Distributional transformations, orthogonal polynomials, and Stein characterizations

Transcription

1 Distributional transforations, orthogonal polynoials, and Stein characterizations LARRY GOLDSTEIN 13 and GESINE REINERT 23 Abstract A new class of distributional transforations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transfored distribution on the test function s higher order derivatives. The class includes the size and zero bias transforations, and when specializing to weighting by polynoial functions, relates distributional failies closed under independent addition, and in particular the infinitely divisible distributions, to the faily of transforations induced by their associated orthogonal polynoial systes. For these failies, generalizing a well known property of size biasing, sus of independent variables are transfored by replacing suands chosen according to a ultivariate distribution on its index set by independent variables whose distributions are transfored by ebers of that sae faily. A variety of the transforations associated with the classical orthogonal polynoial systes have as fixed points the original distribution, or a eber of the sae faily with different paraeter. 1 Introduction The zero bias transforation was introduced in [13]. This apping enjoys properties siilar to those of the well known size biased transforation (see e.g. [15]) on non-negative variables, but can be applied to ean zero rando variables. One ain feature of the zero bias transforation is that its unique fixed point is the ean zero noral distribution, and for this reason it has been applied in Stein s ethod for the purpose of noral approxiation 1 Departent of Matheatics, University of Southern California, Los Angeles CA , USA 2 Departent of Statistics, University of Oxford, Oxford OX1 3TG, UK, supported in part by EPSRC grant no. GR/R52183/01 3 This work was partially copleted while the authors were visiting the Institute for Matheatical Sciences, National University of Singapore, in The visit was supported by the Institute. 1

2 ([13], [11], [12], and [14]). The zero bias transforation is also related to the K i function in the work [16], [5], [22], and [6] and others; for a good overview see [8]. We place the classical size bias transforation and the zero bias transforation in a broader context, showing that both are a particular case of transforing a given distribution X into X (P ) through the use of a easurable biasing function P. To be ore precise, for the given X and P let C denote the collection of functions whose th derivative exists and is easurable on R, and suppressing X on the left hand side, set F (P ) = {F C : E P (X)F (X) < }. We consider transforations characterized by EP (X)F (X) = αef () (X (P ) ) for all F F (P ), (1) where necessarily α = (!) 1 EP (X)X when X F(P ); we insist α > 0. We coin this distribution the X P biased distribution. For discrete distributions the differential operator is replaced by the difference operator; see Sections 4.3 and 4.4. Theore 2.1 provides our ost general conditions on the existence of transforations characterized by (1), where the biasing function P is only required to have sign changes and satisfy certain orthogonality and positivity conditions; we call the order of the resulting transforation. In the particular case where P is a polynoial, the sign change condition can be expressed in ters of the roots and order of the polynoial P, and the orthogonality properties in ters of oents. For exaple, for each = 0, 1,... there exists a distributional transforation which is defined using the Herite polynoial of order as the biasing function, and whose doain are those distributions whose first 2 oents atch those of the ean zero noral; the case = 1 corresponds to the zero bias transforation of [13]. Theore 2.1 in Section 2 shows distributional transforations exist in great generality. In Section 3 we find that there is considerable additional structure for the failies of transforations induced by orthogonal polynoial systes, especially those corresponding to failies of distributions which are closed under addition of independent variables. Corresponding to the Noral, Gaa, Poisson, Binoial and Beta-type distributions, in Section 4 we study the faily of transforations defined using the Herite, 2

3 Laguerre, Charlier, Krawtchouk, and Gegenbauer polynoials, and obtain high order Stein type characterizing equations. Our work here is in the spirit of [9], where other fundaental connections between Stein equations and orthogonal polynoials were first described. The approach in [9] is iterated in [24], cobining it with well-known connections between orthogonal polynoials and birth and death processes, and used as in [17] to describe solutions of Stein equations. We first review soe well known facts regarding the size bias transforation, which is the siplest and best known of all these distributional transforations. For non-negative X with 0 < EX = µ <, the X-size biased distribution X s is defined by the characterizing equation EXF (X) = µef (X s ) for all F F 0 (X). (2) One key feature of the sized bias transforation is the following. If X 1,..., X n are independent non-negative variables with finite positive expectations EX i = µ i and n W = X i, i=1 then a variable with the W -size biased distribution can be constructed by replacing a variable X i, chosen with probability proportional to µ i, by an independent variable Xi s having the X i -size biased distribution. In other words, letting µ i P (I = i) = nj=1 µ j be independent of X 1,..., X n, the variable W s = W X I + X s I (3) has the W -size biased distribution. Letting x + = ax(0, x), size biasing is the case of (1) with biasing function P (x) = x +. This transforation is of order zero, as there are = 0 sign changes of x + on R, and has α = EX + ; when X 0 we have X + = X resulting in the usual characterization (2). The zero bias transforation [13] was otivated by the siilarity between the size bias transforation and the Stein equation [21] for the ean zero noral distribution. In particular, Stein s identity says that Z N (0, ) if and only if EZF (Z) = EF (Z) for all F F 1 (Z). (4) 3

