LOWER BOUNDS FOR THE RATE OF CONVERGENCE
IN
THE CENTRAL LIMIT THEOREM IN BANACH SPACES
V. Bentkus
UDC 519.21
In Banach spaces (if certain natural restrictions hold.) bounds on the rate of convergence in the central limit theorem are known, of order O(n-X/6), where n order O(n -I/6) in
these bounds cannot be improved.
Similar results also hold for the bounds of the rate of
convergence for moments.
Let E c B be a pair of real separable Banach spaces, connected by the identity continuous linear imbedding, with norms ['[E and ['[B, respectively.
Let X, XI, X2, ... e E be
a sequence of independent identically distributed random elements, assuming values in the
space E and having mean zero. We set Sn = Sn(X) = n -I/= (X~ +...+ Xn). We shall assume that
the centered Gaussian random element Y e E and the random element X have identical covariance.
Let the number m > 0, ~ be the largest integer satisfying ~ < m, ~ = m - ~ , x = ( ~ , . , •
be a collection of positive numbers.
We shall say that the function f:B § R: is differentiable in directions from the subspace E, if for any x e ~ h e E
the function ~(O:=f(x+~)
of
the real argument t has derivative ?'(~ :=f'(~ h.
By induction one defines the iterated
derivatives f(k)(x)hl...h k. The function f:B ~ R ~ belongs to the class C$(B/E), if for any
h, h,, h2, ... 9 E, i = lh~lE = [h=IE = ... and all x e B, one has
If(~)(x)h~...h~l<xk,
O~k<~,
[ (/(~)(x + h)-f~)(x))ha..,
Let the c o l l e c t i o n
•
h~[< x~+1 I h I~.
e>O,
be such t h a t •
~(~=c(~r
-k, c ( ~ < ~ ,
l~k<~, z~+~ (@=c(~+l) c-m, c(~+l)<~.
We call the inclusion E c B m-smooth (with collection •
if for all ~ > 0, r > 0 there exists a function g=g~.~eC$(~ (B/E) such that
0~g~1,
g(x) = 1 for Ix]B~g
g(x) = 0 for
]xIB~r+r
We let Fa(r ) = P{[Y + a]B < r}, Pa(r) be the density of the distribution
An(r) = P{[Sn[B < r} -- P{ YIB < r}.
THEOREM i. Let 2 < m ~ 3 , E] X ] ~ < ~,
constants d < = such that
the inclusion
supp, (r) < c (1 +
E=B
be m-smooth.
function Fa(r) ,
If there exist
la [~),
(1)
r
then
sup I A.(r) I
=
0 (n1/m-a/~).
T
If B = E, m = 3, under the assumption that the norm of the space B is thrice
Remark.
differentiable in the sense of Frechet, Theorem I was proved first by Paulauskas [9]. For
m = 3 Theorem i is a special case of the somewhat more general Theorem 31 of Bentkus and
Rachkauskas [2]. If 2 < m < 3, Theorem i can be proved by making relatively small changes
in the proof of Theorem 31 of [2]. A. Rachkauskas recently showed (oral communication) that
the assertion of Theorem i is preserved if (i) holds only for a = O, which improves an
(unpublished) result obtained previously by V. Bentkus by a different method:
IAn(r) I =
O(n~-~/6),
~ > O, if for a = 0 (i) holds, the random element X e E
is bounded, and m = =.
Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR.
Translated from Litovskii Matematicheskii Sbornik
(Lietuvos Matematikos Rinkinys), Vol. 25,
No. 4, pp. 10-21, October-December, 1985.
Original article submitted November 16, 1984.
312
0363-1672/85/2504-0312509.50
9 1986 Plenum Publishing Corporation
THEOREM 2_._. Let
space-B,
m>2,
the sequence
0.$0.
There exist a Hilbert
and a symmetric random element X ~ E
, such that:
the spaces
EcB
are connected
there exists a constant
by an m-smooth
space E, a Banach
inclusion;
c < ~ such that with probability
one
IXIE = c~
there exists a sequence n s + ~ such that for s and r satisfying
2 s-2 r>|,
one has
(2)
where ~ ( O = e x p {-2/r-r2[2};
for k = i, 2, ...,
if 0 < a < b < ~, then for infinitely many n,
inf { A. (r) : a ~ r ~ b } ~ cn TM-*t~ O.,
where
(4)
c = inf {~(r)/2:a~r~b}.
the constant
The proofs of this and other theorems will be given below.
