natural exponential families of probability distributions and exponential-polynomial approximation

Natural Exponential Families of Probability Distributions and Exponential-Polynomial Approximation

Clyde Martin and Victor Shubov

Department of Mathematics

Texas Tech University

Lubbock, TX 79409

ABSTRACT

We consider expansions of regular functions in series with respect to infinite systems of Dirichlet polynomials, i.e., combinations of polynomials and exponents. We show that for every natural exponential family of probability measures on a real line and an infinite sequence of complex numbers, whose real parts belong to the natural parameter space of the above family, one can associate an expansion of any sufficiently regular function with respect to certain Dirichlet polynomials. We give an explicit formula for the remainder term in this expansion. It turns out to be an expectation of convolution of one of the above Dirichlet polynomials with certain differential operator of the function. The Dirichlet polynomials in question are combinations of exponents and Appel polynomials generated by members of the exponential family of distributions. Dirichlet series and expansions with respect to Appel polynomials are particular cases of the above expansions.

1. INTRODUCTION

Approximation of regular functions by Dirichlet polynomials is one of the classical problems of the theory of functions which is also of considerable interest in numerical analysis. (See, for example, monograph [2] for detailed exposition of the subject and list of references.) A Dirichlet polynomial is a finite linear combination of the functions

eAkx, XeAkX,. . . , *mk-lehx (k = 1,2,...),

where {@y= i is a sequence of complex numbers and {m,]T= i is a sequence of positive integers (multiplicities). Expansions with respect to systems of exponents and polynomial expansions are particular cases of expansions with respect to Dirichlet polynomials.

APPLIED MATHEMATICS AND COMPUTATION 59:275-297 (1993)

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276 C. MARTIN AND V. SHUBOV

In this paper we continue the work initiated in our paper [l]. Generalizing a beautiful idea of 0. V. Viskov ([3]), we have shown in [l] that for every probability measure on a real line with finite moments there exists a naturally associated sequence of polynomials which are, in fact, classical Appel polynomials. We derived an explicit formula for the remainder term in the expansion of any smooth function in series with respect to these polynomials. In this paper, we generalize these results to the case of expansions with respect to Dirichlet polynomials.

In order to formulate our results, we have to recall some basic facts about Appel polynomials and main results of [l].

Let { cr,,}rZO be a sequence of complex numbers. Appel polynomials corresponding to this sequence are defined by the formula

It is clear that the same polynomials can be uniquely defined by conditions

(n = O,l,...).

PA+1(x) = P,(x), P,(O) = ff,, (n = O,l,...).

Introduce formally the function A(t) = Cz= a o!,tn. Then A(t)e’” is a generating function for Appel polynomials, i.e.

A(t)etx = 2 P,( x)t”. n=O

It was shown in [l] that it is useful to establish certain natural connection between systems of Appel polynomials and probability measures on a real line. Let Al. be such a measure. Assume that the moment generating function

F(t) = E,( et”) = (m etXdp( x) (1.1)

exists for t E 0, where 0 is certain interval of a real axis. (By Ef = E,f(x)

we denote the expectation of a function f with respect to p.) We associate to

Exponential Families of Probability Distributions 277

the measure I_L a sequence of Appel polynomials defined by the generating function F(t)P’e’x, i.e.

F(t)-‘ef* = 5 P,(x)t” n=O

and

(1.2)

The same polynomials can be uniquely characterized by the following conditions

PIi+, = P,(x) (n 2 O), P,( x> = 1, (1.4)

EP,, = 0 (n > 1). (1.5)

(1.4) is obvious. Taking expectations of both sides (1.2), we obtain 1 = I~=O(EPn)tn, which implies (1.5). It was shown in [l] that using (1.41, (1.5) it is possible to give explicit formulas connecting the numbers CY, = P,(O) and the moments of the measure /..L. These formulas together with the solution of the classical moment problem imply that the correspondence between measures p and polynomials (1.3) is one-to-one.