4 Coparing (4) to (2), for a ean zero, positive variance variable X, we say that X z has the X-zero biased distribution if EXF (X) = EF (X z ) for all F F 1 (X). (5) Note that (5) for zero biasing is the sae as (2) for size biasing, but with variance replacing ean, and F replacing F. That the noral distribution with variance is the unique fixed point of the zero bias transforation follows iediately fro the characterization (4). It was shown in [13] that the zero bias distribution X z exists for all X that have ean zero and finite positive variance. Its existence follows also fro Theore 2.1, as the special case of (1) for the function P (x) = x, having = 1 sign changes on R, and α equal to the variance of X. The zero bias transforation was introduced and used in [13] to obtain bounds of order n 1 in noral approxiations for sooth test functions under third order oent conditions, in the presence of dependence induced by siple rando sapling. In [11] it is used to provide bounds to the noral distribution for hierarchical sequences generated by the iteration of a so called averaging function, in [12] for noral approxiation in cobinatorial central liit theores with rando perutations having distribution constant over cycle type, and in [14] the extension of the zero bias transforation to higher diension is considered. The zero bias transforation enjoys a property siilar to (3) for size biasing. In particular, it was shown in [13] that a su of independent ean zero variables with finite variances can be zero biased by replacing one variable chosen with probability proportional to its variance by an independent variable fro that suands zero biased distribution. Precisely, let X 1,..., X n be independent ean zero variables with variance i = EXi 2 > 0, W = X X n, and I a rando index, independent of X 1,..., X n with distribution P (I = i) = i nj=1 j. (6) Then W z = W X I + X z I (7) 4

5 has the W -zero biased distribution, where Xi z is a variable independent of X j, j i having the X i zero biased distribution. This construction is extended to the failies of transforations associated with orthogonal polynoial in Theore 3.1. In particular, in Section 3 we see that for higher order transforations sus of independent variables are transfored by replacing ultiple variables chosen according to soe distribution (e.g. ultinoial, ultivariate hypergeoetric) with independent variables possessing distributions transfored by the sae faily. In Section 2 we give the oent and sign change conditions on P which guarantee the existence of the X P distribution and provide an explicit construction. In Section 3 we treat the special case where P is a eber of a faily of orthogonal polynoials. The generalization to higher order of the replace one variable zero and size bias constructions is based on the identity (25) expressing an orthogonal polynoial of a su as a su of like polynoials with suands having no larger order, and is given in Section 3. In Sections 4.1, 4.2, 4.3, 4.4 and 4.5 we treat the Herite, Laguerre, Charlier, Krawtchouk and Gegenbauer polynoials, corresponding to the Noral, Gaa, Poisson, Binoial and Beta-type distributions respectively. Special instances of the Beta-type distributions we consider are the unifor U[ 1, 1], the arcsine, and the sei-circle distribution. 2 Transforations in General We begin our study with the following existence and uniqueness theore for the types of distributional transforations under consideration. We say the easurable function P on R is positive on an interval I if P (x) 0 for all x I with strict inequality for at least one x, and siilarly for P negative on I. We say P has exactly = 0, 1,... sign changes if R can be partitioned into + 1 disjoint subintervals with non-epty interior such that P alternates sign on successive intervals. Though the choices for the endpoints of such intervals ay be soewhat arbitrary when there are intervals where P is zero, we will nevertheless say that a sign change occurs at the interval boundaries; the uniqueness guaranteed by Theore 2.1 shows that the X P biased distribution constructed in the proof of Theore 2.1 is the sae for all interval boundary choices, and Exaple 2.1 gives soe additional explanation of this phenoenon in the context of a particular exaple. We note that for existence in general, regarding boundedness, 5