Remark.
Results similar to the assertion of Theorem 2 in the case m = 2 were obtained
by Bentkus [5], Rhee and Talagrand [12], Borisov [7].
In the case m = 2 Theorem 2 improves
a result of [7].
A. Rachkauskas (oral communication), using some elements of the construction of [7], found a result which is essentially equivalent to the assertion of Theorem 2 in
the case m = 2. Apparently the constructions of I. S. Borisov and A. Rachkauskas do not
admit generalization to the case m > 2.
COROLL~
Under the conditions
of Theorem 2, for infinitely many n one has L(ISnlB,
]Y]B)~On n ljm-*/2 , where L is the Levy metric.
It is easy to show that from the bound of Theorem ! it follows
that
L (l S. f., f r 1.) = 0 (.~1~-~1~).
COROLLARY 4. Let the conditions of Theorem 2 hold.
There exists a function
such that O ~ f ~ l , l f ( s ) - - f ( t ) l ~ I s - t l
and for infinitely many n~
f:R ~ § R I
~. : = e f ( 1Y 1.) - E f ( IS. I.) ~ O. n ~ t " - * l t
We note that with the help of integration
of Theorem 1 that Bn = O(n~/m-*/2).
by parts,
it is easy to derive
from the bound
Remark.
In Bentkus and Rachkauskas [4] upper and lower bounds are found for the rate
of convergence in the central limit theorem in Banach spaces in Prokhorov--Levy and type
metrics.
Corollaries 3 and 4 improve the lower bound of [4] in the sense that in the case
of metrics of Prokhorov--Levy type the class of sets with respect to which the metric is constructed consists only of balls with center at zero (or, as is easy to show, one can assume
that this class consists only of one ball of radius i)~ and in the case of the metric ~ a
lower bound is given for the expectation of the individual function f, satisfying a Lipschitz
condition; here the level sets of f are balls with center at zero.
THEOREM 5. Let m > 2 ,
y > 0, the sequence
O,$0.
There exist a Hilbert
Banach space B, and asymmetric random element
X eE
such that:
the spaces
there exists
EcB
are connected
a constant
for infinitely
by an m-smooth
space E, a
inclusion~
c < = such that with probability
one
IXIE = c;
many n,
(3) holds.
One can show (we omit the proof) that the norm of the space B in Theorem 5 on the set
1 < IX]B < 2 cannot belong to the class C$(B/E) with some collection • . However the norm
of the space B can be approximated by differentiable functions.
Namely, as follows from the
results of Bentkus and Rachkauskas [7], the m-smoothness of the inclusion E c B
is equivalent
with the existence of constants c, ci, ..., c8+, < ~ such that for each s > 0
313
inf [sup {Ifx I,- g (x) t: x~B } :g~C~(,~ (B/E) } <~c ~,
where the collection ~ (~)=(oo, c~, c~ .~, ,., c~, -~+~, c~+~ *~'~).
If the norm of the Banach space is Frechet differentiable a suffidient number of times,
bounds on the rate of convergence for moments are found in Zalesskli and Sazonov [8], Bentkus
[ 3 , 6].
Le~ R ~ be the space of all real sequences x = (x~, x~, ..,) with the standard basis
{e~}~.
Let the sequence ~k > 0. We set
ix/...,.--(Z
2'
'
\1/~'
x /xq ,
txio=( l
k=l
")"',
l,
k=l
and we d e n o t e b y ~ , ~ t h e H i l b e r t
space of sequences
xeR ~ with finite
norm [ x I ~ , ~ < ~, by
~ p k , l<~k<~o~,]~--[~,
the standard k-dimensional
B a n a c h s p a c e o f s e q u e n c e s x = ( x t , . . . , Xk, O,
0, ...) a R ~ with finite norm )X]p < ~.
THEOREM 6.