It is easy to see from (1.41, (1.5) that the polynomial P,,(x) (n = 0, 1,. . .) given by (1.3) can be defined as the unique solution of the following problem

c)(lI+ 1) = 0 (1.6) Ev = Eu’ = . . . = E+“-1) = 0, J+(n) = 1. (1.7)

(uCk) means the k th derivative of 0.) This definition turns out to be useful. Using it and the idea of Viskov [3] we proved in [l] the following result. Any smooth function can be represented in the form

f(x) = 5 E(f’“‘)P,(x) + E,/*P& + 77 - t)fcN’(t) dt (1.8) n=O 7

formula us expansion a f respect polynomi- P,(x) an expression the term.

particular of (1.8) are discussed in [l].


In this paper, we prove a generalization of (1.8) to the case of expansions with respect to Dirichlet polynomials. We would like to mention that here we study only formal properties of these expansions and do not consider esti- mates for remainder terms.

In concluding the introduction, we recall the definition of natural exponential families of probability measures (see [4]) and corresponding Appel polynomials. This information will be used in the following.

A probability measure CL, whose moment generating function (1.1) is defined on an interval 0, generates a natural exponential family of distributions

d/_$(X) = F(B)-‘eexdp(.), OE 0. (1.9)

The moment generating function of pe is

= F(B)-'F(t+ 0). (1.10)

We denote the corresponding Appel defined by the expansion

polynomials by I’,( X, 0). They are

F,(t) -let’ = i P,( X, 6)t”. n=O

It follows from (1.11) and (1.10) that

P,(x, 0) = $ i n{Fo(t)e’e’“}I,_,, .( 1 =- ij(i)n{F(t + e)e("fB)x}lt=,F(B)e-e"

=- ~~(-$)"{F(B)~'eR'JF(B)e~s'

(1.11)

(1.12)

We notice, that the function F(8) given by (1.1) can be continued as a regular function into a strip of a complex plane defined by the condition Re0 E 0. The measure /_L* and Appel polynomials P,(x, 0) are well defined by (1.9) and (1.12) for any complex 0, such that Re0 E 0 and F(8) # 0.


However, in the case of complex 8 the measure pLe is no more a probability distribution, it is a complex valued measure. In the following we will use the notation

for “expectation” with respect to pe even if 0 is not real

2. FORMULATION OF MAIN RESULTS

In this section we fix a probability measure or. on a real line and use the same assumptions and notations as in the introduction.

Let .{ hi}:= i be an infinite sequence of complex numbers such that Rehi E

0 and F(hi) # 0, i = 1,2, . . . . These numbers are not obligatory all distinct. Define a sequence of differential operators

L,=fi &hi) i 1 n = 1,2,... . i=l

(2.1)

We consider now the following problem: for any positive integer n, determine a function u(x) which satisfies the conditions

L,u = 0, (2.2)

Eu = Eu’ = . . . = ,&(“-2) = 0, Eu(“-l) = 1. (2.3)

Notice, that (2.3) is equivalent to the conditions

Eu = E( L,u) = ..a = E( L,_,u) = 0, E( L,_,u) = 1. (2.4)

It will be shown later (see Theorem 2.2) that the solution of the problem (2.21, (2.3) or, equivalently, (2.21, (2.4) is unique. Denote this solution by @Jr>, n = 1,2,. . . . We are now in a position to formulate our first result.


THEOREM 2.1. Any smooth function f can be represented in the form

f(x) = ~~~E(L,-,fR(4 + E,j*%(x + 77 -t)(&J-j(t) dt. (2.5) 7

This theorem is a direct generalization of the result from [l]. Really, if all A, = 0, the problem (2.2) (2.3) coincides with (1.6), (1.7), Qn(x) = P,_,(x) and the representation (2.5) turns into (1.8).

Our next result deals with an explicit formula for Qn(x) in terms of A’s and the measure /L. Before we formulate it, let us consider briefly the case when all Ai are equal: Ai = 8, i = 1,2, . . . ; Re0 E 0, F(B) # 0.

In this case it is natural to look for a solution of the problem (2.2) (2.4) in the form U(X) = e”g(x). After substitution we obtain the following problem for v:

lP( x) = 0 (2.6)

E(@, = E(B)V’ = . . . = j$@,(n-2) = 6, E(tijV(“-r) = F(o)-‘. (2 7)

Comparing (1.6), (1.7) and (2.6) (2.7) we see that V(X) = F(8)-‘P,_ ,(x, 01,

where P,_ I( X, 0) are Appel polynomials corresponding to the measure pe and defined by (1.12). We can now substitute D,,(x) = F(0)-lesrP,~l(x, 0)

into (2.5). If we also introduce the function g(x) such that f(r) = eexg(x),

then (2.5) can be easily transformed to the form

g(x) = f E(B)(g(n-l))P,_l( x, e) n=l

+ Ey’ jxPN- 1( x + 77 - t, e)&“)(t) dt D

(2.8)

(2.8) is the same expansion as (1.8). The only difference is that now we deal with the measure we and the corresponding polynomials P,( X, 0) instead of Al. and P,,(X).

