [0, [; see [Ko2], [BH], [Tr].
∞
Using this Jacobi-convolution, we now generalize the Markov processes (d(Sk,eK))
n n 0
above as follows: Fix a measure ν M1([0, [), and consider a time-homogeneous Jaco≥bi
∈ ∞
4
(α,β)
random walk (S ) on [0, [ with law ν of index (α,β), i.e., a time-homogeneous
n n 0
≥ ∞
Markov process on [0, [ starting at 0 with transition probability
∞
P(S(α,β) A S(α,β) = x) = (δ ν)(A) (x 0, A [0, [ a Borel set).
n+1 ∈ | n x ∗(α,β) ≥ ⊂ ∞
This notion agrees with that for (d(Sk,eK)) in the hyperbolic case above. We shall
n n 0
≥
derive all CLTs above in this more general setting. We shall do this in Section 3 for growing
dimensions and in Section 4 for fixed dimension k. In this way, Theorems 1.2, 1.1, 1.3, and
1.4 are just special cases of Theorems 4.3, 4.2, 4.7 and 3.1 below respectively. The proofs of
all theselimittheorems willbebebasedonseveral limitresultsfor Jacobifunctionswhichwe
will derive in Section 2. The basis of all these limit results will be the following well-known
Laplace integral representation for the Jacobi functions; see Section 5.2 of [Ko2]:
1.5 Theorem. Let α β 1/2. Then, for λ C and t 0,
≥ ≥ − ∈ ≥
1 π
ϕ(α,β)(t) = cht+reiϕsht iλ ρdm (r,ϕ)
λ | | − α,β
Z0 Z0
with
ρ:= α+β+1 0
≥
and the probability measure dm introduced in (1.4),(1.5) and (1.6) respectively.
α,β
We mention that Theorem 1.5 admits an analogue for Jacobi polynomials due to Koorn-
winder [Ko1], and that this integral representation also leads to limit results. In particular,
one can revisit Hilb’s formula for Jacobi polynomials (see [Sz]) in order to prove CLTs for
Markov chains on Z associated with orthogonal polynomials; see [Ga], [V1], Section 7.4 of
+
[BH], and references cited there for the topic.
We also mention that parts of this paper can be extended to certain families of Heckman-
Opdam hypergeometric functions of type BC which include the spherical functions for the
symmetric spaces SU(p,q)/(SU(p) SU(q)). For the background on these functions and the
×
associatedconvolutionstructuresonWeylchambersoftypeBwereferto[H],[HS],[O]and[R].
In [RV2] we generalize the Harish Chandra integral representation for SO (p,q)/(SO(p)
0
×
SO(q))forF = Rof[Sa]tothemoregeneralsettingconsideredin[R];thisintegralrepresenta-
tionissimilartothatinTheorem1.5andleadswithmoretechnicalefforttomultidimensional
extensions of some of the result in the present paper.
2 Limit relations for Jacobi functions
We start with two major results where α or both parameters α,β converge to infinity.
2.1 Proposition. Let β 1/2. Then there exists a constant C = C(β) such that for all
≥ −
t 0, α > max(β,0), and λ R
≥ ∈
λ min(1,t)
ϕ(iρα,βλ)(t)−eiλ·ln(cht) ≤ C| |· α1/2 .
−
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Using Theorem 1.5, we consider the difference
(cid:12) (cid:12)
1 π
R := ϕ(α,β)(t) eiλln(cht) = cht+reiϕsht iλ cht iλ dm (r,ϕ) ,
iρ λ − · | | −| | α,β
(cid:12) − (cid:12) (cid:12)Z0 Z0 (cid:16) (cid:17) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
5
which satisfies
1 π
R g(reiϕ,t) 1 dm (r,ϕ)
α,β
≤ | − |
Z0 Z0
with
g(reiϕ,t) := 1+reiϕ sht/cht iλ = eiλln(1+reiϕsht/cht).
