6.
7.
8.
9.
i0.
B. V. Gnedenko and A. N. Kolmogorov, Limit Distribution for Sums of Independent Random
Variables, Addison-Wesley (1968).
H. Cramer, Mathematical Methods of Statistics, Princeton Univ. Press (1946i.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series.
Elementary
Functions [in Russian], Nauka, Moscow (1981).
I. S. Gradshtein and I. M~ Ryzik, Tables of Integrals, Series and Products, Academic
Press (1966).
M. Abramovits and I. Stigan (eds.), Handbook on Special Functions, Dover (1964).
ASYMPTOTIC EXPANSIONS FOR DISTRIBUTIONS OF SUMS OF INDEPENDENT
RANDOM ELEMENTS OF A HILBERT SPACE
UDC 519.214
V. Yu. Bentkus
Let Sn be the sum of n independent random elements of the Hilbert space H, w:H + R i be
a second-order polynomial.
In this paper we find asymptotic expansions for probabilities
of the form P{w(Sn) < r}.
9I.
Basic Results
All the results of the paper are obtained for differently distributed summands (cf. Sec.
2 and 3). Here we give the formulations for identically distributed sunmmnds.
We introduce notation: H is a real separable Hilbert space with scalar products (., ")
and norm I'I; u:H + R l is a continuous linear functional; C:H + H is a bounded symmetric
operator; w(x) = (Cx, x) + u(x); X, X l ..... Xn are independent identically distributed random
elements of the space H, EX = 0, ElXl 2= I; Y, YI, -.., Yn . . . . are independent identically
distributed Gaussian random elements having covariance X, coy Y = c o v X, EY = 0; Xv~ =
X~{IxI ~ / ~ } , x/~ = xI{Ix I > / ~ } are the truncation of X at level /~; Sn = (X i + ... + Xn)/
Vn.
Let d be some class of measurable subsets of the space H, invariant with respect to
translations in directions from H, ~: ~ + R i be a set function.
We shall say that the function # is differentiable on the set A e ~
in the direction h e H, if the limit lim t-:(~
t+0
(~(A--th)--#(A)) = dh~(A) exists.
If the derivative dh ~(A) exists for all h e H and A e ~ ,
then we can define the repeated derivative dhx, dho~(A).
By induction one defines the higher
derivatives d h l . . . h ~ ( A ) = dh~Ldhl...dhZ-1#)(A) (cf. Sec. 4 for more details on derivatives of
set functions)-. Further, let the map [h i .... ' h l) +~ dhl...h~(A) (for each A e d ~) be a
symmetric semilinear form, s = i, 2, .... Then for each polynomial P(z I ..... zs with real
coefficients in s commuting variables z I, .o., zs we can well define in a natural sense the
differential operator P(dh I ..... dhs
acting on the set function ~.
If the operator A:H + H has no less than k eigenvalues (counting their multiplicities),
exceeding the number 8 > 0, then we shall write A e o(~, k). We also let A~,r = {x:w(x + a) <
r}, # be the distribution of the Gaussian vector Y, A 0 = coy X be the covariance operator
of the random element X.
T H E O R ~ i.I. There exists an absolute constant c such that if (CA0) 2 e o(~, k) with
k ~ c, then one has the decomposition
P{w(S.+a)<r}=~(Ao, O+rx,
where the remainder r I admits the estimate
ir~l~c(~, lul, I Cl)O +laD{efXr
~+n-~"El-,t'%x p}.
Let oq = (EIXVnlq)i/(q--2), q > 2.
THEOREM 1.2. Let q > 2, g > 0, (CAo) 2 e o(~, k). There exist
and c 2 = c2(e, q, 6, lul, ICI) such that if k > cl(s), then
constants
cx = cx(e)
Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR.
Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 24,
No. 4, pp~ 29-48, October-December, 1984.
Original article submitted July 5, 1983.
0363-1672/84/2404-0305508.50
O 1985 Plenum Publishing Corporation
305
P{ w(S.+a)<r } =~(Aa.,)+ i/(6Vn) Ed~vT*(A,,~+rx,
where the remainder r I admits the estimate
Irxl ~c~{ ~/n)X-'+(l +!a[~)(EIXr
+n-* EIXV'; I9 }.
If in addition EIX[ 3 < ~, then
jEdge;* (a~ J- Ed}~(A, ~ l(c~O + lal~)eIX C; I~
Definition 1.3. We shall say that the random element X e H satisfies the K-condition of
Cramer, if there exists a nonnegative operator K:H ~ H and numbers r I > 0, p < 1 such that
sup(;=.~,~[ Ee'(x'~l=p.
Remark 1.4. If the operator K is the identity,
classical Cramer condition
(i.i)
then we get a "generalization"
of the
suplx~"t%lEe'(~'x){=P"
(1.2)
However, in the case of an infinite-dimensional space H, (1.2) is devoid of content, since
for each r ~ 0 and each X e H, one always has sup }Ee'(X.m[=l. This follows from the weak
Ixl
=,
sequential continuity of the characteristic function of the elements X and the fact that
the weak sequential closure of the unit sphere contains the point zero.
Remark 1.5.
operator.
As the operator K it will be sufficient for us to choose a finite-dimensional
Let X r be the syrranetrization of X~ ~ .
3
3
Definition 1.6. By the characteristic x(t) =:x(t, n, X, C) of the sum Sn we mean the
quantity
k
1=1
J--k+t
Let s = s
= (s
.'', ~s) be a positive integral multiindex, s
= s163
[s
= s + ... + s
The symbol E' in what follows will denote the summation over all
multiindices s
satisfying 3 s s ~ P, 3 ~ s $ p - Z~ + 2 , .... 3 ~ s ~ P - (s + ... +
s
+ 2(s - i). We also denote by @~ the distribution of ~Y~
Theorem 1,7.
Let the i n t e g e r p ~ 2, the numbers 0 < e < l , q > 2, B > O. Let the operat o r (CAo) 2 e o(B, k).