/s,zclz
The inclusion
is m-smooth,
m~>2,
if the series
cc
(5)
z~'l~'-~)i< oo
k=l
converges.
Theorem 6 for integral m > 2
follows from Theorem 7 of Bentkus and Rachkauskas [i]. In
the case of nonintegral m one can prove Theorem 6 by making relatively small changes in the
proof of Theorem 7 of [i].
THEOREM 7.
It is impossible to increase the exponent m/(2m -- 2) in (5) in Theorem 6.
The rest of the paper is devoted to the proofs of the results formulated above.
8 and 9 below will be useful in the proof of Theorem 2.
Lemmas
LEMMA 8 (Senatov [i0]). Let the independent, identically distributed random elements
X, X,, ..., X n assume values in the space 12 k and have zero mean and identity covariance
matrix, the Gaussian random element Yel~ be such that the mean and covariance of X and Y
coincide.
Then for any convex set A c ~
where the constant c~ is absolute.
LEMMA 9. Let the integer k>4, I=~i> ..>~k=~>0,
the independent real random variables
q*, ..-, qk be distributed normally with parameters 0 and i, the Gaussian vector Y = (o,q~,
9 .., Okqk).
Then the distribution function F(r) = P{IYI~ < r} has continuous density p(r)$
here, for r > 0
p (r) :> k -3~ exp { - 2/0 2 } exp ( - l / r - r~/2 }.
Proof.
I t i s e a s y tO v e r i f y
t h e bound [ ~ ( t ) I ~ < 8 / ( ~ - a ~ e n l t l 3 ) .
We p r o v e
that
if
that the characteristfc
function
~(t)=Eexp{itIY[1}
admits
Consequently,
t h e d e n s i t y p ( r ) e x i s t s and i s c o n t i n u o u s .
l>~l~>...>:a~=~>0,
then
for
0<r<~2
one h a s
the bound
p (r) >1k - 2 exp { - 2[~ 2 } exp { - 1/r }.
(6)
L e t q ( x ) , x~l~ , b e t h e
density of the Gaussian vector
Y e l ~ , t h e n u m b e r A > 0. I t i s e a s y
t o v e r i f y t h a t on t h e s e t A={xel~: r<~ [x]z~<r+A }
the smallest value of the function x § exp
{--Ix[~[(2~2)}
is equal to exp{--(r
+ A) 2 / ( 2 o 2 ) } .
Hence, noting that k>~4,2~<k',a,~<l,
for
x e A we get
q (x) >~k -k cxp { - [ x ]~/(2~~) } >1k -~ exp { - (r + A)2/(2~D }.
We l e t
V(r)
= 2krk/k!
r<~2, k l ~ k k, 2k~>l, k:>l,
be the volume of the
we g e t
p (0 = lira
f
A,~O r < I x l ~ r + A
314
set
{x el~: [xh~<r}.
q (x) dx >I k-
Applying
a v(,).
~exp{ - 21,: } -#V
( 7)
(7),
noting
that
(8)
d
It is clear that -~
Hence
(8) implies
V(r))k_g~-x.
Moreover,
for
O~r~2, k)4
one has
~ - x ) k - ~ exp { - l / r } .
(6).
We prove that for
r>2
one has
p(O~k-Z~exp{-2/~}exp{-r'12}.
(9)
the density of the distribution
function F~(r)
We denote by px(r) = 2(2~) -x/~ e x p { - - r ~ / 2 }
P{IB~I < r}, by pa(r) the density of the distribution function F2(x) = P{~21r~2l + . . . + ~ k "
I~kl < r i, Then
px(y)p2(r-y)dy>~pl(r) f p2(r-,-y)dy.
p(r)=
0
(io)
r-2
We let
p (x) = (k - 1) -a (k-a) exp { - 2/6 ~ } exp { -
l/x }.
A c c o r d i n g t o ( 6 ) , f o r O<x<~2~p~(x)>~p(x) ( i f o n e n o t e s t h a t t h e G a u s s i a n v e c t o r
Oknk) s a t i s f i e s
the conditions
u n d e r w h i c h (6) was o b t a i n e d ) .