6 the orthogonality conditions required by Theore 2.1 are only relative to P and required only up to a finite order; such conditions ay not ipose boundedness on any of the power oents of X, as illustrated in Exaple 2.1. Theore 2.1 Let X be a rando variable, {0, 1, 2,...} and P a easurable function with exactly sign changes, positive on its rightost interval and 1! EXk P (X) = αδ k, k = 0,...,, (8) with α > 0. Then there exists a unique distribution for a rando variable X (P ) such that EP (X)F (X) = αef () (X (P ) ) for all F F (P ). (9) Theore 2.1 says that X is in the doain of the distributional transforation of order defined using the biasing function P having sign changes when the powers of X saller than are orthogonal to P (X) in the L 2 (X) sense, that is, when P (X) {1, X,..., X 1 }, and EX P (X) > 0. As noted above, the existence of both the size and zero bias transforations are both special cases. Proof of Theore 2.1. We give an explicit construction of the variate X (P ). By replacing P by P/α, it suffices to prove the theore for α = 1. Label the points where the sign changes of P occur as r 1,..., r, and let Q(x) = (x r i ), (10) i=1 adopting the usual convention that an epty product is 1. By construction Q and P have the sae sign, so letting µ X denote the distribution of X, dµ Y (y) = 1! Q(y)P (y)dµ X(y) (11) is therefore a easure, and since (8) with k = iplies that EQ(X)P (X) =!, a probability easure. Now with Y and {U i } i 1 utually independent with Y having distribution µ Y and U j having distribution function u i on [0, 1], with r 0 = Y and r +1 = 0, we clai that X (P ) = +1 k=1 ( ) U i (r k 1 r k ) (12) i=k 6

7 satisfies (9), thus proving the existence of the X P biased distribution. We begin by noting that for any F for which either side below exists, and so for k = 0,...,, letting EF (Y ) = 1 EF (X)Q(X)P (X), (13)! R k (x) = i=k+1 (x r i ), a polynoial of degree k, by (13) and (8) we have k E(1/ (Y r i )) = 1 i=1! ER k(x)p (X) = δ k,. (14) We show the clai by induction. In particular, for k 1 letting +1 V k = U i, W k = i=k j=k V j (r j 1 r j ), (15) and taking X (P ) as in (12), we show that for all F Cc, the collection of infinitely differentiable functions with copact support, and k = 0,..., { F EF () (X (P ) ( k) } (V k+1 (Y r k+1 ) + W k+2 ) ) = k! E Vk+1 k ki=1. (16) (Y r i ) We see the expectation on the right hand exists since F and all its derivatives are bounded, V k+1 is independent of Y for all k, EUi k < for i k + 1, and use of (14). The case k = 0 is the stateent that X (P ) = V 1 (Y r 1 )+W 2, which follows fro definitions (12) and (15). Assue (16) holds for soe 0 k <. Using V k+1 = U k+1 V k+2 in (16) and taking expectation over U k+1, with density (k + 1)u k k+1, we obtain EF () (X (P ) ) = (k + 1)!E 1 0 { F ( k) } (u k+1 V k+2 (Y r k+1 ) + W k+2 ) u k k+1 V k+2 k ki=1 u k (Y r i ) k+1du k+1. Cancelling u k k+1 and integrating, we obtain { F ( (k+1)) (V k+2 (Y r k+1 ) + W k+2 ) F ( (k+1)) } (W k+2 ) (k + 1)!E Vk+2 k+1 k+1. (17) i=1 (Y r i ) 7

8 Using the independence of V k+2 and Y for any k, and that W k+2 is independent of Y for all k 0, the second ter in the expectation (17) vanishes by (14), since k+1 1. The induction is copleted by noting that definitions (15) give that V k+2 (Y r k+1 ) + W k+2 = V k+2 (Y r k+2 ) + W k+3. Now applying (16) for k = and using V +1 = 1, W +2 = 0 and r +1 = 0 we obtain { } F (Y ) EF () (X (P ) ) =!E = EP (X)F (X) Q(Y ) by (13). That is, the equality in (9) holds for all F C c. For F F (X), by replacing F by F (x) 1 j=0 F (j) (0) x j j! if necessary, we ay assue, in light of (8), that F (j) (0) = 0 for j = 0,..., 1, and hence, with If = x 0 f, F (x) = I f for soe easurable function f. Since F = F 1 F 2 where F 1 (x) = I f + and F 2 (x) = I f, it suffices by linearity to consider f 0. Letting 0 f n f we have I f n = F n F, and hence the equality in (9) holds for F F (X) using the onotone and doinated convergence theores on the right and left sides of (9), respectively. The distribution X (P ) is unique since (9) holds for all F Cc, which is separating. The existence of the X (P ) distribution also follows fro the Riesz representation theore upon deonstrating the positivity of the linear operator T defined by T f = EP (X)F (X) with F (x) = I f over f Cc 0, the space of continuous functions with copact support. The signed easure dµ = P dµ X has the property x j dµ = EX j P (X) = 0 for j = 0, 1,..., 1, and now the sign change property of P allows us, when on the finite interval [a, b], to invoke Theore 5.4 in Chapter XI of [19] (see also Exaple 1.4 in Chapter XI) to conclude T is positive and hence T f = b a fdµ() for soe easure µ (), which is a probability easure since 8