There exist constants c z = c~(e, p), c 2 = c~(~, p, q, [C l, B), c~ =
c,(r
p, q,
Ic[, lul, 8) such
that if k ~ c~, then one has the decomposition
p {w(S.+a)<r }=OO(Aa, ,)+ax+ . . . +ao-z+rx,
(1.3)
where
an,= ~
s=l
I'
I/l(')!CSnn-l""ll2EQ/-~#l/1--~.(Aa,J'
(1.4)
i 1(8) i =2~.+m
and the remainder r I for each T > 0 admits the estimate
[ rx t ~<c,(1 +[ a [3v-z) {(In T) sup {x(t):cz(n/eDl-*<~ [t I 6T} +T-*+(~,/n)B+E[Xv.;, t'+n-(O-*)l"EI Xr
Moreover,
,+~ ).
one has the estimates
la,, <<.c(m, [ CI, [ul, ~)(I +IaI~=)(EI Xv'~ IZ+n-=/~ EI XW'; ['+9.
(1.5)
We note that if the operator (CA0) 2 is sufficiently many-dimensional, then the function
r 4-+ am(r) is sufficiently many times differentiable, and its derivatives decrease faster than
any power as Ir[ ~ ~. Cf. Sec. 4 for more details on the properties of the terms of the
asymptotic expansions.
If the Cramer K-condition holds, then one can estimate sup {x(t):c2(n/o~) l-e < Itl --< T}.
However, in contrast with the finite-dimensional case, this estimate is not trivial (cf.
Lemma 2.9). Namely, one has:
306
THEOREM 1.8. Let the integer p > 2, the numbers q > 2, A > 0. Let the operator (CA0) 2
a(8, k) and let the Cramer K-condition hold, where KCAoC e o(8, k). There exist constants
c I = cl(p, A) and c 2 = c2(~, p~ q, A, r0~ p, ICI, lul, IKI) such that if k > c~, then one
has the as)nnptotic expansion (1.3) of Theorem 1.7, where the remainder r I admits the estimate
I ~ I <- c~ (t + l a t"-~) { (~/~D -~ + E I x r
The terms e l ,
....
I~ + ,,- (" -~,t~ ~lxr
~ i,+~}.
a p- 2 o f t h e a s y m p t o t i c e x p a n s i o n s g i v e n above e x i s t
i f E l X i ~ < ~ and
depend on the moments of the truncated random element X ~ .
If EIXIP < ~ these terms can be
replaced by quantities depending on the moments of the random element X, and can also be
expressed in terms of the Edgeworth--Cramer polynomials.
We recall that the Edgeworth--Cramer polynomials Pk in the formal variables do ..... dk+2
(these variables are sometimes called "instantaneous") are defined as the coefficients of
tk in the expansion of the function
=o
j=2
in a power s e r i e s
in t ( c f .
[1]).
We w r i t e t h e p o l y n o m i a l Pk in t h e form / ~ = ~
and we define new polynomials Pk in the variables ml,
Pk(ml,
We shall also call the'polynomials
Po=l,
a~,...~d~...d~,
.... mk by means of the equation
m~)= Z ai~ 9 m i.
.m i'.
Pk Edgeworth-Cramer polynomials.
We note that
Pz=m?/6, P2=m~/24-m~m~/8+m~m~/72.
Let
(d)~- d~)...,_~, _~,.
-
THEORKM 1.9. Let EIX]P < ~, the operator (CA0) z ~ o(~, k), m = i ..... p - 2. There
exist constants ci = c1(m), c2 = c~(p, ICI, {u I, 8) such that if k ~ c I, then thequantity
?n
bin= Z
Z'
1/lC~)!C~n-~t~'),/2EQ~Cbr
(A., ,)
s=l ii(S)l=2s+ra
admits the estimate
Ibm [ <~ c2 (1
q- [ a ]am) n-m/~ E [ X [m+ 2
Further, one has
lain--bin
p -2
p --2
m=l
rn=l
[ ~<C2(1 q - l a lzra)n-ml2 g t i X'F" _n ] . + 2 ,
., dx.) q~(A.,, I ~ ~, (1 + l a e ' - D n-("-~/~ (EIX/~ 1"+ n-~/~E1 X/~ I'+q,
I
!EP,.(dxl, .~
dx.)@(A.,~)] <~c~(l +la~")n-';z E]Xt "+~"
In [2] the asymptotic expansions
p--2
P { I g . [ < r } - - P { I Y { < r } + ~ Gn~/2+O(n-~P-I)tD
k=l
are found, where Ck = Ck(r) are certain functions with known Fourier transform.
Here one imposes the condition EIXI2(P +l) < ~, and also certain conditions on the covariance operator
of the random element X, whose description we omit here.
In order to guarantee the decrease
of the characteristic function (in our notation x(t)) in [2] the following condition is
imposed: the quantity
rn
! <~m<~[nl2]
j 1
=
for Itl ~ n l-s satisfies
307
P { 8 (t) > 1 - [(p - I) In n]/n} = 0 (n-<'-x)t')
(we give (1.6) in a somewhat simplified version).
( i. 6 )
It is easy to see that •
satisfies
z(t)~E~"/'(t)~p{)(t)> 1 -[(p- 1)Inn]~} +{ I -[(p- l)Inn]/n}"l'fO~-('-x)/z).
It is also noted in [2] that (1.6) holds for many statistics, which can be realized as special
cases of the central limit theorem in Hilbert space. Thus, in view of what was said above
and Theorem 1.8, the estimate of the remainder r I in Theorem 1.7 can be assumed to be meaningful.
The scheme of construction of the asymptotic expansions given above recalls the scheme
of the analogous constructions in the finite-dimensional case [16, 17]. Our proofs are
essentially based on the results of [3]~ where asymptotic expansions are found for characteristic functions. In Yurinskii [4], under the assumption EIX[ s < - the estimate A(~) = P x
{[Sn + a[ < r} - P{IY + a I < r} = O(n "z/2) is found. In [5] it is proved that if E[XIP < ~,
2 < p < 4, then A(0) = o(n-(P--2)/2). In [6], the question of the structure of the constants
in the estimate A(al) = O(n -I/2) is investigated. The methods developed in the papers mentioned allow one to estimate the characteristic function on an interval of the form It[ ~ c.
n i'E, ~ > 0, and also to get the expansions of Theorem i.i and 1.2.
We note that the basic results of the paper were reported at the 24th conference of
the Lithuanian Mathematics Society and published in [7].
2.
Estimates of the Decrease of Characteristic Functions
In [3] the characteristic x(t, s, s of the sum Sn of independent random elements of a
Hilbert space was defined and in terms of this the characteristics of the asymptotic expansion of the characteristic functions of sums Sn were found. In the present section we investigate the question of the decrease of the characteristic •
s, s
In combination
with the familiar Esseen inequality this allows us in Sec. 3 to find asymptotic expansions
for probabilities of the form P{w(Sn) < r}.
