Askey--Wilson polynomials, quadratic harnesses and martingales

0.39 MB·English

by  Włodek Bryc

ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES
AND MARTINGALES
0 WL ODEKBRYCANDJACEKWESOL OWSKI
1
0
2 Abstract. We use orthogonality measures of Askey-Wilson polynomials to
construct Markov processes withlinear regressions and quadratic conditional
l
u variances. Askey-Wilson polynomials are orthogonal martingale polynomials
J fortheseprocesses.

9
2

]
R 1. Introduction
P
Orthogonal martingale polynomials for stochastic processes have been studied
.
h by a number of authors, see [5, 13, 18, 22, 24, 25, 27, 28]. Orthogonal martingale
t
a polynomials play also a prominent role in non-commutative probability [1, 2], and
m can serve as a connection to the so called “classical versions” of non-commutative
[ processes. On the other hand, classical versions may exist without polynomial

martingale structure, see [6]. In [8] we identify intrinsic properties of the first two
2
conditionalmoments ofaprocesswhichguaranteethatastochasticprocesshasor-
v
7 thogonalmartingalepolynomials. Theseproperties,linearconditionalexpectations
5 and quadratic conditional variances, which we call the quadratic harness proper-
6 ties,alreadyleadtoanumberofnewexamplesofMarkovprocesses[9,10,11]with
0
orthogonalmartingale polynomials. Random fields with harness property were in-
.
2 troduced by Hammersley [14] and their properties were studied e. g. in [19, 32].
1 In this paper we use measures of orthogonality of Askey-Wilson polynomials to
8
construct a large class of Markov processes with quadratic harness property that
0
includes previous examples either as special cases or as “boundary cases”. The
:
v main step is the construction of an auxiliary Markov process which has Askey-
i
X Wilson polynomials [4] as orthogonal martingale polynomials. The question of
probabilistic interpretation of Askey-Wilson polynomials was raised in [12, page
r
a 197].
The paper is organized as follows. In the reminder of this section we recall
backgroundmaterialonquadraticharnessproperty,Askey-Wilsonpolynomials,we
state our two main results, and conclude with some applications. In Section 2 we
give an elementary construction that does not cover the entire range of parame-
ters, but it is explicit and does not rely on orthogonal polynomials. The general
constructionappearsin Section3; this prooffollows the method from[9]andrelies
onmartingalepropertyofAskey-Wilsonpolynomials whichextends projectionfor-
mula for Askey-Wilson polynomials [20] to a large range of parameters. Section 4
Date:December3,2008.
2000 Mathematics Subject Classification. Primary: 60J25;Secondary: 46L53.
Key words and phrases. Quadratic conditional variances, harnesses, orthogonal martingale
polynomials,freeL´evyprocesses,hypergeometricorthogonal polynomials.
1
2 WL ODEKBRYCANDJACEKWESOL OWSKI
contains another elementary but computationally more cumbersome construction
of a purely discrete quadratic harness which is not covered by Theorem 1.1.
1.1. Quadratic harnesses. In [8] the authors considersquare-integrablestochas-
tic processes on [0, ) such that for all t,s 0,
∞ ≥
(1.1) E(X )=0, E(X X )=min t,s ,
t t s
{ }
E(X ) is a linear function of X ,X , and Var[X ] is a quadratic function
t s,u s u t s,u
|F |F
ofX ,X . Here, isthetwo-sidedσ-fieldgeneratedby X :r (0,s) (u, ) .
s u s,u r
F { ∈ ∪ ∞ }
We will also use the one sided σ-fields generated by X :r t .
t t
F { ≤ }
Then for all s<t<u,
u t t s
(1.2) E(X )= − X + − X ,
t s,u s u
|F u s u s
− −
and under certain technical assumptions, [8, Theorem 2.2] asserts that there exist
numerical constants η,θ,σ,τ,γ such that for all s<t<u,
(1.3) Var[X ]
t s,u
|F
(u t)(t s) (uX sX )2 uX sX
s u s u
= − − 1+σ − +η −
u(1+σs)+τ γs (u s)2 u s
− (cid:18) − −
(X X )2 X X (X X )(uX sX )
u s u s u s s u
+τ − +θ − (1 γ) − − .