9 EX P (X) =!. This arguent is siilar to the one used in [13] to prove the existence of the zero bias distribution for a ean zero, finite variance X by noting that when f 0 the function F = If is non-decreasing, and hence X and F (X) are positively correlated, and so the operator is positive. T f = EXF (X) EXEF (X) = 0 Exaple 2.1 Consider the application of Theore 2.1 where P (x) has exactly = 1 sign change at r 1 = 0. Then for the non constant X to be in the doain of the transforation characterized by EP (X)F (X) = αef (X (P ) ) (18) we require EP (X) = 0 and α = EXP (X) > 0. We have Q(x) = x in (10) and, recalling the X variable in the proof was rescaled to have α = 1, the Y distribution in (11) is dµ Y (y) = xp (x)dµ X (y)/α. Fro (12) with = 1, r 0 = Y, r 2 = 0 and U j with distribution function u j on [0, 1] X (P ) = +1 k=1 ( ) U i (r k 1 r k ) = U 1 (r 0 r 1 ) + (r 1 r 2 ) = U 1 Y. i=k Hence X (P ) is absolutely continuous, and one can directly verify that its density is given by f (P ) (x) = α 1 E[P (X); X > x]. (19) When x 0 P (u)du is finite for all x and c = exp( α 1 x 0 P (u)du)dx <, the transforation (18) has a fixed point at the distribution with density ( f(x) = c 1 exp 1 x ) P (u)du ; α 0 for instance, when P (x) = x, f is the ean zero noral density with variance α. 9

10 Taking P to be the sign function P (x) = 1(x > 0) 1(x < 0) provides an exaple of a transforation given by a discontinuous P, and shows that generally the orthogonality conditions ay not reduce to restrictions on the oents of X, in particular, (8) for k = 0 requires X to have edian 0. If in addition α = E X is finite, iposed by (8) for k = 1, Theore 2.1 gives that X is in doain of the transforation characterized by (18). The density of the transfored variables are, by (19), f (P ) (x) = { P (X > x)/e X x > 0 P (X < x)/e X x < 0. (20) For this choice of P the Y distribution in (11) becoes dµ Y (y) = y dµ X (y)/e X, which is the X size biased distribution. Hence, the X P biased distribution is obtain by ultiplying Y µ Y by an independent U[0, 1] variable. The transforation has a fixed point at the Laplace distribution with density f(x) = 1 2α exp ( 1 α x ). Taking P (x) = 1(x > 1) 1(x < 1) gives a transforation having doain those variables X with α = E( X 1( X > 1)) < and satisfying P (X > 1) = P (X < 1). (21) Since P (x) = 0 in the set [ 1, 1] the sign change can be said to occur at point in ( 1, 1) and the polynoial Q in the proof of Theore 2.1 can be taken to be Q(x) = x r 1 for any r 1 ( 1, 1). As assured by uniqueness, the distribution constructed in the proof of Theore 2.1 does not depend on choice of r 1 ; in fact, in this case (21) iplies that the du Y distribution in (11) is the sae for all r 1 ( 1, 1). 10

11 3 Transforations using orthogonal polynoials We consider a syste of polynoinals orthogonal with respect to a non-trivial faily of distributions Z L indexed by a real paraeter. Condition 3.1 For soe 0, the polynoials {P k (x)} 0 k are onic, have degree k, are orthogonal with respect to the distributional faily Z L, and satisfy E[P k (Z )] 2 > 0. Note that since P k is onic and orthogonal it has k distinct roots and is positive as x (e.g. [1]); furtherore, we have EZ k P k (Z ) = E[P k (Z )] 2, k = 0,...,. When studying transforations using an iplicit faily of orthogonal polynoials, we index the transfored distribution by say, X (k), that is, by the paraeter and order k of the polynoial. Applying Theore 2.1 in this fraework, we obtain the following Corollary 3.1 Let Condition 3.1 be satisfied with EZ 2 k set Then for all X M k, where <, and for 0 α (k) = 1 k! EZk P k (Z ). (22) M k = {X : EX j = EZ j, 0 j 2k}, there exists a rando variable X (k) such that for all F F k (P k ) EP k (X)F (X) = α (k) EF (k) (X (k) ). (23) Proof: By Condition 3.1 and orthogonality we have for 0 j k, 1 k! EXj P k (X) = 1 k! EZj P k (Z ) = 1 k! EP j (Z )P k (Z ) = α (k) using X M k. Now invoke Theore 2.1. δ j,k 11

