On a conjecture of J. ZinnLet (£„) be a sequence of real, symmetric independent random variables defined on some probability space (Q, F, P

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVIII (1989)

Pa w e l Hit c z e n k o

(Warszawa)

On a conjecture of J. Zinn

Let (£„) be a sequence of real, symmetric independent random variables defined on some probability space (Q, F, P). Let F0 = ( 0 , Q), F„

= < 7 (£i, £„), n = 1, 2, ... and Foo = < 7 ((JFn). Denote by (£;) a copy of (£„) independent of F Let (X, || ||) be a Banach space. If (vn) is a sequence of X- valued random vectors, then (i>„) is said to be predictable (relative to (F„)) if, for every positive integer n, vn is F„_ j-measurable. In this paper, (r„) denotes a Rademacher sequence independent of all other random variables under consideration.

It was conjectured by J. Zinn that for every Banach space X and all sequences (r„), (£,,) and (ЗД as above, ^ v k^krk converges a.s. if and only if Yuvk£krk converges a.s. It follows(L) that this is true if X = lp, 1 ^ p < oo.

The aim of this note is to show that in'general such equivalence is false.

More precisely, we show that the implication J]vk£krk converges a.s-=>Yavk^krk is bounded as. fails down in the Banach space c0.

We shall use the following notation. If A is a finite subset of the set of positive integers N, then |Л| denotes the cardinality of A. Let be a random variable distributed by the rule:

P(V — 0) = 2_ 1, P ( i/= 1) = P(rç = -1 ) = 2 " 2.

Let (rjk) be a sequence of independent copies of rj and denote by (rjk) an independent copy of (rjk). If A = \ait ..., ap\, ax < ... < ap, щеЫ, i

= 1, . . . , /V then (Л) = И Г 1/2,

v 2

(Л) = \A\' 1/2(1 - 1r,aiI), ..., (A) = \A\- 1/2 (1 - \r,aiI)... (1 - \nUp_, I),

*lj (A) = Vaj, rj'j (A) = ri'aj, Tj (A) = raj, j = 1, 2, ..., p. (* )

(*) J. Z in n (1985), C o m p a r is o n o f m a r t in g a le d iffe r e n c e s e q u e n c e s . P r o b a b ility in B a n a c h s p a c e s V. Proceedings, 1984. Lecture Notes in Mathematics, 1153, 453-457, Springer-Verlag, Berlin, or Seminar Notes on Multiple Stochastic Integration, Polynomial Chaos and Their Applications. Case Western Reserve University, Preprint 85 # 3 4 .

4 — Commentationes Math. 28.2

220 Paw el H it c z e n k o

Note that for every finite subset A cz N one has

(1) ^I Z Ml vk(A)rik(A)rk{A)\ ^\A\ 1/2,

к

Ml Ml

(2) P(| £ vk(A)r,'k(A)rk(A)\ > И11'2) = j“(| s vk(A)

’k(A)\ 3= И 1'2)

k=

k=

==

P{vk{A)

+ Р(гк(Л) = 1, »yHA) = - 1 , к = 1, 2, \А\)

= 2 Р{П1 (А) = ... = щА^ х (А) = 0, п\ (А) = ... = rj\A] (А) = 1)

= 2 -2 "

+ 1 -4~И| = 2-3|j4| + 2.

Let

be the usual basis of c0 and denote by {

*} a sequence of coordinate functionals on c0.

E

. There exists a sequence |pk} of c0-valued random vectors predictable relative to {rjk} such that

Z

converges a.s.

and

Yjvk4krk is unbounded a.s.

P ro o f. Let {gk} be any sequence of positive integers such that:

(3) 9 i = 0 , (1 —2~2,k + 2)qk + 1 ^ 2-1 , к = 1 ,2 ,...

and set

n k = <?i + . . . +<?fe, k = 1 , 2 , . . .

Let {Л„} be a sequence of subsets of N such that

(4) Ao = 0 , A„ — { Z Ц/1 + 1» Z Иу 1 + 2 , ..., Z 14/1+ Ип1}»

; = i j= i j= i

/1 = 1 , 2 , . . . , where

(5) \A„\ = к for ик < n ^ ик+1, к = 1, 2, ...

Define random vectors vk as follows: if к = И 0|+ ... + |Л р_ 1|+у for some p

^ 1 and j = 1, ..., |Лр|, then rk = t>j(4p)ep. It is clear that {rk} is a sequence

of c0.-valued random vectors and that it is predictable relative to {rjk}. To

prove that Z vk 4k rk converges a.s. note that for all positive integers n, m and

Conjecture of J. Zinn 221

j such that m > n the following inequality holds:

m m л ( \ А 0 \ + ... + \A j \)

( 6 ) Ie j ( Z vkrjkrk)| = \ef ( ' Z vk*lkrk)

k = n

\ A j \

k = nv( MqI + ••• + M y - l l + 1)

< I Z Vk(Aj)lk(Aj)rk{Aj) |.

k= 1

For £ > 0 choose a positive integer j 0 such that

(7) И ; 0 Г 1/2 < e .

If n > H ilF ... +\Aj ^ then (1), ( 6 ) and (7) imply that for each m > n one has

\ef ( Z ^{vk Пк} rk)|- ^{< e, j =} 1,2,...

k = n

Consequently,

n + p

lim P (sup || £ vk rjk rk|| > £) = 0

n p k = n

and thus Z

converges a.s.

n

To see that sup|| £ ukrçkrk|| = oo a.s. observe that for each j = 1 ,2 ,...

( 8 )

n k = 1

M 0 l + ... + Mjl

Mol + •••+

Ief ( Z vk Пк rk)| = \ef ( Z vk Vk rk)|

k = Mol + • ••+ M j - l l + 1 k = 1

= I Z

vk(AjWk(Aj)rk(Aj)\.

fe= l

Mjl

It follows from (4) and (5) that £ vk (Aj) rjk ( Aj) rk (Aj), j = 1 ,2 ,... are k = 1

independent random variables and that for each p = l , 2 , ...,

Mjl

Z vk(Aj)r\'k(Aj)rk(Aj), np <j ^ np+ i are identically distributed. Therefore, k=i

by (2), (3) and (5) we have

P( max \ef ( jj vk rj'k rfc)| ^ pl/1)

np < J ^ np + 1 k = 1 Mjl

= P ( max I £ vk (Aj) r\’k (Aj) rk (Aj)| ^ p112)

np < J ^ np + 1 k = 1 M " p + l l

к = 1

= l - ( i - 2 ~ 3p+2)qp+1 ^ 2~1,

222 Paw el H it c z e n k o

whenever ÏI > Ml| + ••• + И «р+11

00 \Aj\

(9) i n max | Y

P=1 »p<j ^ np+ i k= ¹

\Aj\

Since ]max | Y

;Vk(Aj) П'к (Aj)

np < J ^ np + l k = 1

( 8 ), (9) and the Borel-Cantelli lemma we obtain that P(sup\\ Y vkt]'krk|| = oo) = 1,

П k = 1

The proof is completed.