Consequently,
( 2~2,
|174
2
f p2(r-y)dy>~ f ~(x)dx>.p(1),
r--2
which with
(i0) implies
1
(9).
(6) and (9) imply the bound from the conditions
of the lemma.
The lemma is proved.
Proof of Theorem 2. In proving the theorem, without loss of generality we shall assume
that the sequence @~ satisfies
lim n t/(2"' @. = ~ .
(li)
n
If this is not so, then the sequence On
can be replaced by some new sequence @~>O,,
satisfying (Ii), the theorem proved with the sequence @~ in place of @, and it can then
be noted in this case that the assertion of the theorem remains valid for the sequence G~.
We choose integers k s + ~ such that the numbers ks,
following conditions (12)-(15) hold:
s~2,
are divisible by 128 and the
k0=O, k l = 1, k~>~24(~-2~, s>~2;
(12)
2cl N 5i~<~n~l(~m)0, ,
(13)
where we have let n s = ks/128, N = kl +...+ ks-l, and the absolute constant c~ is defined in
Lemma 8 ;
Such a sequence k s exists,
since
8re<~k~nl/(2~)-I/~@~;
(14)
N -an 4 -~ exp {- 2~_i } (128)-~+~/~,>I|
(15)
0.$0
and (ii) holds.
We define blocks I s of length k s of the series of natural numbers,
I,={m:O<rn-ko-kl-...-k~_l<~k,},
s=l,
by setting
2....
We give the construction of the Banach spaces E and B. As the space B we take the space
11. Further, let Z1(Is) be the smallest linear subspace of the space Zx, containing the set
{e~:i~f~}.
The space It(Is) has dimension ks and is naturally isometric with the space ~i ks.
Each element x ell can be written uniquely in the form
eo
s=l
Let ~s = (2S-lks)-2 (m-i)/m.
It is obvious that
315
~ k , ~/tl (m-xn< oo.
(]6 ~,
We set ~i = ~s, if i ~ Is and we consider the space 12.~, corresponding to the weight sequence %=(%0~:. As the space E we take the space ~ , ~ . It is obvious that condition (5),
guaranteeing the m-smoothness of the inclusion EcB,
coincides with (16).
We give the construction of the random elements X and Y. Let a s -- (4S-~ks) -(m-:)/m.
We define a symmetric random element A~') e l1(f,) with the help of the equation
P{XO)=a~e~}=P{X("= -a~ e, } =(2g) -~,
i~l,.
oo
We define the random element X ~ I I as a series 7f= I 76o); here let us assume that the toms=!
ponents X (~), X (=), ... are independent.
where we have set q -- 4 -(m-1)/m.
Obviously with probability one
~o
oo
s=l
s=l
It is easy to verify that the Gaussian random element Y can
oo
be written as a series Y = I ~ I ~.~e~, where the variance Os 2 = as2/ks, and the random
s 1 i~I,
v a r i a b l e s h i , s , i e l ~ , s = 1, 2, . . . , a r e i n d e p e n d e n t in a g g r e g a t e and a r e normal d i s t r i b u t e d
with parameters 0 and i.
=
We show that if M<as]/kJ(2rO,"
then
e{I yo, I~<M} ~<4=//q.
(17)
Let the real independent random variables n~, ..., qks b e n o r m a l ~ distributed with parameters
0 and i. We let ~i = ]niI--El~il and we note that El~il = 2/~2~. Hence,
P {1Y(')11< M } =P { ~i + - . - + E.k,< Mk~l~/as-2kJV~)
(we are using the inequality MksX/2/as < k s / 2 ~ , we apply Chebyshev's inequality, and we
remark that E-.[/2~<2)
.<P { E.I+... + $- < - k~/VY~ } ~<2=~7 =E(~ +... + ~)' ~<4=/~.
(17) is proved.
We let
s--I
Us~[ ~
s--I
S(n~ I1' Vs=[ ~ Y(i)']il
Ms=nl/~
i--I
i~l
al"
i=s
We prove that
nI/2as <~M, <~2n I/2a,.
The right inequality in (18) can be gotten, by sub-
The left inequality in (18) is obvious.
stituting in the series ~
at
(18)
the values of the numbers al, estimating
kt~>ks
for i > s,
summing the geometric progression obtained, and noting that m>~2.