12 We say the faily of distributions Z is closed under independent addition if for independent Z i L i, i = 1, 2 we have Z 1 +Z 2 L There is special structure when the transforation function in Theore 2.1 is a eber of an orthogonal polynoial syste corresponding to such a faily. In particular, the following Theore 3.1 generalizes (3) and (7) in showing how a su of independent variables can be P transfored by replacing a randoly chosen collection in the su by variables with distributions transfored using the sae orthogonal polynoial syste. For n = 1, 2,..., consider a ulti-index = ( 1,..., n ), and with = ( 1,..., n ) and x = (x 1,..., x n ) let and set α () n = = i, n = i i=1 i=1 n = α ( i) i and P (x) = n i=1 i=1 P i i (x i ). (24) Theore 3.1 Let Z, > 0 be a faily of rando variables closed under independent addition with EZ 2 <, and suppose the associated orthogonal polynoials {P k (x)} 0 k satisfies Condition 3.1 and, for soe weights c, the identity P (w) = : = c P (x), (25) where P (x) is given in (24) and w = x x n. Then α () defined in (22) and (24) respectively, satisfy α () = : = and α () c α (), (26) and we ay consider the variable I, independent of all other variables, with distribution α P (I = ) = c () α (), =. (27) 12

13 Furtherore, for any positive 1,..., n and independent variables X 1,, X n with X i M i and n W = X i, i=1 the variable has the W P distribution. W () = : = (X i ) (I i) i Proof. Since X i M i, we have for 0 k 2, and independent Z i L i and Z L, that n n EW k = E( X i ) k = E( Z i ) k = EZ. k i=1 i=1 Hence W M, and the W () distribution exists by Corollary 3.1. Equality (26) follows by ultiplying (25) by W = ( i X i ) taking expectation, and using independence and orthogonality. By (26), for any F C c, α () EF () (W () ) = E Using independence and successively applying the identity c α () F () (W () ). (28) α ( i) i EF (q) ((X i ) ( i) i + y) = EP i i (X i )F (q i) (X i + y), (29) we see that the right hand side of (28) is equal to E c P (X)F (W ) = EP (W )F (W ), (30) by (25). Coparing (28) to (30) we have α () EF () (W () ) = EP (W )F (W ), for all F C c, and hence W () has the W -P biased distribution. 13

14 For the possibly infinite syste of onic polynoials {P (x)} orthogonal with respect to L, define the generating function φ t (x, ) = 0 P (x) t!. (31) Though the constants α () can be found using F (x) = x in (23), squaring (31) and taking expectation using orthogonality gives the alternative ethod E[φ t (Z, )] 2 = 0 α () t 2!. (32) Theore 3.2 applies in the special cases considered in Sections 4.1 through 4.4. Theore 3.2 If the polynoial generating function φ t (x, ) in (31) satisfies n φ t (w, ) = φ t (x i, i ) (33) i=1 for w = x x n and = n, then (25), and hence (27), in Theore 3.1 are satisfied respectively by c = ( ) Proof: Rewriting (33), 0 and P (I = ) = t! P (w) = = = n i=1 i 0 P 1,, n t! 0 ( ) () α α () P i i (x i ) t i i! =, =. (x) t1+ +n 1! n! ( ) P (x), giving (25) with the values claied. We also note that squaring (25) and taking expectation, using independence and orthogonality, results in α () = = ( ) 1 c 2 α, (34) 14