The above remark suggests that the most interesting case is when each hi can occur in the sequence {Ai};= I only finite number of times. From this moment we assume that in the sequence {hi}:= r some consecutive numbers may be equal but the number of equal members is always finite. In this case, we introduce a new sequence { t&}y= 1 which consists of the same numbers,


but now all 8, are distinct and we assign to each Ok its multiplicity mk. We

are in a position to formulate our second result.

THEOREM 2.2. Let n = Cizlrnk and the operator L, defined by (2.1)

has the form

(2.9)

where all 8, are distinct, Reek E 0, F(B,) # 0. Then the problem (2.2>,

(2.4) has a unique solution a,,(x) = Qn,< x; 8,, . . . , es>, which is given by the

formula

CD,(X) = i Qk(x; e,,...,es)F(ek)-lee~x. k=l

(2.10)

Here Qk(x; e,,..., 6,) are polynomials with respect to x defined by

m-1

Qk(x;el>...> es) = C c!k)(e, ,..., es)pi(x, e,), (2.11) i=l

where P,( x, e,) are Appel polynomials corresponding to the measure ,uoLe,:

P,(x, 0,) = A G i(F(B)-leL’}l,=BF(Bl)e~““x, i i

i = O,l,..., mk - 1. (2.12)

The coejkients C!k’(8,, . . . , 0,) have the form

c,‘~)( e,, . . . , 0s) = crnk -

i = O,l,..., mk - 1. (2.13)

The formula (2.9) shows that Q,(X) are Dirichlet polynomials of a special type. So, representation (2.5) is, in fact, an expansion of a function f with


respect to certain Dirichlet polynomials in which we have an explicit formula for the remainder term.

Let us discuss now the corollary of the above theorems in the case when all multiplicities mk = 1, k = 1,2,. . . , i.e. all numbers in the sequence {hi}:= r are distinct.

COROLLARY 2.1. In the case when all A, (k = 1,2,. . .) are distinct, the solution Q”(x) of the problem (2.2), (2.3) is given by the formula

Q,,(X) = fi (A, - Aj)-l F( hk)-leAkx (2.14) k=l j=l

j+k

Substituting (2.14) into (2.5) and changing the order of summation we obtain

f(x) = f ciN)eAkr + E, ‘DN( 1c + 77 - t)( LNf)(t) dt, (2.15) k=l

/ 17

where

ciN) = F(Ak)-l $ fi(A, - Aj)-‘E(L,_,f). n=kj=l

j#k

(2.16)

We see that in the case of unit multiplicities Theorems 1 and 2 give us an approximation of any smooth function by linear combinations of exponents and an explicit expression for an error.

To combine Theorems 1 and 2 in the general case of arbitrary finite multiplicities we should introduce additional notations. Recall that we replace the original sequence {hi}:= r by the sequence {Ok}~=r which consists of the same numbers, but now all numbers are distinct and mk is the multiplicity of 8, in the sequence (A$‘= r.

For every integer n 2 1 define an integer s(n) > 1 by the rule:

s(n)- 1 s(n)

C m, <n, Cm,>n. i=l i=l

For every integer k 2 1 define

t(k) = min{nls(n) = k}.


it is clear that s(t(k)) = k for every k > 1. Now the relation between sequences {A$‘= r and {Ok}:= r can be described as follows:

A, = Q+ 8, = $k) = At(k)+1 = *** = *t(k)+fn-l; k,n = 1,2 ,... .