Let: Xx, ..., Xn be independent random elements with values in the space H, having zero
mean and satisfying P{Xj = 0} > 0, j = i, ..., n; Sn = X~ + ... + Xn; Y~, ..., Yn be independent Gaussian random elements such that EYj =EXj, covYj = covXj; Zn = Yl + ... + Yn; X6=
XI{]x]<~}, X 6 = XI{]x] > ~ } ; X b e t h e symmetrization of X; ~* = ( X 1 ~ ; 02 = n max E[Xj[2;
l<j__<n
c:a = max (n '~l~E [ X] [a)xl(a-~), q > 2.
1 <-j~-.
Condition 2.1. On the random elements Xl, ..., Xn and the polynomial w(x) = (Cx, x) +
u(x) we impose the following condition: there exists a nonnegative operator A0:H ~ H such
that cov Xj ~ A0/n, j = i, .., n, and(CA0) 2 e o(8, k).
Throughout the entire paper (if nothing is said to the contrary) we shall assume that
all random elements occurring are independent in aggregate.
n
Definition 2.2 [3]. Let V=(V, ..... V,)e N {XJ, Yj} , the integers s and s be nonnega/=i
tive, the set N = {I .... , n}, the sets A l .... , A~+~ c N be disjoint, the set D c N. By the
characteristic z(t, s, s = z(t, s, s X l .... , Xn, C) of the sum Sn we mean
9
V
c.ardD<~5 AtU
By theGaussiancharacteristicxg(t,
x,(t, s, ')= sup
...
uAI+L=N~D I<~q<~IBcAQ
s, s
inf
= x(t,
i, i), •
k~A~B
of the sum Sn we mean
sup
= xg(t,
>,)}
inf El/'exp { 2it ( Z CYj,
carflD<~s At U... UAt+x=N\D 1 ~q<~l t$~A~
We also set •
j~-B
]~B
j~Aq\B
i, i).
It is easy to verify that •
s, s ~ •
s + i, Z), x(t, s, s
Analogous inequalities are also valid for x~g(t, s, s
308
& x(t, s, ~ + i).
LF24MA 2.3 (cf. [8, p. 84]). Let A and B be bounded linear operators in H, and let A g 0.
The collections (counting multiplicities) of nonzero eigenvalues of the operators AB, BA, and
A~/2BA~/2 coincide.
To the end of making the account complete we give a proof. Let ~ # 0 and H%(AB) c H
be the subspace of all eigenvectors of the operator AB, corresponding to ~. Then
AII~B:Hz(AB)-->Hx(AIJ~BAII~);
(2.1)
in addition the operator A I/2 B in (2,1) has no kernel. In fact, h e Hh(AB), h ~ 0, implies
AI/2BAX/2(A:/2Bh) = XAI/2Bh and if AI/2Bh = 0, then 0 = Abh = lh. Analogously, the operator
AI/2:HA(AI/2BAI/2) ~ HA(AB) is an injection and hence dim Hx(AB) = dim H%(AZ/~BA~/~). The
lemma is proved.
LEMMA 2 . 4 . Let n ~ s + s N = [(n - s)/s (here [~] is the greatest integer in the
numbe~ ~ a~d let condition 2.1 hold. Then xg(t, s, Z) ~ c(8, k, N/n)(l + t~) -k.
Proof.
S= E
C9~,
Let the sets Bz, B= c {I, ..., n} be disjoint, N = card B~ = card B=.
R= S Yj
Let
According to the definition of the characteristic xg it suffices to
show that the quantity ~: = E exp {2it(S, R)} admits an estimate just like that of xg from
the conditions of the lemma. Noting that
Err exp { 2it (S, Yj) },
A= E H
j~B,
using the well-known formula for the characteristic functional of a Gaussian vector, applying
the inequality coy Yk = coy Xk ~ A0/n [condition (2.1)], we get
A ~ Eexp { 2~ F2--~/n (Zo, S)},
where Z 0 is a centered Gaussian vector with covariance cov Z 0 = A o.
to the preceding, we have
Continuing analogously
A ~Eexp { - 8#(N/n)'lZt ~ },
where Z is a centered Gaussian vector with covariance cov Z = (CAoC)I/2Ao(CAeC)I/2. According to Lemma 2.3, the collections of eigenvalues of the operators cov Z and (CA0) 2 coincide.
Using the explicit formula for E exp {-slZl 2} (cf. [9. Chap. V, Sec. 6]) and keeping in mind
condition 2.1 (CA0) = e o(~, k), we get the estimate needed for h. The lemma is proved.
The rest of the section is devoted to estimating the decrease of the characteristic x(t,
s, s
We note that Lemma 2.5 and 2.7 are adapted to our goals of modifications of similar
assertions in [4].
LEMMA 2.5. Let the random elements ~ ..... $s
be symmetric, independent, and suppose
either ISj[ $ 6 with probability one, or Sj is Gaussian, 1 ~ j $ m + s
Let
l
s=
I+m
R=
j =1
a==ovS,
B==o R,
j=l+l
be centered random elements, U and V be Gaussian, A = c o v U, B = c o v V, the numbers p and y
be positive. Then for all r, satisfying 21Cldpl~ ! ~ i, and all T, satisfying 2ICIT6 max
{7, p} $ i, one has
Eexp { ix (CS, R) } ~ 6 exp { - p~/(TrA + B~ } + exp { --:/TrB } + ~ (s),
where
~(s)=Eexp{is(CU,
Proof.
10/2}, s = m i n { l * t, r } .
We have
t+m
Eexp{i~(CS,
R)}=E IX E~exp{i'r(CS, ~)}.
]=l+1
If the random element Sj is Gaussian, then obviously
E s exp { iv (CS, Cs)} ~<exp { - "r' (Aj CS, CS)/4 },
(2.2)
309
where AJ = coy ~j.
cos x S i - x=/4,
Now if the random element {~jj{ $ 8, then applying the elementary inequality
s
if Is{ < w e get
{ i.c(CS, ~) } <.n,(1-.d(CS, ~)'I4)= I -.'(A,C$, CS)14,
6p ~ r
s + m.