15 so that the conclusion of Theore 3.2 can also be seen to hold by equating coefficients of (26) and (34) when α takes on sufficiently any values. We end this section with a result about the potential for iterated biasing. Theore 3.3 Let Condition 3.1 be satisfied, and suppose that the the distributional faily at Z is closed under transforation with respect to P k (x), that is, there exists µ(, k) such that (Z ) (k) = Z µ(,k). Then if X M we have X (k) M k µ(k,) for k. In particular for non-negative j with 0 k + j, the distribution (X (k) )(j) µ(,k) exists. Proof. Let 0 j 2( k) and F (x) = x k+j /(k + j) k, where (x) k = x(x 1) (x k + 1). Then α (k) E(X(k) = EP k (Z )F (Z ) = α (k) )j = α (k) EF (k) (X (k) ) = EP k (X)F (X) EF (k) ((Z ) (k) ) = α(k) E(Z µ(,k)) j. Thus the first 2( k) oents of X (k) atch those of Z µ(,k), and the existence of the distribution (X (k) )(j) µ(,k) follows fro Corollary Special Orthogonal Polynoial Systes In Sections we specialize to the classic Herite, Laguerre, Charlier, Krawtchouk and Gegenbauer orthogonal polynoial systes, corresponding to the Noral, Gaa, Poisson, Binoial and a Beta like faily, respectively. All these failies correspond to a collection of orthogonal polynoials satisfying Condition 3.1, and except for the last case, have a generating function which satisfies (33). The Noral and Poisson distributions are fixed points of their associated transforations. In the Gaa, Binoial and Beta-type cases the transforations ap to the sae faily, but with a shifted paraeter. For further connections between probability distributions and such polynoial syste generating functions, see [2] and [3]. 15

16 4.1 Herite Polynoials For σ 2 = > 0, define the collection of Herite polynoials {H (x)} 0 through the generating function or equivalently, the Rodriguez forula e xt 1 2 t2 = H (x) t =0!, (35) H (x) = ( ) e x2 2 d x 2 dx e 2. (36) These polynoials are orthogonal with respect to the noral distribution N (0, ) with density (2π) 1/2 exp( x 2 /(2)). For F Cc and Z N (0, ), applying the Rodriguez forula (36) we have Hence, EH (Z )F (Z ) = = = ( ( ) e x2 d 2 ( ) ( d x 2 dx e 2 ) x 2 dx e 2 F () (x) e x 2 2 dx 2π ) F (x) e x2 2 2π dx 1 F (x) dx 2π = EF () (Z ). (37) (Z ) () = Z, that is, for each = 0, 1,..., the noral Z N (0, ) is a fixed point of the th order transforation induced by H (x). Fro (37) we see that α () =, which we could find alternatively using (32) and E[e Z t 1 2 t2 ] 2 = e t2 = t2 0!. Now since the generating function (35) satisfies the conditions of Theore 3.2, the distribution of the rando index I in Theore 3.1 is ultinoial 16

17 Mult(, ). For zero biasing and = 1, this ultinoial distribution reduces to the pick an index proportional to variance as specified in (6). Lastly, we indicate two ways in which the classical Stein equation can be generalized to the Herite case. With N h = Eh(Z), the standard noral expectation of h, both the equations and f (x)h 1 1 (x) H 1 (x)f(x) = h(x) N h (38) f () (x) H 1 (x)f(x) = h(x) N h (39) reduce to the usual Stein equation when = 1 (see [21], [22]) f (x) xf(x) = h(x) N h, (40) and in particular the expectations on the left hand sides of each evaluated at a rando variable W are zero for all f Cc if and only if W is standard noral. 4.2 Laguerre Polynoials For > 0, let {L (x)} 0 be the collection of Laguerre polynoials defined by the generating function { } xt (1 + t) exp = 1 + t or equivalently, the Rodriguez forula L (x) t =0!, (41) L (x) = ( 1) x +1 e x d dx x+ 1 e x, (42) which are orthogonal with respect to the Gaa distribution with paraeter, having density x 1 e x /Γ(), x > 0. For F Cc and Z with this density, applying the Rodriguez forula (42) yields EL (Z )F (Z ) = 0 ( 1) x +1 e x ( d dx x+ 1 e x ) F (x) x 1 e x dx Γ() 17

18 = ( 1) Γ() = ( d 0 Γ( + ) Γ() where () is the rising factorial, 0 dx x+ 1 e x ) F (x)dx F () (x) x+ 1 e x Γ( + ) dx = () EF () (Z + ), (43) () = ( + 1) ( + 1) = Γ( + ). Γ() Hence Fro (43) we see that α () using (32) and E (Z ) () = Z +. = (), which we could find alternatively [ ( )] (1 + t) Z t 2 exp = (1 t 2 ) = () t2 1 + t 0!. Since the generating function (41) satisfies the conditions of Theore 3.2, the rando index I in Theore 3.1 has distribution P (I = ) = ( ) ni=1 ( i ) i = () ( ) ni=1 i ( ) i + 1, which we recognize as the ultivariate hypergeoetric distribution with paraeters and ,..., n + n 1, see [18], p.301. Though the Gaa is not a fixed point of the Laguerre transforations as the noral is for the Herites, nevertheless there exist Stein equations for the Gaa paralleling (40) for the noral which can be used for studying distributional approxiations for the Gaa faily; for details, see [20]. In particular, we have the Stein characterization that X Γ(, 1) if and only if E(X )f(x) = EXf (X) for all soooth functions f. Using that L 1 (x) = x, the X (1) order one Laguerre transforation is characterized by E(X )f(x) = Ef (X (1) ) 18