(u s)2 u s − − (u s)2
− − − (cid:19)
In this paper it will be convenient to extend [8] by admitting also processes
defined on a time domain T which is a proper subset of [0, ), and by relaxing
∞
slightly condition (1.1). (Compare [8, Proposition 2.1].) Accordingly, we will say
thatasquare-integrablestochasticprocess(X ) isaquadraticharnessonT with
t t T
∈
parameters (η,θ,σ,τ,γ), if it satisfies (1.2) and (1.3) on T.
Under the conditions listed in [8, Theorem 2.4 and Theorem 4.1], quadratic
harnesses on (0, ) have orthogonal martingale polynomials. Although several
∞
explicit three step recurrences have been workedout in [8, Section 4] and for some
of them corresponding quadratic harnesses were constructed in a series of papers
[9,10,11],the generalorthogonalmartingalepolynomialshavenotbeenidentified,
and the question of existence of corresponding quadratic harnesses was left open.
It has been noted that the family of all quadratic harnesses on (0, ) that sat-
∞
isfy condition (1.1) is invariant under the action of translations and reflections of
R: translation by a R acts as (X ) e aX and the reflection at 0 acts
t − e2at
∈ 7→
as (X ) (tX ). Since translations and reflection generate also the symmetry
t 1/t
7→
group of the Askey-Wilson polynomials [21], it is natural to investigate how to re-
late the measures of orthogonality of the Askey-Wilson polynomials to quadratic
harnesses. The goal of this paper is to explore this idea and significantly enlarge
the class of available examples. We show that quadratic harnesses exist and are
Markov processes for a wide range of parameters (η,θ,σ,τ,γ). The basic Markov
processweconstructhasAskey-Wilsonpolynomialsasorthogonalmartingalepoly-
nomials. The final quadratic harness is then obtained by appropriate scaling and
deterministic change of time.
Since Askey-Wilson polynomials together with their limits are regarded as the
most general classical polynomials, it is natural to inquire whether all quadratic
harnesses can be obtained by taking limits of processes with Askey-Wilson transi-
tion probabilities.
ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES AND MARTINGALES 3
Theorem 1.1. Fix parameters 1 < q < 1 and A,B,C,D that are either real or
−
(A,B) or (C,D) are complex conjugate pairs, and such that ABCD,qABCD <1.
Assume that
(1.4) AC,AD,BC,BD,qAC,qAD,qBD,qBD C [1, ).
∈ \ ∞
Let
((A+B)(1+ABCD) 2AB(C+D))√1 q
(1.5) η = − − ,
− (1 AC)(1 BC)(1 AD)(1 BD)(1 qABCD)
− − − − −
((D+C)(1+ABCD) 2CD(A+B))√1 q
p
(1.6) θ = − − ,
− (1 AC)(1 BC)(1 AD)(1 BD)(1 qABCD)
− − − − −
AB(1 q)
(1.7) σ = p − ,
1 qABCD
−
CD(1 q)
(1.8) τ = − ,
1 qABCD
−
q ABCD
(1.9) γ = − .
1 qABCD
−
Consider the interval J =(T ,T ) of positive length defined by
0 1
γ 1+ (γ 1)2 4στ
(1.10) T = max 0, − − − , τ ,
0
( p 2σ − )
1 γ 1+ (γ 1)2 4στ
(1.11) = max 0, − − − , σ .
T1 ( p 2τ − )
Then there exists a bounded Markov process (X ) with mean and covariance
t t J
∈
(1.1) such that (1.2) holds, and (1.3) holds with parameters η,θ,σ,τ,γ. Process
(X ) is unique among the processes with infinitely-supported one-dimensional
t t J
∈
distributions that have moments of all orders and satisfy (1.1), (1.2), and (1.3)
with the same parameters η,θ,σ,τ,γ.
The assumptionsonA,B,C,D aredictatedby the desireto limit the number of
casesinthe proofbut do notexhaustallpossibilities where quadraticharness(X )
t
with Askey-Wilson transition probabilities exists; see Proposition 4.1.
Remark 1.1. In Theorem 1.1, (γ 1)2 >4στ. To see this, write (γ 1)2 4στ =
− − −
(1 ABCD)2(1 q)2/(1 qABCD)2. Togetherwiththeupperboundin[8,Theorem
− − −
2.2] this shows that to construct a quadratic harness on [0, ) from Theorem 1.1,
∞
we must have 1<γ <1 2√στ.