Hence Ixl
6p
lclao
Consequently, for
~6,
Eexp { i':(CS, R) } ~<P {[SI > p } +Eexp
-< vE (2.2) holds for s + 1 $ j =<
{-.d(BCS, CS)[4 },
(2.3)
Since
E exp { -- x" (BCS, CS)[4 } = E exp { i'r (CS, I0[ ]/'2 },
arguing analogously to the proof of (2.37, f o r lm{
lcl6~ <- ~
we get
Eexp{-'v~(BCS, CS)]4} <~P{II,'t> y}+Eexp{-.?(ACV, cr')/8 }.
(2.4)
We estimate the probability P{IS] > p} in ( 2 . 3 ) .
Let S I be the sum of the Gaussian
summands in the sum S, S 2 be the sum of the nongaussian summands in the sum S. We have S =
S I + S 2 and
P{IsI>p}<.P{ISaI>H2}+I"{Is,I>H2}.
We estimate the probability P{lSzl > p/2} with the help of the analog of Bernshtein's inequality in Hilbert space, which in our case assumes the form (cf. [i0])
P { I S~ ] > p/2 } ~<2 exp { - p2/(8Tr cov $2 + 88p) }.
To estimate the probabilities P{{Sz[ > p/2} and P{{V] > 7) we apply the inequality P{{Z] > r} -<_
4 exp {-r2/(4Tr covZ)},which holds for each centered Gaussian element Z. We note in addition
that Tr coy Si < Tr coy S, i = i, 2. Collecting the estimates listed and using the fact that
the right side of (2.3) decreases as {~{ increases, we complete the proof of the lemma.
LEMMA 2.6.
Let Tr cov Sn = i, the number q > 2, and let condition 2.1 hold.
Let
n
(V1..... V~e N
{(X))~' ~J} ' the sets 8z, 02 c {i .... , n} be disjoint,
]=1
l=cardOz, m=cardO3, A=cov S
Vt, B=cov S
ke@,
Vk.
keO,
There e x i s t c o n s t a n t s c i = c i ( q , 8, o, ICI), i = l , 2, such t h a t CBCA e o(s
n and 6 satisfy the inequalities ~ >_- C2Oqn-lf2, nmaxE{XjiI~ex.
-2, k) whenever
l~j~.
P r o o f . I t i s known ( c f . [11, Chap. X, See. 4]) t h a t i f t h e o p e r a t o r S e ~(28, k) and
(Px, x) => (sx, x) - slxl =, then the operator P e o(8, k) also. Hence (if in addition one
keeps Len~na 2.3 in mind) the lemma will be proved if we verify the inequalities
((CBC)~/"A (CBC)W'x, x)>~21n-*((CBC)XI"Ao(CBC)X/'x, x) - c i xl=mln-* { max E [ X n l ' + n - q " 8 ' - ' ~ q - ' } ,
(2.5)
I <~j,~.
21n-*(AIol2CBCAIo/2X, x)>~4lmn-~(A~12CA.CAIJgx, x) -clxl=lm,,-~{max etx~,l~+.-'l,a,-,~-,},
(2.6)
where t h e c o n s t a n t c = c ( q ) .
L e t t i n g z = (CBC)L/2x, we have
((cscw'x (cat3','
x. x) = (az, z) = ~
g(F,. z),.
(2.7)
Further,
E(Irj, z)=2E(Yj, z)=>~2n-Z(Aoz, z),
E(($'))',z)'~>E(.?J,z),-Izl~el(~))gl"
~>2e(xJ,z)=12(EX),z)~-lzl~a"-"El~)i'>~
>>.2.-x(aoz, z)-4Lzl~EIX~ll~-e[zi'a2""~ -',
310
(2.8)
(2.97
the constant c = c ( q ) .
Since
IB]<~ ~ ElVjl~<<.4m~2n-~,
j~O,
Izl~<le:lnltx:,
one has
lz[ ~<c(*' ICl )mn-xlx:"
Summing e s t i m a t e s ( 2 . 8 ) , ( 2 . 9 ) i n ( 2 . 7 ) c o n s i d e r i n g
be verified similarly.
The lemma i s p r o v e d .
LEMMA 2 . 7 .
L e t Tr c o y Sn = 1,
and l e t c o n d i t i o n 2 . 1 h o l d . Then f o r
c i = c i ( e , A, p, q, ~, o, t o t ) , i =
~ ( ~ , k) w i t h k ~ c a , t h e n f o r n and
(2.10)
( 2 . 1 0 ) we g e t ( 2 . 5 ) .
Estimate
(2.6)
can
t h e c o n s t a n t s c 1 = c l ( p ) , cz = c 2 ( p ) , t h e number q > 2,
a l l r > 0, A > 0, t h e r e e x i s t c o n s t a n t s c 3 = c 3 ( e , A ) ,
4, 5, 6, s u c h t h a t i f i n c o n d i t i o n 2 . 1 t h e o p e r a t o r (CA0) z
t satisfying
[t[<c~/e~)l-%
n maxEtXnl~,
l~j~n
one has
~(t,
c~(p),
c~(p))<.c~(n/~)-"+c~(1+t9-<
Proof.
I n t h e c o u r s e o f t h e p r o o f o f t h e lemma, i n o r d e r t o s i m p l i f y t h e n o t a t i o n a l l
c o n s t a n t s w i l l be d e n o t e d by t h e one l e t t e r c, and we s h a l l n o t n o t e t h e d e p e n d e n c e o f t h e
constants on parameters.
Moreover, one can assume that the number n/o~ is sufficiently
large, since otherwise the estimate of the lemma is obvious.
Let the numbers s m >_- 1 be integral,
0 I= Z, card @2 = m and let
the sets 01, 02 c {I, ..., n} be disjoint,
card
n
.... ,v )e l] {%, g}j=l
According to the definition of the characteristiclx(t , c1(p) , c2(p)) it suffices to prove
that the quantity
jE@,
jEO,
admits an estimate j u s t
o f t h e lemma.
like
that
of the characteristic
y.(t,
cl(p),
c2(p))
We n o t e t h a t due t o t h e s y m m e t r y o f t h e random e l e m e n t Vk, f o r a l t
one h a s E exp { i ( z , Vk)} ~ E exp { i ( z , V~)}.
Hence
Al~l+ra,cninf E e x p {
2it(Z
CV],
i~O~
whereV~ =V~, i f v j
--:~,
Z
from the condition
z e H and a l l
6 > 0
V])},
jeO,
andVj =Vj, i f V j - - ~ j .