19 for all sooth functions f. Coparing these two equations we see that X Γ(, 1) if and only if for all sooth functions f, EXf (X) = Ef (X (1) ); in other words, X Γ(, 1) if and only if X (1), the first order Laguerre transforation of X, equals its size bias transforation X s. 4.3 Charlier Polynoials For > 0, let {C (x)} 0 be the collection of Charlier polynoials defined by the generating function e t (1 + t) x = C (x) t =0!, (44) or, equivalently, with (x) k = x(x 1) (x k + 1), the falling factorial, C (x) = k=0 ( ) (x) k ( ) k, (45) k giving a faily orthogonal with respect to the Poisson distribution P() with ass function e k /k!, k = 0, 1,.... Fro (45) one can derive the Rodriguez forula ( C (x) = ( 1) Γ(x + 1) x x ), (46) Γ(x + 1) where f(x) = f(x) f(x 1), the backward difference. Since the transforations in Theore 2.1 defined using derivatives of test functions yield absolutely continuous distributions when 1, no discrete distribution will be a fixed point. However, parallel to (9), for an integer valued rando variable X we can define the discrete X P biased distribution via EP (X)F (X) = αe F (X () ) for all F F (P ), (47) where f(x) = f(x + 1) f(x), and again suppressing dependence on X, F (P ) = {F : R R : E P (X)F (X) < }. 19

20 That for all = 0, 1... the Poisson P()-distribution is a fixed point (Z ) () = Z of the discrete transforation (47) with P replaced by C follows. For Z P(), by the Rodriguez forula (46) and can be seen as b k a k = ( 1) b k a k, (48) k=0 k=0 we have EC k (Z )F (Z ) = e k=0 k! C (k)f (k) ( ) = ( 1) e k F (k) k! Fro (49) we see that α () (32) and k=0 = e k F (k) k! k=0 = E F (Z ). (49) =, which we could find alternatively using E[e t (1 + t) Z ] 2 = e t2 = t2 0!. Using the existence of the Charlier biased distributions and that (29) holds with derivative replaced by difference, it is easy to see that the arguent and hence conclusion of Theore 3.1 holds in this discrete case. Now since the generating function (44) satisfies the conditions of Theore 3.2, the distribution of the rando index I in Theore 3.1 is ultinoial Mult(, ), as in the noral case. As the order one Charlier polynoial is C 1 (x) = x, Stein characterizations of the for (38) or (39), with Herite replaced by Charlier and derivatives replaced by differences, generalize the Stein equation for the Poisson distribution with paraeter given in [7], and extensively studied for exaple in [4]. 20

21 4.4 Krawtchouk Polynoials With = 1, 2,... and p (0, 1) fixed, let {K (x)} 0 be the collection of Krawtchouk polynoials defined by the generating function (1 + qt) x (1 pt) x = =0 t! K (x) (50) where p + q = 1, giving the faily of polynoials orthogonal with respect to the Binoial B(, p) distribution. In contrast to the previous exaples, the Binoial is not infinitely divisible and has support on a bounded set. Following the approach set out in [3], the polynoials can also be given by the Rodriguez forula K (x) = ( 1)! ( ) p x q x {( ) ( ) x } p ), x q and so for Z B(, p), 0 and bounded F, ( x EK (Z )F (Z ) = EF (Z ) ( 1)! ( ) p Z q Z ( ) ( ) Z p ) q = =! ( k k=0 ( ) ( Z )p k q k F (k) ( 1)! ( ) p k q k p q ( 1) k=0 Z ( ) k ( F (k) k ) ( p q ( k ) k. ) ( ) k p q Using (48) and letting () again be the falling factorial, we write the last expression as ( ) ( ) () p q k p F (k) k=0 k q ( ) = () (pq) p k q k F (k) k k=0 = α () E F (Z () ), 21