− −
Remark 1.2. Intermsoftheoriginalparameters,the end-pointsoftheintervalare:
CD(1 q)
(1.12) T = max 0, CD,− − ,
0
− 1 qABCD
(cid:26) − (cid:27)
1 AB(1 q)
(1.13) = max 0, AB, − − .
T − 1 qABCD
1 (cid:26) − (cid:27)
If CD < 0 or AB < 0 then the third term under the maximum contributes for
q <0 only.
4 WL ODEKBRYCANDJACEKWESOL OWSKI
Remark 1.3. From (1.9), the construction will give γ > 1 when ABCD < 1.
−
Multiplying (1.7) and (1.8), we see that this may occur only when στ <0, i.e. for
harnesses on finite intervals; compare [8, Theorem 2.2].
1.2. Martingale property of Askey-Wilson polynomials. For a,b,c,d C
∈
such that a=0, and
6
(1.14) abcd,qabcd,ab,qab,ac,qac,ad,qad C [1, ),
∈ \ ∞
theAskey-Wilson[4]polynomialsw¯ (x)=w¯ (x;a,b,c,d)aredefinedbythe recur-
n n
rence
(1.15) (2x (a+a−1))w¯n(x)=Anw¯n+1(x) (An+Cn)w¯n+Cnw¯n 1(x), n 0,
− − − ≥
where
(1 abcdqn 1)(1 abqn)(1 acqn)(1 adqn)
−
(1.16) A = − − − − ,
n a(1 abcdq2n 1)(1 abcdq2n)
− − −
a(1 qn)(1 bcqn 1)(1 bdqn 1)(1 cdqn 1)
− − −
(1.17) C = − − − − .
n (1 abcdq2n 2)(1 abcdq2n 1)
− − − −
Theinitialconditionsarew¯ =0andw¯ =1. Wealsonotethatrecurrence(1.15)
1 0
−
is symmetric in parameters b,c,d but a is distinguished.
Askey-Wilson polynomials w¯ (x;a,b,c,d) can be defined also for a = 0; the
n
definition appropriate for our purposes uses the limit as a 0 of the normalized
→
(monic) form of the three step recurrence:
1 1
(1.18) xwn(x)=wn+1(x) (An+Cn (a+a−1))wn+ An 1Cnwn 1(x).
− 2 − 4 − −
Recall that polynomials r (x;t) : n Z ,t I are orthogonal martingale
n +
{ ∈ ∈ }
polynomials for process (Z ) if:
t t I
∈
(i) E(r (Z ;t)r (Z ;t))=0 for m=n and t I.
n t m t
(ii) E(r (Z ;t) )=r (Z ;s) for s6 <t in I a∈nd all n=0,1,2....
n t s n s
|F
The following result shows that Askey-Wilson polynomials are orthogonal mar-
tingalepolynomialsforafamilyofMarkovprocesses. IfA,B,C,D areasinPropo-
sition 2.1, this is a re-interpretation of the projection formula from [20].
Theorem 1.2. Suppose that A,B,C,D satisfy the assumptions in Theorem 1.1.
Let
1
(1.19) I =I(A,B,C,D,q)= max 0,CD,qCD ,
{ } max 0,AB,qAB
(cid:18) { }(cid:19)
with the convention 1/0 = . (The last terms under the maxima can contribute
∞
only when q <0 and CD or AB are negative.) If A=0, let
6
√1 q
(1.20) r (x;t)=w¯ − x;A√t,B√t,C/√t,D/√t .
n n
2√t
(cid:18) (cid:19)
(If A=0, use monic w instead of w¯ .) Then
n n
r (x;t):n=0,1,2...,t I
n
{ ∈ }
are orthogonal martingale polynomials for a Markov process (Z ) which satisfies
t
(1.2) and (1.3) with η =θ =σ =τ =0 and γ =q.
Remark 1.4. Process(Z )fromTheorem1.2iscloselyrelatedtotheMarkovprocess
t
(X ) that appears in the statement of Theorem 1.1, see (2.27).
t
ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES AND MARTINGALES 5
Remark 1.5. If A,B = 0 we can choose either A or B as the distinguished first
parameterinw¯ . The6correspondingpolynomialsr differbythefactor An(BC,BD)n
n n Bn(AC,AD)n
whichdoes not depend ont, so both sets ofpolynomials areorthogonalmartingale
polynomials for the same process (Z ).
t
1.3. Someworkedoutexamples. ThissectionshowshowTheorem1.1isrelated
to some previous constructions, and how it yields new examples. From examples
that have been previously worked out in detail one can see that the boundary of
the range of parameters is not covered by Theorem 1.1; in particular it does not
cover at all the family of five Meixner L´evy processes characterized by quadratic
harness property in [31]. On the other hand, sometimes new examples arise when
processes run only on a subinterval of (0, ).