We show that if 2Ms < a s / ~ ,
then
P{I Yll<r} ~4~/k~+P{ V ~<r-2M~i .
(19)
We have
eflYl,<r}~<e{v,+l r"~11<r}~<
(we consider the disjoint events IY(s) Ix < 2Ms and IY(~)II~>2M, , and we make use of the additivity of the probability)
<~p{~Y(~)I,<2M~}+P{V.,<r-2M~},
from which (19) follows in view of (17).
316
We verify that
P { IS. ll<r}>~P { V s < r - M s } - e l n - l l i
NS/~,
(20)
where the absolute constant ci is the same as in Lemma 8, N = kl +...+ ks-1.
one has IS~~
1t2 ai
and P { ] S , [ I < r } > ~ P { U s < r - M s } .
The estimate
P{ Us<r-M~}>~P{Vs<r-M,}-cln
after application of a suitable transformation
proof of (20).
!X(ol1<~a~,
-1raN 51~
follows from Lemma 8, which completes
2Ms<~asVk-~/(ir:),
It follows from (19) and (20) that if
Since
the
then
A,(r)>~ p { r _ 2Ms ~ Vs.< r _ Ms }_4rc/ks_c~ n-ll~ N~m.
We s h o w t h a t
n<<.ks/128,
if
and
s and r are
such
2 ~-2 r ~ l ,
that
(21)
then
A. (r)>/N-aN 4-~ exp { - 2k 4_ ~ } V (r) n ~12k; -1+I I,. _ 4re/k. - ci n-lP N~tZ.
where
N=k1+...+k~_1,
~(r)=exp
{-2]r-r~/2}.
(22 )
If 2M s < r/2, then Lemma 9 implies
P { r - i M s ~ < V s ~ < r - M s}>~M s inf
p(r) >~M sN-3uexp{-2/~ z_~}p(r).
r/2<x<r
According to the left inequality in (18) and the definition of as,
M~ >/n 1{2a~ i> nV~ 4 -~ ks ~1+1/".
Applying
(12), we get
~;-, =a~-llk~-i >12 -4(~-2) ks~3~>1kL4-,.
Combiningthe
estimates, we derive from (21) that (22) holds,
M,~rl4 ,
if
2 M : ~ a ~ V ~ / ( 2 n ).
(23)
But according to (18), the right inequality in (23) follows from n<~ks/128. Analogously, the
left inequality in (23) follows from (18), the estimates n<<.ks/128,m>.2
and the inequality
2 ~-2 r~>l. Estimate (22) is proved.
We estimate the first, second, and third summands on the right side of (22) with the
help of (15), (14), and (13), respectively.
We get that if n<~k~/12$,2i-ir~>i, then
A (r)'~ n 112(]~/128) -1+1/'" 6)a#z8 [~ (r) - 2 -1 n-~l ~ (k./12g)~li-ll (~") - 2 -1 n -1 (k~]i28)1-'-;(2"')}.
Letting n = n s = ks/128 , we get the estimate
(2) from the conditions of the theorem.
(4) obviously follows from (2).
(3) is easy to get by estimating the characteristic
Theorem 2 is proved.
function ~(0=L'ezp{it[Y+aIa}.
Proof of Corollary 3. Let c=sup{po(r):r>~O], where po(r) is the density of the distribution function Fo(r) = P{[YII < r}. Then as is easy to verify, &, (r)~<(l -'-c) L(!S~:[~ IY]~),
and
the assertion of the corollary follows from (4).
Proof of Corollarv 4.
We choose f(t) = 0 for
t~<1, f(t) = t -- 1 for 1 < t < 2, f(t) =
t
1 for
t~>2.
Then if we pass to distribution
functions and integrate by parts, ~,,= !" A,(t)dt
0
and the assertion of the corollary follows from (4).
The following lemma will be applied in the proof of ql'neorem 5.
LEMMA i0. Let y > 0, M>~0, the centered random elements Z and W assume values in the
space ~ k, and have identical covariance. P{]Z!Z<2}=Io
the random element W has Ganssian
distribution.