22 yielding α () = () (pq) and (Z ) () = Z. Hence, siilar to the Gaa faily, the Binoial distribution is not a fixed point of its own transforational faily, but the transfored distribution is a eber of the sae faily. One can calculate α () alternatively using (32), (50), and series expansion of E(1 + qt) 2Z (1 pt) 2 2Z = (1 + pqt 2 ). As for Exaple 4.3, the conclusion of Theore 3.1 holds, and since the generating function (50) satisfies the conditions of Theore 3.2 the distribution of the rando index I in Theore 3.1 is given by P (I = ) = ( ) (pq) i ni=1 ( i ) i () (pq) = ( ) ni=1 ( i ) i = () ni=1 ( i i ) ( ), which we recognize as the ultivariate hypergeoetric distribution with paraeters and 1,..., n, see [18], p.301. Fro [10] we have the Stein characterization that X B(, p) if and only if pe( X)f(X + 1) = qexf(x), for all functions f for which these expectations exist. Using the first Krawtchouk polynoial is K 1 (x) = qx p( x), we obtain that the first order Krawtchouk transforation is characterized by qexf(x) pe( X)f(X) = pqe f(x (1) ). Cobining these equations yields that X B(, p) if and only if pe( X) f(x) = pe( X)(f(X + 1) f(x)) = qexf(x) pe( X)f(X) = pqe f(x (1) ). Putting g(x) = f( x) we see that X B(, p) if and only if X (1) has the ( X)-size biased distribution, that is, if and only if X (1) B( 1, q) + 1, which is equivalent to X (1) B( 1, p). 22

23 4.5 Gegenbauer Polynoials In this last section, we consider a polynoial syste orthogonal with respect to a continuous distribution with copact support. For > 1, let (see [3]) 2 α () = Γ()Γ(2 + )Γ( + 1) 2 2 Γ( + + 1)Γ( + )Γ(2), and the collection of Gegenbauer polynoials G (x) be defined via the Rodriguez forula G (x) = ( 1) α () Γ( + 1 )Γ( + + 1) 2 Γ( + 1)Γ( + + 1)(1 x2 ) 1 d 2 dx {(1 2 x2 ) }. Then G (x) are onic, have degree, and satisfy the orthogonality relation where Z g with 1! EGk (Z )G (Z ) = α () δ k, k = 0,...,, g (x) = 1 Γ( + 1) π Γ( + 1)(1 x2 ) 1 2, x 1. 2 In particular Corollary 3.1 obtains, proving the existence of the faily of Gegenbauer transforations. This faily of distribution is a special case of the centered Pearson Type I-distributions, soeties also called Beta Type I-distributions, see [23], p.150. We note that for = 1/2 we obtain the unifor distribution U[ 1, 1], for = 0 the arcsine law, and for = 1 the sei-circle law [25], [26]. Considering the action of the G transforation on Z g, for F Cc we have EG (Z )F (Z ) = 1 Γ( + 1) π Γ( + 1 α () Γ( + 1 )Γ( + + 1) )( 1) Γ( + 1)Γ( + + 1) 2 1 = 1 α () Γ( + + 1) 1 π Γ( + + 1) F () (x)(1 x 2 ) = α () EF () (Z + ), F (x) d dx {(1 x2 ) } 23

24 yielding (Z ) () = Z +. Thus, for = 0 we obtain that the first order Gegenbauer transforation of the arcsine distribution is the sei-circle law. Lastly we note that since the above Beta-type distributions are not invariant under addition, Theore 3.1 and its construction do not apply. However, as G 1 (x) = x, we recognize the first order Gegenbauer transforation as the zero-bias transforation, so that for sus of independent rando variables the construction given in (7) applies. Acknowledgeent. The authors would like to thank the organizers of the progra Stein s ethod and applications: A progra in honor of Charles Stein and the Institute of Matheatical Sciences in Singapore for their ost generous hospitality, and the excellent eeting where this work was copleted. Also we would like to thank an anonyous referee for very helpful coents. References [1] Abraowitz and A.A. Stegun, eds. (1972) Handbook of Matheatical Functions. M Dover, New York [2] Asai, N., Kubo, I., and Kuo, H. (2003). Multiplicative renoralization and generating functions I. Taiwanese Journal of Matheatics 7, [3] Asai, N., Kubo, I., and Kuo, H. (2004). Multiplicative renoralization and generating functions II. To appear in Taiwanese Journal of Matheatics. [4] Barbour, A.D., Holst, L., and Janson, S. (1992). Poisson Approxiation. Oxford University Press. [5] Bolthausen, E. (1984). An estiate of the reainder in a cobinatorial central liit theore. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, [6] Cacoullos, T., Papathanasiou, V., and Utev, S. (1994). Variational inequalities with exaples and an application to the central liit theore. Ann. Probab. 22,

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