∞
1.3.1. q-Meixner processes. Theorem 1.1 allows us to extend [10, Theorem 3.5 ]
to negative τ. (The cases γ = 1 which are included in [10] are not covered by
±
Theorem 1.1.)
Corollary 1.3. Fix τ,θ R and 1<γ <1 and let
∈ −
0 if τ 0
≥
T = τ/(1 γ) if τ <0, γ 0
0 
− − ≥
 τ if τ <0, γ <0.
−
Then there exists a Markov process (Xt) on [T0,∞) such that (1.1), (1.2) hold, and
(1.3) holds with parameters η =0,σ =0.
Proof. Let q =γ, A=0, B =0, and
θ+√θ2 4τ θ2 4τ θ √θ2 4τ θ2 4τ
C = − 2√1 −q ≥ , D = − −2√1 −q ≥
(−θ+2√i√1−4τq−θ2 θ2 <4τ (−θ−2√i√1−4τq−θ2 θ2 <4τ
− −
Then (1.4) holds trivially so by Theorem 1.1, (X ) is well defined on J =(T , ).
t 0
∞
Straightforwardcalculationoftheparametersfrom(1.6),(1.8),(1.9)endstheproof.
(cid:3)
When τ <0,the univariatelaws ofX arethe “sixth” family to be addedto the
t
fivecasesfrom[10, Theorem3.5]. The polynomialsorthogonalwithrespectto the
law of X satisfy recurrence
t
xp (x;t)=p (x;t)+θ[n] p (x;t)+(t+τ[n 1] )[n] p (x;t),
n n+1 q n q q n 1
− −
where [n] = (1 qn)/(1 q). So the polynomials orthogonal with respect to the
q
− −
standardized law of X /√t are
t
θ τ
(1.21) xp˜ (x;t)=p˜ (x;t)+ [n] p˜ (x;t)+(1+ [n 1] )[n] p˜ (x;t).
n n+1 q n q q n 1
√t t − −
The same law appears under the name q-Binomial law in [23] for parameters
n = t/τ N, τ = p(1 p) [ 1/4,0). When q 0, this law is discrete
− ∈ − − ∈ − ≤
two-point law at t= τ .
| |
AjustificationofrelatingthislawtotheBinomialcanbegivenforq =0. Inthis
case,recurrence(1.21)appearsin[7,(3)]withtheira= θ andtheir b= τ. By [7,
√t t
Proposition2.1],the lawν of 1 X isafree convolution t -foldpowerofthe two-
t √τ t τ
| | t/τ ⊞
point discrete law that corresponds to t=−τ. That is, νt =ν−τ| | ; in particular,
6 WL ODEKBRYCANDJACEKWESOL OWSKI
at t = nτ, X /√τ has the law that is the n-fold free additive convolution of a
t
−
centered and standardized two-point law.
1.3.2. Bi-Poisson processes. Next we deduce a version of [9, Theorem 1.2]. Here
we again have to exclude the boundary cases γ = 1 as well as the case 1+ηθ =
±
max γ,0 .
{ }
Corollary 1.4. For 1 < γ < 1, and 1+ηθ > max γ,0 there exists a Markov
− { }
process (X ) such that (1.1), (1.2) hold, and (1.3) holds with σ =τ =0.
t t [0, )
∈ ∞
Proof. Let A=0, B = η , C =0, D = θ . Then BD = ηθ <
−√ηθ+1 q −√ηθ+1 q ηθ+1 q
1. The condition qBD < 1 is a−lso satisfied, as we assum−e ηθ+1 > 0 when q−< 0.
Thus(1.4)holdsandwecanapplyTheorem1.1. Fromformulas(1.5)through(1.9),
the quadratic harness has parameters η,θ,σ =0,τ =0,γ, as claimed. (cid:3)
1.3.3. Free harness. Next we indicate the range of parameters that guarantee ex-
istence of the processes described in [8, Proposition 4.3]. Let
η+θσ ητ +θ
(1.22) α= , β = .