Applying Lena 2.5, we get that if
41C160I~ I < i, 41CIT6 max {7, p} < I one has
A4 ~<6 inf
l+m~en
where s = m i n
{[t[, T}, ~(s) = E
[exp { - p'/(TrA + 8p) } + exp { - y'/TrB } + 9 (s)],
exp
{-s2lZl2},
(2.11)
the centered Gaussian vector Z has covariance
covZ=(CBC)I/'~A(CBC)I/2, A=cov ~ V], B=cov ~ V].
jr
jeO,
We choose 6 = c n-i/2oq.
Then according to Lemana 2.6, coy Z e o(~m~n -e, k).
viousiy Tr A ~ 4Zo2n -I, Tr B _-< 4mo2n -I. Since
Moreover,
ob-
ao
Ee-S Iz
,' = H
(1 + 2s 2 ~j)-,:2
j=l
311
(where 8J are t h e elgenvalues of the operator coy Z), it follows from (2,11) that if 41C I.
ttt~o ~ 1, 4tclz~ max {~, O} ~ i one has
A'gc inf [~xp{-p'/(41aSn-*+8~)}+~xp{-y'n/(4m=9}+?(s)],
where s = mln{lt I, T}.~ We pass from n, s m to new (normalized) variables n, s m with the
help of the equations n = 6 -= = cn/~, s = n~/n, m = nm/n. We get that if c[t[p $ n~/~,
oTmax {7, D} [ ~ / 2 one has
A ~ < e MftexP{-~/(4l~'l~+p~-~/')}+exp{-y,h/(4~e)}+(l +s'f~h-')-~.
(2.12)
I+m~r
Choosing T = [t[, s = m - cn, p = ~ = c(A in ~)-t/2 in (2.12), we get that for sufficiently
large n the estimate from the condition of the lerama holds in the interval [tI~c(~in~)*/a.
On the interval c(n in ~)i/z ~ |t I ~ c~1-e the estimate A # ~ cn-A is easily found by choosing
- cn~, m - on, T - c(n i n n)~
p=c~'-11n~)*/', T=c(InBp/'
in (2.12).
The lemma is proved.
Condition 2.8. We shall say that the random elements X~, ,.., Xn satisfy the Cramer Kcondition if there exist a bounded nonnegative operator K:H + H and numbers r0 < | p < 1
such that
sup
sup
IEcxp { i(x, XO } [= p.
LEMMA 2.9. Let Tr coy Sn = i, let Condition 2.1 hold (in particular, (CA0) 2 e o(8, k)),
let Condition 2.8 of Cramer hold, where the operator KCAoC e a(~, k). For each number A > 0
there exist constants c I = c1(A) and ci = ci(8, k, o, q, A, r 0, p, IKI, ICI), i = i, 2, such
that if k, n, and t satisfy
k~cx, n max ElXjxl'<c,,Itl~/~)at2,
then IE exp {itw(S~ + a)}l -< ca(n/o~)-A.
Proof.
Let S--
CXJ, R= I
E
1~j~.12
tion
inequality
(of.
i')
Applying t h e analog of the familiar symmetriza-
.12<],~.
[4, Lemma 2 . 1 ) ] ,
we g e t
t Z exp { inr (S'. + a) } I' ~<Z exp { 2it (S, R) } < E exp { 2it (S 8, R) },
where
Ss= ~
(CX})s , t h e number ~ > 0 i s a r b i t r a r y
(the last
inequality
( 2.13 )
in (2.13)
is a
I ,~j~nl2
consequence of the syranetry of X~). Continuing (2.13), we have
Eexp{itw(Sl"+a)}l<<'E H
IEx,exp{2it(SS, xJ)}12<<.E H
nt2 <J <<.n
[IEx, exp{2it(SS, xz)}I+2P{ILI >~1}]'~<
n/2 <j <n
(we apply the estimate P{IXk i _-> i} _-< ElXk[ 2 -< o2/n and we make use of the Cramer K-condition)
~<(p+2a~/n)"+(l
+ 2az/.) P { 4t2 (KS~, Sa) <~r~./~ }.
(2.14)
Applying the inequality 1 + t _-< et, we have
(p+2a21n)'<~p"c(p, ~)~<n-ac(p, ~, A)<(n/~)-ac(p, ~, A, q),
since oq > c(o, q). We also have (i + 2o2/n) n --< c(o). Hence, in view of (2.14), to complete
the proof of the lemma, it remains to estimate the quantity
A: = P { 4t'(KSs, Sn)<~4 nln~},
appropriately, where 6 > 0 is arbitrary.
312
(2.15)
In view of Lemma 2.7, we can assume that Itl >_- (n/o~) a+I/2 with some number ~ = ~(A),
0 < a < i/4, which we choose in the course of the remainder of the proof.
Hence it follows
from (2.15) that
A < ~ P { ( K S ~,
S~)~*z},
(2.16)
where we have written E = r0(n/o~)-~/2.
We shall assume below that E < i, since otherwise
the estimate from the condition of the lemma is obvious.
We show that if we choose ~ = cn-i/2oq,
then
coy (/O/~S~)~a (~/2, k),
(2.17 )
c(~, k)<~ Trcov(Ka/~ S~)~ 4 [KI [ CI".
(2.18)
The l e f t inequality in (2.18) follows from (2.17), and the right inequality is an obvious
consequence of the relation Tr cov Sn = i. With the help of arguments analogous to those
given above at the beginning of the proof of Lemma 2.6, it is easy to derive (2,17) from
the inequality
(cov ( I01 ~ S 8) x,
Jr .>i( K al2 CAo C-~zl~ x, x) - cn ]x }~ [ max E ] X sl ]~ + ~ - q n -q/~ ~-~]
and the condition KCAoC ~ o(~, k).
(we let z = K1/2x)
( 2.19 )
It is easy to derive (2.19) from the following estimates:
1 <~j <<.n/2
E (( CX)) ~, z)~= E (2), Cz)~ - l z i= E l ( CX))n p ~ 2E (X), Cz)~ - 2 (EX), Cz)~ - I z t= ~'-q l C l~ E l kJ [q >~
>12n -~ (IO 12 CA o CK al~ x, x) - el x t~ { E IX n (~+ 8 ~-~ n -~/~ ~r~q-~ }.