· | · |
| · |
As eix 1 √2 x for x R, we have
| − | ≤ ·| | ∈
g(reiϕ,t) 1 √2 λ ln(1+reiϕ sht/cht ).
| − | ≤ ·| |·| | · | |
Moreover, as for z C with z < 1
∈ | |
ln(1+z ) ln(1+z) z + z 2+ z 3... = z /(1 z ),
| | | | ≤| | ≤ | | | | | | | | −| |
and as 0 sht/cht min(1,t) for t 0, we obtain for 0 r 1 and t 0
≤ ≤ ≥ ≤ ≤ ≥
r sht/cht r
g(reiϕ,t) 1 √2 λ · √2 λ min(1,t) .
| − | ≤ ·| |· 1 r sht/cht ≤ ·| |· 1 r
− · −
Therefore, as r [0,1],
∈
r
g(reiϕ,t) 1 2√2 λ min(1,t) .
| − | ≤ ·| |· 1 r2
−
Now consider the probability measure m for α > β > 1/2 introduced in Theorem 1.5.
α,β
−
We conclude that
2 min(1,t) Γ(α+1) π 1
R 2√2 λ · · sin2βϕdϕ (1 r2)α−β−2r2β+2dr.
≤ ·| |· Γ(1/2)Γ(α β)Γ(β +1/2) · −
− Z0 Z0
Using standard formulas for the beta-integrals on the right-hand side and finally
Γ(α+1)/Γ(α+1/2) = O(√α) (α ),
→ ∞
we obtain
Γ(β +3/2) Γ(α+1)
R 8 λ min(1,t) = λ min(1,t) O(1/√α)
≤ ·| |· Γ(β+1) · (α β 1)Γ(α+1/2) | |· ·
− −
as claimed. The case β = 1/2 follows in the same way from the definition of m .
α,β
−
We now consider a variant where α and β tend to infinity in a coupled way. We note
that the convergence results 2.1 and 2.2 (without error estimates) correspond to well-known
limit transitions for Jacobi polynomials which can be found e.g. in [Ko3] or [Sz].
2.2 Proposition. Fix constants c > 1 and d > 0 and put α := cβ +d. Then there exists a
constant C = C(c,d) such that for all t 0, β > 0, and λ R
≥ ∈
λ min(1,t)
ϕ(iρcβ+λd,β)(t)−eiλ·ln√ch2t−(1/c)sh2t ≤C| |· β1/2 .
−
(cid:12) (cid:12)
(cid:12) (cid:12)
The proof will b(cid:12)e based on the following observati(cid:12)on which is likely well-known.
6
2.3 Lemma. Consider a continuous function f : [0,1] [0, [ such that there exist x
0
→ ∞ ∈
[0,1] and constants 0 < c c 1 such that f(x) 1 c (x x )2 holds for all x [0,1],
1 2 2 0
≤ ≤ ≤ − − ∈
and f(x) 1 c (x x )2 for all x [0,1] in a suitable neighborhood of x . Moreover, let
1 0 0
≥ − − ∈
g : [0,1] [0, [ be continuous with g(x )> 0. Then, for continuous n ,
0
→ ∞ → ∞
1 1 1
x x fn(x) g(x)dx = O fn(x) g(x)dx .
0
| − |· · √n ·
Z0 (cid:18) Z0 (cid:19)
Proof. For n sufficiently large, we obtain by continuity arguments and omitting parts of the
left hand integral that
1 1/√c1n g(x ) 1
fn(x) g(x)dx (1 c x2)ng(x )/2dx = 0 (1 y2/n)ndy
1 0
· ≥ − 2√c n −
Z0 Z0 1 Z0
where the integral on the right hand side converges to some positive constant. On the other
hand,
1 x0 1
x x fn(x) g(x)dx = + x x fn(x) g(x)dx
0 0
| − |· · | − |· ·
Z0 (cid:18)Z0 Zx0(cid:19)
1
2 x(1 c x2)n h(x)dx
2
≤ − ·
Z0
2 √c2n
= (1 y2/n)n h(y/√nc )dy
2
n√c − ·
2 Z0
for some continuous function h depending on g where the integral on the right hand side
converges to some finitepositive constant. Acombination of bothresults leads to the lemma.