For every n we introduce

s(n)- 1 rii,=n- C mi*

i=l

Now we can write the solution of the problem (2.21, (2.3) in the form

s(n)

W4 = R(x; k..., %(J = C Qk(x; e,,..., es(nJF(ek)-lee~~, k=l

where

Qk(X;h..T es(,)) = CC) (0 l,...,e,(,))Pi(X,ek),

and the summation is from i = 0 to i = mk - 1 in the case 1 < k < s(n) - 1

and from i = 0 to i = hi, - 1 in the case k = s(n). The coefficients Cik’ are given by (2.13). Combining these two formulas we obtain

s(n)-1 q-1

Q,,(X) = c c c:k’(&..., os(,))‘i(‘, ek)F(ek)-les”x

k=l i=O

i=O

Notice that the operator L, can be written in the form

Now we can substitute (2.17) into (2.5). Changing the order of summation we arrive at the following result.


COROLLARY 2.2. Eve y smooth function f can be expanded with respect to the Dirichlet polynomials Pi( x, Bk)eekr as follows:

s(N)-1 q-1

f(x) = kFI c A$y’P,(x, %)eBkX i=O

N i,-1

+ C C BniPi( X, BSC,))ees(n)~ n=l i=O

+ E, / ( ‘@, x + TI - t>( &f)(t) dt, (2.18) 7

where

A\:) = qe,j-’ l’“‘, %(,,)E(L,f), (2.19) n=t(k+l)

kj = F(e~(,,)~‘c!s(n))(e,,..., e,,,,)E(k-,f) (2.20)

3. PROOF OF THE THEOREM 1

In this section we prove Theorem 1. It will follow, in particular, from this proof that the solution of the problem (2.2) and (2.3) [or (2.4)] exists and is unique. In fact, we will give an explicit construction of this solution by means of a certain inductive process. The proof will be divided in several steps.

Step 1. Let A be a complex number such that Reh E 0 and F(A) # 0. Consider the following problem: find u such that

(3.1)

Eu = 0, (3.2)

where f is a given function of x.

LEMMA 3.1. The solution of (3.11, (3.2) is given by the formula

u(x) = F( A)-l EV/‘eA(r+7-t)f( t) dt. I)

(3.3)

Exponential Families of Probability Distributions

PROOF. The general solution of (3.1) has the form

285

u(x) = jxe *(“-“‘f(t) dt + Ce*', 0

(3.4

where C is an arbitrary constant. Substituting (3.4) into (3.2) we easily find that

C = -F( A)-‘E,/‘eAcq-‘lf(t) dt. 0

Substitute this expression for C into (3.4) and recall that F(h) = E,(eAV). We have

u(x) = /xe”‘xe’)f(t) dt - F( h)~‘e”“E~~~e*(~ - t)f(t) dt 0

= F(A) -’ ET( ehn)~xeA~z~f)f( t) dt - En~‘eAcx+v-‘if( t) dt] [

Eq_/xe”(x+n-t)f( t) dt - En_/‘eA(*f~-ilf( t) dt 0 0 I

= F( A)-lEq~xeA~X+n~‘)S(t) dt.

11

The lemma is proved. Step 2. For any A such that Reh E 0, F(A) # 0 introduce two operators

bA=-&A (3.5)

(u*f)(x) = F(A)-lE,~XeMX”~t)f(t) dt. 11

(3.6)

It follows from the above lemma that a,f is the solution of the problem (3.1),

(3.2). Therefore, bhahf = f for any f, and uAbhu = u if Eu = 0. These relations mean that a, is inverse to the restriction of bA to the subspace of functions u such that Eu = 0.


Let us compute a,b,u for arbitrary u. We have

ahbAu = F(h)-’ Es_/-xeA(x+~~‘)(u’(t) - Au(t)) dt 11

= F( A)-‘E,e *(*+qU(X)e-“X - u($eP”]

= F(A)-‘[E,(e%(x)) - E,(e*“u(v))]

= u(x) - F( A)-‘eA”Eu.

So, the relations between a, and bA can be represented in the form

bh%f = f) (3.7)

[(I - a,b,)f]( x) = F( A)-‘e*“Ef, (3.8)

for any f. Here Z is an identity operator. Step 3. Now we come back to the original sequence {hi}y==l. Denote

a, = aAc, bi = bA , i = 1,2,. . . . Notice, that the operator L, given by (2.1) can be written in the form L, = b, . . . b,. Let cp,( x) = eAnx, then (3.8) for a,,, b, can be written in the form

(1 - a,b,)f= F(kV’(Ef)cp,. (3.9)