Let the nonincreasing function ?:R ~ ~ R ~ be infinitely differentiable, ?((-~, i]) = i,
?([2, ~)) = 0. Let 7 = T r c o v ( K ~ / ~ S ~) and we consider the function ~a(x) = ~ (a-•
x ~ H. it is clear that the function ~E is infinitely Frechet differentiable and one has
y)~-% m=O, 1. . . .
[l +~") (x) [i <<.c (m,
(2.20)
It follows from (2.16) that
To estimate the quantities E~E(7-Z/iK~/iS6), we apply the. asymptotic expansion for smooth
functionals (cf. [3], the estimates and expansion of Theorem 3.2, and the remark on Theorem
3.1).
For this we let V = Vz+ ... + Vn/2, Vj =
U = U~ + ... + Un/2 be a sum
of Gaussian random elements EUj
0, cov Vj = coy Uj. Writing the asymptotic expansion and
estimating the terms of this expansion with the help of the estimates from the formulation
of the theorem mentioned, we get
y-~/iK~/~(C~[~)6,
m--2
A ~ E +. (y-1/~ KII~ S~) <~c ( I K [, I c l , Y,
}
m) I./ I r
s=O
{I Z Ut ~<2~-'Y-~/=}+=-z,,+s~,,-~ ,
(2.22)
jeO~
O=
where we have estimated
s247
~, tcl,
IKI)~'EIX)I', ~ fiX)I"<.].
i
The upper bound in (2.22) is taken by all subsets @s c {i . . . . , n/2}, card Os >-- n/2 - c(s).
Analogously
to (2.17) it is easy to verify that
coy Z
Uje=(c(m)~, k) .
Consequently,
JeO 5
JeOs
Moreover,
estimated
6 = cn-i/2cq (cf. (2.18)).
Hence,
in (2.18), we get from (2.22) that
if one recalls that ~ = Tr cov (KI/2S 6) is
(2.23)
313
where the constant c(...) depends on parameters Just like the constants in the formulation
of the lenma. Since r = c(n/o~) "a (cf. (2.16)), taking a = ~(m) such that r
= i,
~
2 - A , and finally taking k large
then taking m = m(A) large enough
that (n . i / 2 Oq)' m . 2 ~ (n/o o)
enough that ck -sm+' $ (n/a~) "A, we get from (2.23) that A ~ c(n/o~) "A. In view of (2.15) the
lemma is proved.
'
3.
Asymptotic Expansions for Distributions of Sums of Differently Distributed Summands
We adhere to the notation introduced in Sec. 2.
] --1
Moreover, we shall write
] --!
S~=XI+...
+Z~;
]'|
Z,(/('))=Z.U, .... , J,)=YI+...
+ Y.,
where the summands with indices Jl, ..., js are omitted; On,j(s) is the distribution of Zn x
(j(s)); Aa,r = {x e H:w(x + a) < r} the symbol E' denotes slnmnation over all integral multiindices s
= (s
..., s
satisfying 3 ~ s
p, 3 ~ s & P - s + 2, ..., 3 ~ s ~ p (s + . + s
+ 2(s - 17; the symbol Z" denotes stummation over all integral multiindices
j(s) = ii~, ..., js). satisfying 1 ~ js < Js-l < ..- < J2 < Jl ~ n. The differential operators
(d&
Q, = Q, ( ! %
THEOREM 3 . 1 .
j(,)) = -
~
""
L e t T r cov Sn = 1, t h e i n t e g e r
l,
(%.-%/
"~
"~"
p ~ 2, t h e numbers q > 2, 0 < e $ 1, A > 0,
B > 0. Let the random elements X~ .... , Xn satisfy Condition 2.1, in particular, the operator
(CA0) = e s(8, k). There exists a constant c = c(p, ~, A) such that if k ~ c, then one has
the asymptotic expansion
~{s.~., ,}=P{z,~.,
,} + a , + . . .
+a,_, +a,
where
,j
a~--~
~'
~" I/!(",EQ~',,,/.,(A.,,).
s = 1 I/('} I = v + 2 s
One has
ta, I ~<c.(1 +[a[3")(A~+Lv+~)r
the constants c v = cv(8,
the estimate
Icl,
lul),
= i,
..., p - 2.
(3.17
For each T _>- 0 the remainder R admits
]Rl<~cR(l+]al,P-3)[A2+Lp+,+(n/~,)-a+{nmaxElXn(~}B+l/T+ln(1 +/')sup{lEe uw~s.+a)
2,-.< l t l < T } ] ; ( 3 2)
" [:(n/cry)
1 <],~n
the constant cR = CR(A, B, e, p, q, 8, o, lul, IC[).
Proof. To prove the theorem we apply the familiar Esseen inequality, the asymptotic
expansions of the characteristic functions found in [3], and the estimates of the decrease
of the characteristic functions of Sec. 2.
Let
F(r)=e { s.~,.,}, FI(,)=e { s.~A,.,),
a0(r)=p{z.~A..,},
~(,)=a0(r)+ ... +a,_,(,).
and let f(t), g(t), Ag(t) be the Fourier-Stieltjes transforms of the functions of bounded
variation F I, G, a v respectively, v = 0, ..., p - 2.
Let ~(x) = e itw(x+ a).
It is proved in Sec. 4 that
A,(t)= ~
s=l
314
~'
i10) 1 . 2 # + v
~" I/F')!EQ~?(Z,(J('))+a)9
According to Theorem 4.6 of [3],
[A,(t)t<~c~(It]+]t]8~)(l +[alag(As+L~+,)x~(t, v, 3v+l),
= i .....
(3.3)
p - 2, the constants cv = cv(lu I, [C[)-
IA~ (t) I~<~ (t)
(3.4)
is a consequence of the familiar symmetrization inequality (cf. [4], Lemma 2.1).
mate
The esti-
~,(t, c~(~), c~(9).<c(~, k, ~)(i+t9-~
(3.5)
is proved in Lemma 2.4 of Section 2.
Expression (3.1) from the condition of the theorem follows from the inequality la~(OI~<
+~
f itI-11.4v(t)~dtand
the estimates (3.3) and (3.5).
To complete the proof of the theorem, it remains to estimate the remainder R. Here,
in view of (3.1), one can assume that the numbers o~/n and n max EIXjll = are sufficiently
l_-<j<n
small since otherwise (3.2) is obvious. To the end of simplifying the notation below we
shall not note the dependence of the constants on parameters.