Proof of Proposition 2.2. Precisely as in the proof of Proposition 2.1 we obtain that
1 π
R := ϕ(α,β)(t) eiλln√ch2t (1/c)sh2t 2λ lnA dm (r,ϕ)
iρ λ − · − ≤ | | | | α,β
(cid:12) − (cid:12) Z0 Z0
(cid:12) (cid:12)
for
(cid:12) (cid:12)
cht+reiϕsht
A:= | | . (2.1)
cht+i 1 sht
| · √c |
As for r [0,1] and c > 1, we have (1 r)/2 A 1+r, we obtain
∈ − ≤ ≤
A 1 max(r,(r+1)/2) = (r+1)/2.
| − | ≤
Therefore, using ln(1+z ) z /(1 z ) for z = A 1 1 as in the preceding proof,
| | | | ≤ | | −| | | | | − | ≤
we conclude that
A 1 2 1 A2 2 4
lnA | − | | − | 1 A2 1 A2 .
| |≤ 1 A 1 ≤ 1 r · 1+A ≤ 1 r ·| − |≤ 1 r2 ·| − |
−| − | − | | − −
Moreover, defining τ := sht/cht min(t,1), we have
≤
(1+rτ cosϕ)2+r2τ2sin2ϕ τ2(1/c r2)+2rτ cosϕ
1 A2 = 1 = −
| − | − 1+τ2/c 1+τ2/c
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 1/c r2 (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
τ(cid:12) 2 cosϕ + | − | 2 (cid:12)τ c(cid:12)osϕ + r 1/√c . (cid:12)
≤ | | 1+τ2/c ≤ · | | | − |
(cid:18) (cid:12) (cid:12)(cid:19)
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
(cid:12) (cid:12)
7
In summary, we have
1 π cosϕ + r 1/√c
R 16λ τ | | | − | dm (r,ϕ)
≤ | | · 1 r2 α,β
Z0 Z0 −
2Γ(α+1)
= 16λ τ
| | · Γ(1/2)Γ(α β)Γ(β +1/2)
−
1 π
( cosϕ + r 1/√c ) ((1 r2)r2/(c 1))β(c 1) 2 r1+4/(c 1)(1 r2)d sin2βϕdrdϕ.
− − − −
· | | | − | · − · − ·
Z0 Z0
We now apply Lemma 2.3 to g(r) := r1+4/(c 1)(1 r2)d and f(r) := (1 r2)r2/(c 1) with
− −
− −
the maximum value of f on [0,1] at r = 1/√c and notice that dm (r,ϕ) is a probability
0 α,β
measure. This yields
1
r 1/√c) ((1 r2)r2/(c 1))β(c 1) 2 r1+4/(c 1)(1 r2)ddr = O(1/ β).
− − − −
| − | · − · −
Z0
p
As a similar argument also yields
π
cosϕ sin2βϕdϕ = O(1/ β),
| |·
Z0
p
we obtain R 16λ τ O(1/√β) as claimed.
≤ | | ·
We next turn to a limit concerning Bessel functions. Recapitulate that the normalized
Bessel functions
∞ ( 1)n(t/2)2n
j (t) := F (α+1; t2/4) = Γ(α+1) −
α 0 1
− · n!Γ(n+α+1)
n=0
X
with j (0) = 1 for α> 1/2 admit the integral representation
α
−
Γ(α+1) 1
j (t) = eitu(1 u2)α 1/2 du. (2.2)
α −
Γ(1/2)Γ(α+1/2) −
Z 1
−
In order to compare j with the Jacobi functions, we rewrite it as
α
1 π
j (t)= eitrcosϕdm (r,ϕ) (2.3)
α α,β
Z0 Z0
for α > β 1/2. In fact, the right hand side of (2.3) can be easily reduced to (2.2) by
≥ −
applying first polar coordinates u = rcosϕ, v = rsinϕ with v [0,√1 u2] and then the
∈ −
transform z := v/√1 u2 [0,1], where the z-integral fits the constants.
− ∈
The following limit result is related to the asymptotic expansion of the Jacobi functions
in terms of Bessel functions in [ST]; see Lemma 3.3 in [V2].