Consider, following the Viskov’s idea (see [3], [l]), an obvious relation

F (al...a,_lbl...b,_l - a,...a,b,...b,) = Z - a,...a,b,...b,, n=l

(3.10)

where we set a, = b, = I. Applying both sides of (3.10) to an arbitrary smooth function f, taking into account (3.9) and the fact that bi bj = bj bi,


a,aj = aja, (i,j = I,&. . .>, we obtain

f= 5 (ul...a,_lbl...b”_l - a,...a,b,...b,)f+ a,...u,b,...b,f 7I=l

= ; E(L,~,f)F(A,)-lu,...u,_,(q+,) + q...a,,,(LNf). (3.11)

n=l

Step 4. To complete the proof of the formula (2.15) it remains to prove the following result.

LEMMA 3.2.

a> The solution @, of the problem (2.21, (2.4) is unique and can be represented in the form

Qn = F(A,)-‘u,...u,~,(cpn), (3.12)

where q,,(x) = eAnX.

b) For any f the following formula is valid

(al... a,f)( x) = E,/‘Qm( x + 77 - t)f(t) dt. (3.13) V

PROOF.

a> Let us apply the operator a, to both sides of the equation (2.2). In the left hand side, we have

u,L,u = anbnL,-Iu = LnPlu - F(A,)-lE(L,_lu)(p,

= L n-1~ - F(A,)-‘cpn.

We used here (3.9) and the fact that E(L,_ ,u> = 1 due to (2.4). We can


conclude that the problem (2.21, (2.4) is equivalent to the following problem

L”_iU = F(A,))‘% (3.14)

Eu = E( L, u) = a.. = E( L,_,u) = 0. (3.15)

Now apply the operator al . . . a,_ 1 to both sides of (3.14). Taking into account that a,b,w = o (i = 1,2,. . . ) for any v such that Ev = 0 we immediately obtain that u = a,, given by (3.12). Notice, that we have shown the uniqueness of the solution.

b) Notice, that the function u = a,. . . u,f in the left hand side of (3.13) is the unique solution of the problem

L,u =f, (3.16)

Eu =E(L,u) = ... =E(L,_,u) =0 (3.17)

To show that it is sufficient to apply the operator aI.. . a, to both sides of (3.16) and reason like in the above case of (3.14), (3.15). So, to prove (3.13) we have only to show that the function in the right hand side of this equality satisfy both 13.16) and (3.17). Denote this function by o:

( r ) = EJW x + 77 - t)f(t) dt. rl

(3.18)

Let us show first that

(Lp)(x) = (b,... bkv)( x) = EqLxth (f - 4)@rt(’ + n - t)f(t)dt,

k = 1,2 ,..., n - 1. (3.19)

To prove (3.19) we use induction. Apply b,, 1 (k = 0, 1, . . . , n - 2) to both sides of (3.19), if k 2 1, or to both sides of (3.181, if k = 0. In both cases we


have similar computations;

We have taken into account the fact that E( L,@,,) = 0 (k = 0, 1, . . . , n - 1) due to (2.4). (L,) = Z-the identity operator). So, (3.19) is shown.

Now we are ready to verify that L,v = f. Let us take (3.19) with k = n - 1 and apply to both sides the operator b, = d/dx - A,,. We have

cLv)(x) = q( $ - A,)/:;_ (f - +?L(X + 77 - t>f(t) dt

We used here the fact that E( L,_ ,@,J = 1 and L,@, = 0 due to (2.2) and (2.4).

To prove that u satisfies (3.17) let us take expectations of both sides of (3.19) (or (3.181, in the case k = 0). We obtain

E(U) = E,E,/*(LlW@ + rl- t)f(t> dt 7

= EXE, [/

x(L@,,)(~ + 77 - t>f(t> dt

‘,c )( ’ Lk@n x + 77 - t)f( t) dt = 0 0 1

The last step here is due to symmetry with respect to transposition of x and

rl. This completes the proof of Lemma 2 and, therefore, of Theorem 1.

C. MARTIN AND V. SHUBOV 290

4. PROOF OF THEOREM 2

We would like to begin with an easy independent proof of Corollary 1, i.e. with a derivation of the-formula (2.14j in the’ case when all A, (k = 1: 2, . . . are distinct.