Let
p . = s u p l P i i S n ( Z ) [ ~ < r } - P i [ W l l < r } t,
r
A , (y) = l E ( IS,, (Z) t~ + M)v - E ( I W 11+ M)v I.
Then
A . (y) 4 c (y) k ~,'-~(1 + M v) p. In v (6/p.).
317
Proof. Let Fn and F be distribution functions of random variables [Sn(Z)I~ and IW]~
respectively. Let R = 40 k~/~ in (6/Pn). Integrating by parts, we get A~(y)<J~+/,,
where
R
i~ = (n + ,~)~,~oaX I F,, ~t)- F(t) I + V [" (t + M)~-~ I F, (t)- ~'(t) I at,
== '
0
J~=~" f (t+M)v-~[P{lS.(Z)[~>~t}+P{IWl~>~t}]dt.
R
S i n c e [F,~(t)-F(t)[<~p,, (a+b)v<~c(y) (av+bv),
one has Jl~<c(y) p,(Rv+mV).
Since Ix t1~<k1/~ Ix ]~,
applying the large deviation in Hilbert space inequality (el.
J u r i n s k y [ii], the corollary on p. 491), for t>~R we get
p { [ s,, (Z) lx >~ t } <~P { f S. (Z) l~>~k-a/2 t } <~2 exp { - k-X/2 t/20 }.
Analogously, P{lWh>~t}<~2exp{~k=if;'>-/2o}~ Hence, elementary calculations lead to the estimate J~<~e(y) kvi~-(l+Mv) p,.
Combining the estimate for J~ and J2, we complete the proof of
the lemma.
The proof of Theorem 5 to a large extent recalls the proof of Theorem 2. Hence we only
give an outline of the proof. We choose the sequences as, ~s, the Banach spaces f e B , and
the random elements X and Y just as in the proof of Theorem 2. About the sequence k s § = we
now assume that ko = 0, kx = 1 and we make the choice of k s more precise below. Us, Vs, M s
are defined above by (18).
In view of the choice of the numbers a i and the inequalities k,~>1,m~>2.
[ X(1) j_
+ X(~-I) ~
. . . .
[~.=al
z ,
T
. 9
. ~
,
2
a~_l
~<2.
Hence, according to Lemma i0,
E (U~ + M~)v~<E (I/'~+ 7ffAv+ c (y) Nvl2 (1 + MT) ~. ]nv (6/t%,),
(2 4 )
where
N=kl+...+k~-x,
,o.=sup[P{U~<r)-P{V~.<r}i.
r
We estimate the size of Pn with the help of Lemma 8" p,<~cxn-Xl~N 51~.
some constant c,(m, y) > 0 one has n-~/~N~/z<.c~(m, y),
then
Consequently, if for
p. In (6/p.) ~<c (m, y) [n-x/z N~/z]x-w".
(25)
~n(y)~ E(Vs+l Y(~)~I~-E(U~+ M~)L
(26)
Obviously
Using t h e i n d e p e n d e n c e o f Vs and
IY(S)l ,
we g e t
g(V.+l Y(~)}x~E(V~+lYc~)lx~,r~,~2M.)~E(V.+ZM.)V-P{IY(~)Ix~2M.}E(V~+2M.)L
(27)
Obviously
E ( Vs + 2Ms)e ~ c ~ ) (1 + MD ( 1+ EFD.
Since for
x~l~
( 28)
one has Ixlx~<NX/'~ Ix[2,
(we a p p l y t h e i n e q u a l i t y
values in a Hilbert space)
EVv~ <<.NY/~E y(n + . . . + y(,-1) I~;<<E]Z[~<~c(y)EY/aIZ[~ , which h o l d s f o r any G a u s s i a n v e c t o r Z w i t h
~<c (y) NY/~Ev/~ [ r(l) + . . . + y(,-x~ I~<~c (7) Nr/~,
(29)
since
El y ( x ) + . . . + y ( ~ - l ) ~ = ~ + . . .
Moreover, if
+a~-x~< 2.
2Ms<<.asV~s[(2~), then [of. (18)].
(30)
P { [ r(~) la ~<2Ms } ~<47z/k,.