1 στ 1 στ
− −
Corollary 1.5. For 0 στ < 1, γ = στ, and η,θ with 2+ηθ+2στ 0 and
≤ − ≥
1+αβ > 0, there exists a Markov process (X ) such that (1.1), (1.2) and
t t [0, )
∈ ∞
(1.3) hold.
Remark 1.6. When 2 + ηθ + 2στ < 0, two of the products in (1.4) are in the
“forbiddenregion”[1, ), so Theorem1.1does notapply. However,the univariate
∞
Askey-Wilson distributions are still well defined.
Proof of Corollary 1.5. Take q =0, and let
α+βσ 4σ+(α βσ)2 α+βσ+ 4σ+(α βσ)2
A= − − − , B = − − ,
− 2q√1+αβ − 2q√1+αβ
β+ατ 4τ +(β ατ)2 β+ατ + 4τ +(β ατ)2
C = − − − , D = − − .
− 2q√1+αβ − 2q√1+αβ
To verify that AC [1, ) we proceed as follows. Note that
6∈ ∞
α+σβ
(1.23) A+B = ,
−√1+αβ
ατ +β
(1.24) C+D = ,
−√1+αβ
(α σβ)2 4σ
(1.25) A B = − − ,
− √1+αβ
p
(β τα)2 4τ
(1.26) C D = − − .
− √1+αβ
p
Multiplying (A+B)(C+D) and (A B)(C D) and using ABCD =στ we get
− −
στ στ (α σβ)2 4σ (β τα)2 4τ
AC+ BC = − − − −
AC − − BC 1+αβ
p p
and
στ στ (α+σβ)(ατ +β)
AC+ +BC+ = .
AC BC 1+αβ
ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES AND MARTINGALES 7
This gives the following quadratic equation for AC:
στ (α+σβ)(ατ +β)+ (α σβ)2 4σ (β τα)2 4τ
(1.27) AC+ = − − − − .
AC 2(1+αβ)
p p
We now note that a quadratic equation x+a/x = b with 0 < a < 1 and complex
b can have a root in [1, ) only when b is real and b 1+a; this follows from the
∞ ≥
fact that x+a/x is increasing for x>a, so x+a/x 1+a for x 1.
≥ ≥
Supposethereforethattherighthandsideof (1.27)isrealandlargerthat1+στ.
Thencalculationsleadto η2 4σ√θ2 4τ 2+ηθ+2στ. Therighthandsideis
− − ≥
nonnegativebyassumption,sosquaringtheinequalityweget(1+αβ) (1 στ)2
p − ≤
0, which contradicts the assumption.
OthercaseswithAD,BC,BD arehandledsimilarly. Since ABCD =στ <1 by
assumption, by Theorem 1.1 the quadratic harness exists.
It remains to calculate the parameters. From AB = σ CD = τ we see that
(1.7) and (1.8) give the correct values, and γ = στ from (1.9). To compute
−
the remaining parameters, we re-write the expression under the square root in the
denominator of (1.5) as
(1 AC)(1 BC)(1 AD)(1 BD)
− − − −
στ στ
= 1+στ (AC + ) 1+στ (BC+ ) .
− AC − BC
(cid:16) (cid:17)(cid:16) (cid:17)
This is the product of two conjugate expressions(see (1.27), andits derivation). A
calculationnowsimplifiesthedenominatorof (1.5)to(1 στ)/√1+αβ. Inserting
−
(1.23) and (1.24), the numerator of (1.5) simplifies to (α βσ)(1 στ)/√1+αβ
− −
The quotient of these two expressions is α βσ = η. Similar calculation verifies
(1.6). − (cid:3)
1.3.4. Purelyquadraticharness. Thequadraticharnesswithparametersη =θ =0,
and στ >0 has not been previously constructed.
Corollary 1.6. For σ,τ > 0 with στ < 1 and 1 < γ < 1 2√στ there exists
− −
a Markov process (X ) such that (1.1), (1.2) hold, and (1.3) holds with η =
t t [0, )
∈ ∞
θ =0.
Proof. Let
4(γ+στ)
q = .
2
1+γ+ (1 γ)2 4στ
− −
(cid:18) q (cid:19)
To see that 1<q <1, note that for γ+στ =0,
− 6
1+γ2 2στ (1+γ) (1 γ)2 4στ
q = − − − − ,
2 (γ+σqτ)
which gives
2 (1 γ)2 4στ
(1.28) q 1= − − − <0,
− 1+γq+ (1 γ)2 4στ
− −
q
8 WL ODEKBRYCANDJACEKWESOL OWSKI
and
2(1+γ)
(1.29) q+1= >0.