The general solution of the equation (2.2) in this case has the form

k=l

where Cl”) are constants. To find these constants

Cramer’s rule. It gives us

F( A,)Cf”’ = DI”‘/D’“‘, (4.3)

where DC”’ is a Vandermond-type determinant of the matrix in the left hand side of (4.2) and Dp) is the determinant of the same matrix with kth column replaced by the right hand side column. It is well known that

DC”’ = I-I (Aj - Ai), (4.4) l<i<j<n

which is not zero, because all Ai are distinct according to the initial assumption. Expanding Dp’ with respect to the k th column, we see that the only nonzero cofactor is again a Vandermond-type determinant. So, we have

Dz”’ = ( _l)n+k II ('j - ‘i) (4.5) lai<j<n

i,j+k


Substituting (4.4) and (4.5) into (4.3) we obtain

Cp) = F(Ak)-‘(-l)“+k l,gcn(Aj - hi)lcQcntAj - ‘i)-l . i,j#k

= F(A,)_l( -l)fl+k fi (5 - A,)-“G(Ak - hJ)-l j=k+l

= F(Ak)-’ Jgl(Ak - h,)-‘, j#k

which together with (4.1) gives us (2.14).

Now we turn to the proof of Theorem 2 in the general case of arbitrary finite multiplicities.

In this case the general solution of the equation (2.2), where the operator L, is given by (2.9), has the form

u(x) = i qk( x)egkx, k=l

(4.6)

where 9k(x) is an arbitrary polynomial of degree mk - 1, k = 1, . . . , s. The problem is to find coefficients of these polynomials using conditions (2.3) or (2.4). Substitution of (4.6) into (2.3) or (2.4) would give us a linear system for the above coefficients. However, now we will not try to write the corresponding system and find its solution using Cramer’s rule. The reason is that the determinants are now very complicated.

Notice, that we can look for polynomials 9k(x) in the form

T?Q-I

qk( x) = F( ek)-l c c,!“‘pi( x, ok), k = l,...,s, (4.7)

where C!k) are unknown coefficients and I’{( x, 0,) (i = 0, 1, . . . , mk - 1) are Appel pdlynomials corresponding to the measure /_L~, and defined by (2.12). It is possible to represent 9k(x) as a linear combination of the above Appel polynomials, because these polynomials obviously form a basis in the space of all polynomials of degree less or equal than mk - 1.

The problem now is to find coefficients C,!k’. We have to show that the conditions (2.3) or (2.4) will be satisfied if C,!k’ are given by (2.13). To do it

292 C. MARTIN AND

let us replace (2.31, (2.4) by the following system of conditions

E ($-ek)'$(~-ej)mju

jzk

1 = 0, 1,. . . , mk - 1; k = 1,. . ., s. (4.8)

V. SHUBOV

(Here 6 mt _ i, l is the Kronecker delta.) (4.8) obviously follows from (2.3). We are going to substitute (4.61, (4.7) into (4.8). This substitution will give us a linear system for coefficients Ci ck) We will solve this system and show that its . solution is given by (2.13). It will be sufficient for the proof of the theorem, because we know from Section 3 that the solution of the problem (2.2>, (2.3) or (2.4) exists and is unique. So, it will not be necessary to verify that the obtained solution satisfy (2.3) (or (2.4)) or to prove that (4.8) is equivalent to one of these conditions.

The advantage of the system (4.8) by comparison with (2.2) or (2.3) consists in the following. If we substitute (4.6), (4.7) into the conditions (4.8) with fixed k, we will obtain a closed system of mk equations for mk unknowns C!k’, i = 0, 1,. . . , mk - 1, with given k. In other words, (4.8) splits into s independent systems. To see this it is sufficient to notice that

because deg qj = mj - 1. Therefore, the substitution of (4.6) into (4.8) gives us

l=o,l,..., mk-l. (4.9)

From this moment we will consider (4.9) for certain fixed k.


Now we have to substitute (4.7) into (4.9). Notice, that

= F( ey T<lC!k)[ (%)‘P,(x, t9k)]eekr

= F( Ok)-’ C Clk)Pi_,( x, Ok)eekX i=l

Wh-l-l

= F( q-l C Cft)P,( x, ek)eokx. i=O

We used here the fact that P,l(x, 0,) = E’_~(x, 6,) (see (1.4)). Taking into account the above relation we obtain from (4.9) the following triangle system for coefficients C!k):

Here

l=O,l,..., m,-1.