+~
IG'(r)l <. f Ig(t)Idt ,
Applying the estimate
from (3.3)-(3.5) we derive
--nO
C=sup 1Gt (OI ..~c (1 +la 1"-9.
r
With the help of the familiar inequality
(3.6)
n
[F(r)--Fa(r)[ ~<2 Z P{tXj[~>I}~<2AI, we get
j-I
IR[<JF(r)-G(r)I<<.2A,+I6(r)-G(r)I.
(3.7)
Applying Esseen f s inequality (cf. [12, Chap. 5, Sec. I, Theorem 2]), for 0 < T I < T we
get
I FI(r ) - G(,)I<~ c { C / T + I , + I , + I 8},
(3.8)
where
f Jtl-,I/(t)-g(t)jdt,
I,=
n=
f ttl-'t:(t)ld,, •=
Tt~ It| ( T
f
ltl-ZIg(t)ldt;
T ~ Itl <T
the constant c is absolute, the constant C is defined in (3.6).
We choose T I = (n/o~) 1-e.
It follows from (3.3)-(3.5) that if k _-> c(p, E, A), then
/3 .< c ("I~D-A
(3.9 )
Since f(t) = E exp {itw(S~ + a)}, obviously
I ~<r}.
I, <<.em(T + 1)sup {lee'Ca+~
(3.10)
We make use of the asymptotic expansions for the characteristic functions (cf. [3,
Theorem 4.6 ])
If(t) - g (t) I ~<c ( ] t t + I t 18n-3) (1 + I a [8,- .~)(A, + L,+ 0 x (t, cx (P), c, (p)).
( 3.11 )
According to Lemma 2.7, if k > c(p, e, A), th-enon the interval It[ < (n/oR)x-g one has
x(t, c,(P), c2(P))<~c(n/~)-a-a~
-A-a~.
(3.12)
}.
(3.13)
Relations (3.11) and (3.12) imply
Iz <<.c(1+ l a ~~
Collecting
(3.6),
(3.9),
(3.10),
(3.13),
{ A,+ Lp+a +(n/r
in ( 3 . 8 ) ,
considering
(3.7)
we get ( 3 . 2 ) .
The
theorem is proved.
315
THEOREM 3.2. Let Tr coy Sn = I, the integer p ~ 2, the numbers q > 2, A > 0, B > 0.
Let the random elements X~, ..., Xn satisfy condition 2.1, and let Cramer's K-condition 2.8
hold (so that, in particular, the operators (CA0) ~, KCA0C ~ a(8, k)). There exists a constant c = c(p, A), such that if k ~ c, then one has the asymptotic expansion
e {s,~.
,}:P{z,~a,
,}+a~+ . + a , _ , + a ,
where t h e terms of t h e e x p a n s i o n a~, . . . , ap-2 a r e d e f i n e d in t h e f o r m u l a t i o n o f Theorem
3.1, and the remainder R admits the estimate
I RI .< c,, 0 +la 1"-' { A, + r~,+~+("/4)-" + [" max E[Xj, I,P ).
The constant cR = CR(A, B, p, q, 8, o, p, r 0, IKi,
lu[,
IC[).
Proof. To prove the theorem one should set E = 1/2 in the estimate of the remainder R
in Theorem 3.1, and with the help of Lemma 2.9 prove the inequality
inf [ 1/T + In (1 + T) sup { IEe ~'w(s'.+ o) [ : (n/a~)*m <~t t [ <<.T } ] ~ c ( n / ~ ) - a.
T
THEOREM 3.3. Let Tr cov Sn = i, the numbers q > 2, 0 < e ~ I, B > 0. Let the random
elements X I . . . . , Xn satisfy Condition 2.1, in particular, the operator (CA0) 2 e o(8, k).
There exists a constant c = c(e) such that if k ~ c, then one has the asymptotic expansions
P{ S,eAa,, } : P { Z , eA,. ,} +R,,
(3.14)
e { s , ~ . r} = e { Zneae, } +a~+a,,
(3.15)
where
J-1
Here #n,j is the distribution of the random element YI + ... + Yj-I + Yj+I + ... + Yn.
has
all~cl(l +lalS)L,.
R~l<cm(l+lalS)[A,+La+(n/~)"*+{nmaxElXn['}B],
One
(3.16)
R, l~ c.,(l+ Ia I')[Az+L,+~/~)'-*+{n max E[XI, l'}B],
the constants c I = c1(B, lu[, ICl), cRi = CRi (B, g, q, 8, a, lul, [C[), i = i, 2.
Proof. In Theorem 3.1 one should set p = 2, p = 3, then choose T = (n/a~) I-E, A = 1 - e
and if p ~ 3 take into account the symmetry of the Gaussian random elements Yj. It is easy
to get (3.16) directly. The theorem is proved.
THEOREM 3.4. Let the integer p ~ 2, Ap < ~, 9 = i, ..., p -- 2. Let the random elements
X I, ..., Xn satisfy Condition 2.1, in particular, the operator (CA0) 2 9 o(8, k). There exists
a constant c = c(p) such that if k ~ c(p), then the q u a n t i t i e s
bv= ~/~ I '
s=l
~" 1,I('),EQ~O /,,(A,.,)
I/(S)l = ~ + v
admit the estimate
Ib~l.<cl(1 + | a l ~ ) ~ ; §
Moreover,
|a,-bvl<<.c,(l + l a ] " ) A ~ + , .
The constant c I = c I (p, 8, lu[, ICI) 9
Proof. In Sec. 4 we find the Fourier-Stieltjes transforms Av(t), B~(t)
a~(r), bu(r ). In Theorem 4.8 of [3] estimates of the quantities |A~(t) I and
IB~(t)l are found in terms of the characteristic xg (t, cl(p), c2(p)). These
the familiar inversion formula for the Fourier,Stieltjes transform imply the
the theorem.
316
of the functions
IAv(t) - B~(t)l,
estimates and
assertion of
Proof of the Theorem of Section I. All the theorems of Sec. 1 (except Theorem 1.9) are
direct consequences of the theorems of the present section, if one considers the identical
distributedness of the summands and renormalizes.
Theorem 1.9 is derived with the help of
the inversion of the Fourier-Stie!tjes transforms of the corresponding asymptotic expansions
for the characteristic functions found in Sec. 1 of [3]. Since this inversion strongly recalls the preceding proofs, we omit it here.
Remark.
quantities
In the estimates of the remainder terms of the asymptotic expansions the
~ = max E[Xj] ~, ~.= max @q/~EIX) ~)l!(q-~)
l~j~n
l~j~n
occur.