Combining ( 2 7 ) - ( 3 0 )
318
and ( 2 4 ) ,
( 2 5 ) , and ( 2 6 ) , we g e t t h a t
if
n -~/~ N ~/~< c~ (m, y), 2M~ ~<a~ Vk-ff(2=),
(31)
then
(32)
ft. (Y) >1J1 - AJ~ - A/k~,
where
j~ = E (v~ + 2M~)~ ~- E (V~ + M~)':,
A = c (m, y) Nvl2 (1 + M~'), Js = c (rn, y) [n-1/2 N~l~]l-ll~
We estimate J~. We denote the density of the distribution
able Vs by p(x).
Then
2
2
function of the random vari-
2
Jl~ f [(x+ 2M,)~-(x+M~)Vlp(x)dx=yM~ : p(x) : ( x + t M ~ ) v - ~ d x .
1
1
I
The function (x + tMs)Y -I depends monotonically on x and t. Substituting x = t = i (or x =
t = 2) for y~I (for Y < i respectively) and estimating the density p(x) with the help of
Lemma 9, we get
A~c~,
m, s - l ,
N, ~ _ I ) M ~ ( I + M ~ - I ) ,
(33)
where the positive function c(m, Y, .,.,.) is independent of the specific choice of the
sequence ks, s~2.
We choose the integers n s proportional to the numbers k s . Namely, we let ns = ks/T,
s~2, with some sufficiently large T > I
For such a choice the right inequality in (31)
holds. Moreover, for s~2
one has c~, T) n~Im-I/2~M~].
Hence, combining (32) and (33), we
get that if nF ll~NS/~el(m, ~,
then
~n (Y)~
c~ n~Im-ll~ -- c~ n71 - c ~ fl12+
ll(~m)
(34)
where the positive constants c2, c3, c, depend on y, m, T, s -- 1, N = T(nl +...+ ns-~),
Tns-~ and are independent of the specific choice of the sequence n s + ~. In view of (34),
it is now easy to choose a sequence n s (consequently also a sequence ks) , such that Bns(Y)
@nn~lm-ll~. '
Theorem 5 is proved.
Proof of Theorem 7. Let us assume the opposite, that one can increase the exponent in
(5) in Theorem 6. Then there exists an m' < m such that the inclusion ~ c f ~
will be msmooth if (5) holds with m' in place of m.
It is easy to verify that in the proof of
Theorem 2 we only used (5)~ consequently, since (5) holds with m' instead of m, for the symmetric random element X of Theorem 2 for infinitely many n one has A~(1)~ ~,nl/m'-a/~
At the
same time, according to Theorem 1 and in view of the m-smoothness of the inclusion f~.>cll,
one has An(r ) = O(nl/m-I/=), which contradicts A.(1)~ ~n n~/~'-I/2 (Theorem 1 for symmetric X
remains valid for 3 < m ~ 4 ,
cf. [2]).
LITERATURE CITED
i.
2.
V. Yu~ Bentkus and A. Rachkauskas, " B o u n d s on the rate of approximation of sums of
independent random variables in a Banach space.
I , " Liet. Mat. Rinkinys~ 22, No. 3,
12-28 (1982).
V. Yu. Bentkus and A. Rachkauskas, " B o u n d s on the rate of approximation of sums of independent random variables in a Banach space.
I I , " Liet. Mat. Rinkinys, 22, No. 4, 8-20
(1982).
3.
4.
5.
6.
7.
8.
V. Yu. Bentkus, " A s y m p t o t i c s of moments in the central limit theorem in Banach spaces,"
Dokl. Akad. Nauk SSSR, 272, No. i, 17-19 (1983).
V. Yu. Bentkus and A. Rachkauskas, " B o u n d s on the distance between sums of independent
random elements in Banach spaces," Teor. Veroyatn. Primen~ 29, No. i, 49.-64 (1984).
V. Yu~ Bentkus, " L o w e r bounds on the accuracy of the normal approximation in Banach
s p a c e s , " Liet. Mat. Rinkinys, 24, No. i, 12-18 (1984).
V. Yuo Bentkus, " A s y m p t o t i c s of moments in the central limit theorem in Banach spaces,"
Liet. Mat. Rinkinys, 24, No. 2, 49-64 (1984).