1+γ+ (1 γ)2 4στ
− −
Noting that (1 q)2+4qστ 4στ(q1 στ)>0, let
− ≥ −
i√2σ
A= B =
−
(1 q)+ (1 q)2+4qστ
− −
r
q
and
i√2τ
C = D = .
−
(1 q)+ (1 q)2+4qστ
− −
r
q
Since A,B,C,D are purely imaginary, we only need to verify condition BC < 1
which reads
(1.30) q+2√στ 1< (1 q)2+4qστ.
− −
q
This is trivially true when q+2√στ 1< 0. If q+2√στ 1 0, squaring both
− − ≥
sides we get 4(1 q)√στ >4(1 q)στ, which holds true as q >1 and 0<στ <1.
− −
Thusquadraticharness(X )existsbyTheorem1.1,anditremainstoverifythat
t
its parameters are as claimed. A straightforward calculation shows that (1.7) and
(1.8) give the correct values of parameters. It remains to verify that formula (1.9)
indeed gives the correct value of parameter γ. Since this calculation is lengthy, we
indicate major steps: we write (1.9) as γ 1=(q 1)(1+ABCD)/(1 qABCD),
− − −
and evaluate the right hand side. Substituting values of A,B,C,D we get
(1 q)2+(1+q) (1 q)2+4qστ
(q 1)(1+ABCD)/(1 qABCD)= − − .
− − 2q
p
Then we use formulas (1.28) and (1.29) to replace 1 q and 1+q and note that
−
since γ <1 2√στ we have γ <1 2στ and
− −
2(1 γ 2στ)
(1 q)2+4qστ = − − .
− γ+ (1 γ)2 4στ +1
p − −
Thiseventuallysimplifies(q 1)(1+ABCDp)/(1 qABCD)toγ 1,asclaimed. (cid:3)
− − −
2. The case of densities
In this section we give an explicit and elementary construction of a quadratic
harness on a possibly restricted time interval and under additional restrictions on
parameters A,B,C,D.
Proposition 2.1. Fix parameters 1 < q < 1 and A,B,C,D that are either real
−
or (A,B) or (C,D) are complex conjugate pairs. Without loss of generality, we
assume that A B , C D ; additionally, we assume that BD < 1. Then
| | ≤ | | | | ≤ | | | |
the interval
D 2 CD 1 CD B 2
J = | | − , − | |
1 AB D 2 B 2 AB
(cid:20) − | | | | − (cid:21)
has positive length and there exists a unique bounded Markov process (X ) with
t t J
∈
absolutely continuous finite dimensional distributions which satisfies the conclusion
of Theorem 1.1.
ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES AND MARTINGALES 9
Remark 2.1. Proposition2.1 is a special case of Theorem 1.1; the latter may allow
to extend the processes constructed here to a wider time-interval:
D 2 CD 1 CD B 2
T | | − and − | | T .
0 ≤ 1 AB D 2 B 2 AB ≤ 1
− | | | | −
The easiestwayto seethe inequalities is tocomparethe endpoints ofthe intervals
(2.9)and(1.19)aftertheMo¨biustransformation(2.26);forexample, D 2 CD
| | ≥| |≥
max 0,CD,qCD , where the last term plays a role only when q <0 and CD <0.
{ }
Therestofthissectioncontainstheconstruction,endingwiththeproofofPropo-
sition 2.1.
2.1. Askey-Wilson densities. For complex a and q <1 we define
| |
n−1(1 aqj) n=1,2,...
(a) =(a;q) = j=0 −
n n
(1Q n=0
∞
(a) =(a;q) = (1 aqj)
∞ ∞ −
j=0
Y
and we denote
(a ,a ,...,a ) =(a ,a ,...,a ;q) =(a ;q) (a ;q) ...(a ;q) ,
1 2 l 1 2 l 1 2 l
∞ ∞ ∞ ∞ ∞
(a ,a ,...,a ) =(a ,a ,...,a ;q) =(a ;q) (a ;q) ...(a ;q) .
1 2 l n 1 2 l n 1 2 l n
∞ ∞
The advantage of this notation over the standard product notation lies both in its
conciseness and in mnemotechnic simplification rules:
(a,b) (b)
n n
= ,
(a,c) (c)
n n
(2.1) (α) =(qMα) (α) ,
M+L L M
and
(2.2) (α)M =(−α)MqM(M2−1) qMqα .