In Condition 2.1 and in Cramer's K-condition 2.8, one imposes restrictions on all
the random elements XI, ..., Xn. One should note that the orders of the remainders are
preserved if the number mn of indices i = 1 .... , n, for which the conditions listed hold,
satisfies lim inf mn/n > 0. The orders of the r e m i n d e r s become worse if we know some
n
lower estimates for the rate of decrease to zero of the quantities mn/n as n ~ ~.
4.
Some Properties of Terms of the Asymptotic ExPansions
Let d be some class of subsets of the space H, invariant with respect to translations
in directions from H. Let #: d § R I be some set function.
We shall say that the set function 0 is differentiabie on the set A e ~
in the direction h e H, if the limit
lira t -a ( 0 (A - th) - 9 (A)) : = 4 ~ (A)
t~O
exists.
If ~ is differentiable in all directions h e H and on all sets A e d , then we shall
say that ~ is differentiable (on the class ~ ) .
If ~ is differentiable, then its derivative
A ~+ dh ~(A) is a set function (for fixed h). By induction one can define the repeated
derivatives
a~...h, 9 (A).
THEOREM 4.1. Let r be a Gaussian centered countably additive measure, having covariance
operator A, and let the operator (CA) 2 be infinite-dimensional.
We let
.ur
A=Ao,,= { x ~ H : w (x + a) < r } : reR ~, aeH }.
Then the measure r is infinitely differentiable on the class ~ ,
where the map
(h~, ..., hO ~ 1 . . . ~ ( A ~ . 3
for fixed A~,r e ~
One has
is a continuous symmetric k-linear form in the variables h I .....
141..., 9 (A,. ~1~ c 0 +lalk)Ihxl... Ih, I,
hk.
(4.1)
where the constant
c=c({~},
~, lul, Icl), o~= f Ixl~O(ax),
B
{~j}j~l is the collection of all eigenvalues of the operator (CA) 2.
Proof.
We show that the function a~+ Oa(r) = #(Aa,r) is infinitely Frechet differen w
tiable and its derivatives admit estimation by the right side of (4.1).
Since dhl...hkO,(r) =
dhl...hk~(Aa,r), this will prove the theorem.
The function r I§ @a (r) is a function of
bounded variation and
f e'ao.(o= (
R1
H
where t h e G a u s s i a n
vector Z had distribution ~.
The function
[3, Lemma 4.4]),
a ~+ Ee itw(Z+a) = ~,(a)
is infinitely Frechet differentiable,
ldh,...~ ~,(a)l<~clhl[. .. [hkl(ltl+ltlk)(l +lay')~g(t, k),
while (cf.
(4.2)
317
where c -- c(k,
[u[,
ICI, o),
k)=Eexp {- 16t2(k+2)-I(CACZ, Z)}= I-I (1 +32~5,/(k+2)) -1
J=l
•
(the last equation is a consequence of the familiar explicit formula for E exp {-s[Ul2}, U
a Gaussian element, and Lemma 2.3). Estimate (4.2) and infinite-dimensionality of the operator
(CA) 2 guarantees the decrease faster than any power as It I + ~ of the function
t ~ ~ , . . ~ ~, (a).
(4.3)
Applying the inversion formula for the Fourier-Stieltjes transform, we see that the inverse
transform of the function (4.3) (we denote it by ghl...hk,a(r)) is an infinitely differentiable function of bounded variation, having bounded derivatives.
We show that
$~...hk @, (O = ~...~. =(r).
To avoid awkward notation we restrict ourselves to the ease k = i.
sion formula, the theorem of the mean, and (4.2), we have
(4.4)
Applying the inver-
IO,+,(r)-Oo(r)-g,,,(r)[<~ f [tl-xlg,(a+h)-%(a)-dhg,(a)ldt<. f [t1-1 sup [4~?,(a+.:h)ldt~c[hl'(l+la[+lh[) ,
Rx
which proves dh |
a'
= gh, a(a).
Estimate (4.1) is derived from (4.4) and (4.2) with the help of the inverse FourierStieltjes transform.
The theorem is proved.
Remark 4.2. By methods similar to those used in the proof of Theorem 4.1, one can show
that the set functions
A,,,~
-~-
cl,(A,,,),
v = l , 2, . . .
are infinitely differentiable, and here as Irl ~ ~, as functions of r, they decrease faster
than any power. We omit the proofs connected with this question.
Remark 4.3. The terms of the asymptotic expansions found in the present paper can be
represented as sums of summands having, for example, the form Edxl...Xk~(A,,r).
In view of
the results of the present section, they are well-defined.
Remark 4.4. The theory of differentiable measures in infinite-dimensional spaces was
developed in the papers of Averbukh, Smolyanov, Fomin [13], Skorokhod [14], etc. Here it
turned out that there do not exist measures which are differentiable in all directions on
all sets of an infinite-dimensional space. But such measures exist if one either restricts
the class of directions, or restricts the class of sets, or both classes at once.
In the
context of the present paper only restriction of the class of sets is admissible. Uglanov
[15] developed the theory of surface measures on smooth surfaces of a Banach space.
In view
of the (generalized) Green's formula [15], one of the consequences of this theory is the
fact that on sets bounded by a smooth surface, a measure can be differentiated a sufficient
number of times in all directions.
This indicates the role of the smoothness of the boundary
of the set for the existence of the terms of the asymptotic expansions.
Remark 4.5. The terms of the asymptotic expansions can be expressed in terms of surface
measures associated with the limit Gaussian measure ~. Let A c H be a set with a sufficiently
smooth boundary F. The surface measure ~o, associated with the measure #, on "good" subsets
B c F can be defined with the help of the equation Co(B) = lim r
where B E is the ee~0
neighborhood of the set B c H in the space H. Then the measure r can be extended as a countably additive measure to the class of all Borel subsets of the surface F. Green's formula
[15] gives
~(Aa, 3= f (nx, h)~a(dx),
(4.5)
~a, r
where nx is the normal to the surface Fa,r at the point x e Ta,r. Analogously to (4.5) the
derivatives of higher orders dhl...hk~(A,,r) can be expressed in terms of linear combinations
of derivatives with respect to r functions of the form
318
r~
f
(n~, h~) . . . (nx, h k ) ~ ( d x ) .
~a, r
In this way one can get corresponding representations for the terms of the asymptotic
expansions.
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319