2.4 Proposition. Let α β 1/2 with α > 1/2 and T > 0 a constant. There exists a
≥ ≥ − −
constant C = C(α,β,T) such that for all λ R, t [0,T], and n 1,
∈ ∈ ≥
ϕ(α,β) (t/n) j (λt) C λ t2/n.
| iρ nλ − α | ≤ ·| |
−
8
Proof. The integral representations in Theorem 1.5 and (2.3) imply that
(α,β)
R := ϕ (t/n) j (λt)
iρ nλ − α
−
(cid:12) 1 π (cid:12)
(cid:12)(cid:12) exp iλn ln c(cid:12)(cid:12)h(t/n)+reiϕsh(t/n) eiλtrcosϕ dmα,β(r,ϕ).
≤ · | | −
Z0 Z0 (cid:12) (cid:12)
(cid:12) (cid:0) (cid:1) (cid:12)
Using the well-known(cid:12)inequality eix eiy √2 x y for x,y R,(cid:12)we obtain
| − |≤ ·| − | ∈
1 π
R λ n ln ch(t/n)+reiϕsh(t/n) trcosϕ dm (r,ϕ).
α,β
≤ | |· · | |−
Z0 Z0
(cid:12) (cid:12)
As (cid:12) (cid:12)
1
ln ch(t/n)+reiϕsh(t/n) = ln (ch(t/n)+rcosϕsh(t/n))2 +r2sin2ϕsh2(t/n
| | 2
= 1ln(cid:0)1+2rcosϕ t/n+t2(1+r2)/n2+O(t3/n3) (cid:1)
2 ·
1 (cid:0) t t2(1+r2) t2 t3(cid:1)
= 2rcosϕ + 2r2cos2ϕ +O
2 n n2 − n2 n3
(cid:18) (cid:18) (cid:19)(cid:19)
uniformly for t [0,T],r [0,1],ϕ [0,π], the claim follows.
∈ ∈ ∈
(α,β)
The next result describes the oscillatory behavior of ϕ (t) in the spectral variable
iρ λ
λ R for fixed α,β, which is uniform in t 0. For this we fo−llow Section 7.2.2 of [BH] and
∈ ≥
define for k N the so called moment functions
∈
∂k ∂k
(α,β) (α,β) (α,β)
m (t) := m (t) := ϕ (t) = ϕ (t)
k k ∂λk iρ+iλ λ=0 ∂λk −iρ−iλ λ=0
1 π (cid:12) (cid:12)
= ln c(cid:12)(cid:12)ht+r eiϕsht k dm(cid:12)(cid:12)α,β(r,ϕ) (2.4)
| · |
Z0 Z0
(cid:0) (cid:0) (cid:1)(cid:1)
for t 0 (where the second last equation follows from symmetry of the Jacobi functions in
≥
the parameter, and last one from Theorem 1.5). In particular, the function m appears as
1
substitute of the additive function x x on the group (R,+) on the Jacobi convolution
7→
structure ([0, [, ); see [Z1] and Section 7.2 of the monograph [BH]. m can be used
α,β 1
∞ ∗
to define a modified drift-part in a CLT on [0, [; see [Z1] [BH] and Section 3 below. We
∞
mention that for the parameters α,β, for which the Jacobi functions are spherical functions
of rank-one, non-compact symmetric spaces, this meaning of m is well-known for a long
1
time in probability theory on on hyperbolic spaces; see, for instance, [KTS], [Tu], [F].
We now derive a result which improves a general result of Zeuner [Z1] for general Chebli-
Trimeche hypergroups on [0, ] in the special case of the Jacobi convolution structures
∞
([0, [, ). It will be used to derive a Berry-Esseen-type CLT below.