I . S . Borisov, " L o w e r bounds on the rate of convergence in the central limit theorem
in Banach spaces," Teor. Veroyatn Primen., 29, No. 3, 604-605 (1984).
B . A . Zalesskii and V. V. Sazonov, " P r o x i m i t y of moments for a normal approximation in
Hilbert s p a c e , " Teor. Veroyatn. Primen., 28, No. 2, 251-263 (1983).
319
9.
i0.
ii,
12.
V. I, Paulauskas, " R a = e of convergence Sn the central llms theorem in certain Banach
spaces," Teor. Veroyaen. Primen., 20, No. 4, 775-791 (1976).
V, V. Senator, "Uniform bounds on the rate of convergence in the mul~idlmensional central limit =heorem," Teor. Veroya~n. Primen., 255, No. 4, 757-770 (1980).
V. V. Jurinsky, "Exponential inequal~ties for sums of random vectors," J. Multivar.
Anal., ~, No. 4, 473-499 (1976).
W. Rhee and Talagrand, " B a d rates of convergence for the central limit theorem," Ann.
Prob., 12, No. 3, 8 4 3 - 8 5 0 (1984).
LIMITING BEHAVIOR OF PRODUCTS OF
RANDOM TRIANGULAR MATRICES
UDC 519.21
A. Grincevicius
This paper is devoted to the study of the asymptotic behavior of distributions of products of independent, identically distributed (i.i.d.) random upper triangular matrices of
order m (m~2) with positive diagonal elements in the most interesting case, when the expectations of the logarithms of the diagonal elements are equal to zero. In the two-dimensional
case this problem was considered in a number of papers [1-2, 4, 9-11], and for matrices of
large dimension, under stringent restrictions in [3] and [5]. The results of the paper are
recounted in the abstracts [12-14].
In what follows we shall assume that ~(k) ~ {~ij(k), i, j = i, ~ r m}, k = i, 2,
are i.i.d, random matrices (r.m.) of order m with probability i, ~ij ~ J = 0 for
and ~ i i ( I ) > O, i = 1, . . . . m,
.o
~
The element in the upper right corner of the product ~,m (n) can be written in the form
~(n)__~lm
.
~
X(1,
.
ix, 1,. kOX(ix,
.
i~, kx+ 1, k2)
X(~, m, k a + l , n),
(2)
where the summation is over all collections
q=l .....
m-l,
1 </1 </~ < . . . </,~=rn,
(3)
1 <~k~<k2< . . . <k~<~n
and
Z(i, j, k, l)=~(k)~(k+Z)
%ii ~ii
"""
~(I-~)E(O
.i1"
"~ii
We l e t x ( i , j , k , k) = ~ i j ( k ) , x ( i , j , k , k - - i ) = i .
In our case the basic contribution
to the asymptotic behavior of the distribution of the random variable ~im (n) will be made by
the part of the sum (2), when q = m -- i, and consequently, ij = j + i, J = i, ..., m -- I,
i.e., the sum
I
over
.-(x)
%11
9
~(k,-l)~(k,)~(~,+n
~12
%22
9 9 ~11
" " "
~(k
-~)m C..(k,.
~+1)
m--l,
~mm-
9 9 9
~(.)
.ram
(4)
t~<k,<k=<...<k~_l~n.
m--I
It
is
easy
to see that
t h e n u m b e r o f summands i n
(2)
is
equal
to
~
C~._2
i-1 C., ~< C~ - l
2 m-2
i~l
for n~>2m-3, and since each summand is equal to the product of independent random variables
with bounded number of different distributions, one can hope that with some reservations the
distribution of the random variable (r.v.) I~im(n)[ behaves asymptotically just like the
absolute value of the largest summand of (2). For second-order matrices Grenander made this
Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR.
Translated from Litovskii Matematicheskii Sbornik (Lietuvos }~tematikos Rinkinys), Vol. 25,
No. 4, pp. 40-52, October-December, 1985. Original article submitted January 4, 1985.
320
0363-1672/85/2504-0320509.50
O 1986 Plenum Publishing Corporation