- ej ( pi( x, ek)eoL*) . I

(4.10)

(4.11)

Now we make the foIlowing observation.

LEMMA 4.1. the form

The coefficients cq(k) given by (4.11) can be represented in

, i = 0,l , . . . . mk - 1. (4.12)

0= 81_


Let us complete the proof of Theorem 2 first, using this result. Then we will come back to the proof of the lemma.

Consider the following obvious relations:

Z=O,l,..., mk-1. (4.13)

Let us differentiate the left hand side of (4.13) using the Leibnitz rule for multiple derivative. We have

= 4fL-lJ; Z=O,l,..., mk-1. (4.14)

Dividing both sides of (4.14) by (mk - 1 - I>! and taking into account that

‘%n, - 1, l/cmk - 1 -I>!= f!&+_r l we obtain relations which coincide with (4.10), where cx,!“) are given by (4.12) and

i 7G Ce - ej)p"' I . (4.15)

‘jzk 0= Ok

Replacing here i + Z by i we see that (4.15) coincides with (2.13). We have shown that equations (4.10) and, therefore, (4.91, (4.8) are satisfied if the coefficients C!k) are given by (2.13). As it was mentioned above, this is sufficient for the proof of Theorem 2. It remains to prove the Lemma.


PROOF OF THE LEMMA. We have to transform (4.11) to (4.12). Notice, that for any function ti

Ex[ F( 0) -%?‘o( x)] = EC%,

where E(O) is the “expectation” with respect to the measure ps. Therefore, we can rewrite (4.11) in the form

a(k) = E(6) 1 x

e-O,r (Pi< x, Bk)&X)

1

(4.16)

j#k

Now we have

Therefore,

JfJ(g -fjj)ml= c i_I (mj)! jfk (y ,,... Gk ,..., 4,):;: tmj - 9j)!(9./)!

. (4.17)

Here the summation is over the set of all ordered (S - l)-tuples of integers

(91,. . . > &, . . . , qs), where the symbol tk means that 9k is missing, 1 < sj < mj (j z k), and we denote

9(k) = i: 9j’ j=1 j#k


Let us substitute (4.17) into (4.16). Before we do it, notice that

Really,

e-Oir ($ - Bkju(‘)(Pk(X, Ok)eOkX) = ( $)9(*)pi(xa ek)

P. I-lCkj(~, 0,) when i 2 9(k); =

0 when i <q(k).

To show (4.18) it remains to recall that

according to one of the main properties of Appel polynomials [see (1.5)]. Substituting (4.17) into (4.16) and taking into account (4.18), we obtain

a!k’ = 1 (8, - 6j)m’-9f. (4.19)

Here the summation, like in (4.171, is over the set of all ordered (s - I)-tuples

(91,. . . > <k>. . . , 9,) with 9k missing, 1 f 9j < mj (j # k), and, in addition to the case of (4.171, satisfying the condition

9(k) = k 9j = i. j=l

jfk

Now we are in a position to complete the proof of the Lemma. Let us show that (4.12) can be transformed to the form (4.19). Using Leibnitz rule for computation of the multiple derivative of the product, we can rewrite

Exponential Families of Probability Distributions

(4.12) in the form

297

a!k) = * (4.20)

where the summation is the same as in (4.19). To show that (4.20) coincides with (4.19) it remains to notice that

i it ; q’ (0 - ej)“Lye=e, = ($“,‘,), (e, - ejy-:

I .

The proof of the Lemma and, therefore, of Theorem 2 is completed.

REFERENCES

C. F. Martin and V. Shubov, Probability measures, Appel polynomials and polynomial approximation. Appl. Math. Comp. (1992).

A. F. Leont’ev, Sequences of polynomials of exponentials. Nauka, Moscow, 1980 [in Russian]. 0. V. Viskov, A noncommutative approach to classical problems of analysis. Proceedings of the Steklov Institute of Mathematics 4:21-32 (1988).

C. N. Morris, Natural exponential families with quadratic variance functions. Ann. Statist. 10:65-80 (1982).

C. F. Martin and V. Shubov, Distributions with quadratic variance and orthogonal

polynomials. Preprint (1991).

natural exponential families of probability distributions and exponential-polynomial approximation

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