(cid:18) (cid:19)M
which often help with calculations. For a reader who is as uncomfortable with this
notation as we were at the beginning of this project, we suggest to re-write the
formulas for the case q =0. For example, (a;0) is either 1 or (1 a) as n=0 or
n
−
n > 0 respectively. The construction of Markov process for q = 0 in itself is quite
interesting as the resulting laws are related to the laws that arise in Voiculescu’s
freeprobability;theformulassimplifyenoughsothattheintegralscanbecomputed
by elementary means, for example by residua.
From Askey and Wilson [4, Theorem 2.1] it follows that if a,b,c,d are complex
such that max a, b, c, d < 1 and 1 < q < 1, then with θ = θ such that
x
{| | | | | | | |} −
cos θ =x
1 1 e2iθ,e−2iθ
(2.3) dx
∞
√1 x2(aeiθ,ae iθ, beiθ, be iθ, ceiθ, ce iθ, deiθ, de iθ)
Z−1 − − (cid:0)− (cid:1) − − ∞
2π(abcd)
= ∞ .
(q, ab, ac, ad, bc, bd, cd)
∞
10 WL ODEKBRYCANDJACEKWESOL OWSKI
When 1 < q < 1 and a,b,c,d are either real or complex and form two conjugate
−
pairs and max a, b, c, d < 1, the integrand is real and positive. This allows
{| | | | | | | |}
us to define the Askey-Wilson density
K(a,b,c,d) (e2iθ) 2
(2.4) f(x;a,b,c,d)= √1 x2 (aeiθ, beiθ, ce∞iθ, deiθ) I(−1,1)(x),
− (cid:12) ∞(cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
(q, ab, ac, ad, bc, bd, cd)
(2.5) K(a,b,c,d)= ∞ .
2π(abcd)
∞
The first two moments are easily computed.
Proposition2.2. SupposeX hasdistributionf(x;a,b,c,d)withparametersa,b,c,d
as above. Then the expectation of X is
a+b+c+d abc abd acd bcd
(2.6) E(X)= − − − − ,
2(1 abcd)
−
and the variance of X is
(1 ab)(1 ac)(1 ad)(1 bc)(1 bd)(1 cd)(1 q)
(2.7) Var(X)= − − − − − − − .
4(1 abcd)2(1 abcdq)
− −
Proof. If a=b=c=d=0, E(X)=0 by symmetry. If one of the parameters, say
a C is non-zero, we note that
∈
aeiθ, ae−iθ = aeiθ, ae−iθ aqeiθ,aqe−iθ
1
∞ ∞
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) =(1(cid:1)+a2 2ax) aqeiθ,aqe−iθ .
−
∞
Therefore by (2.3) (cid:0) (cid:1)
K(a,b,c,d) (1 ab)(1 ac)(1 ad)
E(1+a2 2aX)= = − − − .
− K(qa,b,c,d) 1 abcd
−
Now (2.6) follows by a simple algebra.
Similarly, for non-zero a,b C,
∈
4abVar(X)=E (1+a2 2aX)(1+b2 2bX) E(1+a2 2aX)E(1+b2 2bX)
− − − − −
(cid:2) K(a,b,c,d) K2(a(cid:3),b,c,d)
=
K(qa,qb,c,d) − K(qa,b,c,d)K(a,qb,c,d)
(1 ab)(1 qab)(1 ac)(1 ad)(1 bc)(1 bd)
= − − − − − −
(1 abcd)(1 qabcd)
− −
(1 ab)2(1 ac)(1 ad)(1 bc)(1 bd)
− − − − − .
− (1 abcd)2
−
Again after simple transformations we arrive at (2.7).
If only one parameter is non-zero but q = 0, the calculations are similar, start-
ing with E (1+a2 2aX)(1+a2q2 2aq6X) ; when q = 0 the density is a re-
− −
parametrization of Marchenko-Pastur law [15, (3.3.2)]; we omit the details. If
(cid:0) (cid:1)
a,b,c,d are zero, f(x;0,0,0,0) is the orthogonality measure of the continuous q-
Hermite polynomials [17, (3.26.3)]; since H (x) = 2xH (x) (1 q)H = 4x2
2 1 0
(1 q), the second moment is (1 q)/4. − − (cid:3)−
− −
We need a technical result on Askey-Wilson densities inspired by [20, formula
(2.4)].