α,β
∞ ∗
2.5 Proposition. Let α β 1/2 with α > 1/2. Then there exists a constant C =
≥ ≥ − −
C(α,β) such that for all t 0 and λ R,
≥ ∈
ϕ(α,β)(t) eiλm1(t) C(λ2+ λ 3).
| iρ λ − · | ≤ | |
−
The proof depends on the following elementary observation:
9
2.6 Lemma. For z C with z 1, ε ]0,1], and the Euler number e = 2,71...,
∈ | | ≤ ∈
1
ln 1+z .
| | || ≤ eε(1 z )ε
−| |
Proof. Elementary calculus yields xε lnx 1/(eε) for x ]0,1]. Therefore,
| · |≤ ∈
ln 1+z = ln(1+z) ln(1+z) = z z2/2+z3/3 ...
| | || |ℜ | ≤ | | | − ± |
1
z + z 2/2+ z 3/3 ... = ln(1 z ) .
≤| | | | | | ± | −| | | ≤ eε(1 z )ε
−| |
Proof of the Proposition: Let h(t,r,ϕ) := 1+reiϕ sht/cht . Then, for t 0,
| · | ≥
1 π
eiλm1(t) = (cht)iλ exp iλ ln(h(t,r,ϕ))dm (r,ϕ) .
· α,β
·
(cid:18) Z0 Z0 (cid:19)
Therefore, using the integral representation of the Jacobi functions, we obtain
R := ϕ(α,β)(t) eiλm1(t)
| iρ λ − · |
−
1 π 1 π
= eiλln(h(t,r,ϕ))dm (r,ϕ) exp iλ ln(h(t,r,ϕ))dm (r,ϕ) .
· α,β α,β
−
(cid:12)Z0 Z0 (cid:18) Z0 Z0 (cid:19)(cid:12)
We now w(cid:12)rite down the usual power series for both exponentials and observe that the (cid:12)terms
(cid:12) (cid:12)
of order 0 and 1 are equal in both expansions. Therefore,
1 π
R eiλln(h(t,r,ϕ)) (1+iλ ln(h(t,r,ϕ))) dm (r,ϕ)
· α,β
≤ − ·
Z0 Z0 (cid:12) (cid:12)
(cid:12) 1 π (cid:12) 1 π
+ ex(cid:12)p iλ ln(h(t,r,ϕ))dm (r,ϕ)(cid:12) 1 iλ ln(h(t,r,ϕ))dm (r,ϕ) .
α,β α,β
− −
(cid:12) (cid:18) Z0 Z0 (cid:19) Z0 Z0 (cid:12)
Using the(cid:12)(cid:12) well-known estimates cosx 1 x/2 and sinx x x 3/6 for x R, we(cid:12)(cid:12)
| − | ≤ | − | ≤ | | ∈
obtain eix (1+ix) x/2+ x 3/6, and thus
| − | ≤ | |
R A2λ2/2+A3 λ 3/6+A λ2/2+A λ 3/6
≤ 1 1| | 2 3| |
for A := 1 π ln(h(t,r,ϕ))k dm (r,ϕ), k = 1,2,3. In particular, by Jensen’s inequality,
k 0 0 | | α,β
R R R A λ2+A λ 3/3. (2.5)
2 3
≤ | |
Assume now that α > β > 1/2 holds. Choose some ε ]0,1[ with ε< (α β)/3 and apply
− ∈ −
Lemma 2.6 as well as 1 rsht/cht 1 r (1 r)/2 for r [0,1],t 0. This and the
− ≥ − ≥ − ∈ ≥
definitions of A and h imply for k = 2,3 and some constants C ,C ,C that
k 1 2 3
1 π C
1
A dm (r,ϕ)
k ≤ 1 rsht/cht kε α,β
Z0 Z0 ·| − |
1 π
C (1 r)α β 1 kε(rsinϕ)2β rdrdϕ C (2.6)
2 − − − 3
≤ − · ≤
Z0 Z0
as claimed.
The case α > β = 1/2 follows in the same way by using m .
α, 1/2
− −
Finally, the case α = β > 1/2 can be reduced to the case α > β = 1/2 by the
− −
well-known quadratic transform ϕ(α,α)(t) = ϕ(α,−1/2)(2t) (see Eq. (5.32) of [Ko3]) which in
2λ λ
particular implies m(α,α)(t) = 1m(α,−1/2)(2t).
1 2 1
10