In this section we summarize that a continuous functions obtained by it- erated random functions converge to a standard normal distribution. The machinery which used to prove this result rests on the stability theory de- veloped in Section 2. These techniques are extremely appealing as well as powerful, and can lead to much further insight into asymptotic behavior
of the iterated random function system. Here we will focus on two results:
central limit theorem and quick convergence.
Let g G CQ(TT) be a square integrable function with mean 0, i.e.
/ g d-K = 0 and \\g\\l = / g2 dn < oo. (19) Jx Jx
Consider the sequence
Sn{g) := s(Mi) + - " + 5 ( Mn) , n > 1, (20) which may be viewed as a Markov random walk with driving chain (Mn)n>o-
By constructing a solution h £ C2(ir) to the Poisson equation
h = g + Ph, (21) where Ph(x) := Jx h(y) P(x, dy), and a subsequent decomposition of Sn(g)
into a martingale and a stochastically bounded sequence, Benda (1998) showed that Sn{g)/^/n is asymptotically normal as n —* oo under Px for 7r-almost all x e X, if g e £jCiP(X, R ) ,
ELl < 1 and Ed(Fi(x0),x0)2 < oo. (22) It was observed by Wu and Woodroofe (2000) that these conditions may be
relaxed if the integrability assumption on g is slightly strengthened to g £
C2(-IT) n £r(7r) for some r > 2. Their further assumptions are -Elog+ L\ <
0, (11) and a 7r-square integrability condition on a certain local Lipschitz constant for g with respect to a flattened metric %p o d. The main point is that it allows discontinuous g, for instance suitable indicator functions. As to the above mentioned local Lipschitz constant for g, we will show that its integrability (instead of square integrability) with respect to n suffices.
We will further give sufficient conditions for the /3-quick convergence of n~1Sn(g) to 0. The concept of quick convergence was introduced by Strassen (1967). A sequence (Zn)n>o is said to converge /?-quickly (/? > 0) to a constant /i if
E ( s u p { n > 0 : \Zn-p\ > e})" < oo (23) for all e > 0. Plainly, Zn —ằ ji /?-quickly implies Zn —> /z a.s. Put Ne :=
sup{n > 0 : \Zn - fi\ > e}. Since (23) then reads EiVf < oo for all e > 0, the /?-quick convergence of Zn to fi holds if, and only if,
^2 n^-1P(JVe >n) = ^ n / 3~l p f SUP \ZJ - Ml > e ) < oo (24)
n > l n > l ^ n '
for all E > 0.
Our results will be stated in Theorem 3.1 and Corollaries 3.1 and 3.2.
As in Benda (1998), and Wu and Woodroofe (2000), the bulk of the work is to verify the existence of a solution to the Poisson equation (21). This is the content of Theorem 3.1. The asymptotic normality of Sn{g)/y/n (Corollary 3.1) then follows as in Benda (1998) by applying a martingale central limit theorem; while the /?-quick convergence of Sn(g)/n to 0 for suitable /3 (Corollary 2) will be obtained by using a result from Alsmeyer (1990), and Fuh and Zhang (2000).
Some preliminary considerations are needed before presenting our re- sults:
A. Flattening the metric. In order for solving the Poisson equation (21) for a given function g, the particular given complete separable metric d on the space X will not be essential but may rather be altered to our convenience. This has been observed by Wu and Woodroofe (2000) who therefore consider flattened variations of d obtained by composing d with an arbitrary nondecreasing, concave function tp : [0, oo) —> [0,oo) with ip(0) = 0 and ip(t) > 0 for all t > 0. Let \J> be the collection of all such functions. It is easy to see that d^ := ip o d is again a complete metric.
Possible choices from $> include tpp{t) := tp for any 0 < p < 1 as well as ip*{t) := j^n.. The latter choice leads to a bounded metric d,/,* satisfying
d^<(x,y) < d(x,y) < 2d^,.{x,y) (25) for all x,y £ {(u, v) G X2 : d(u,v) < 1}. This shows that the behavior
of d and d$* is essentially the same for small values. Notice further that tp* o %j) G \J> with
l i m ^ . ! ( 2 6 )
no v(*)
for all V> € * .
B. Integrable local Lipschitz constant. One can further relax the global Lipschitz continuity of g needed in Benda (1998) and instead be satisfied with a 7r-almost sure local Lipschitz continuity (with respect to a flat- tened metric d$) in combination with an integrability condition on the local Lipschitz constant. To make this precise, let ip € * . For a measurable g : X —> R, define its local Lipschitz constant at x € X with respect to d$
l^{g,x) = sup — — r— {it)
y.0<d{x,y)<l a-i>\X,y)
and, for r € [l,oo],
IMIr,* = IMff.Ollr, (28) where || • ||r denotes the usual norm on Cr(n). It is easily seen that || • \\r^
defines a (pseudo-) norm on the space
^V-.oC71") = | s £ £ r(7 r) : / 9(x) n(dx) - 0 and \g\T^ < oo > (29) and that £ ^0( T T ) = £5,.o^i0(7r) w i t h \ II • IkvoV ^ II - llr,v ^ II ' I k v * ^ o n
this space (use (25) and (26)). Possibly after replacing i/> with ip* o ip, we may therefore always assume ip be bounded when dealing with elements of
Plainly, all global Lipschitz functions, i.e. all g S £ L ,P( X , R ) , are ele- ments of £JJ, 0(7r) for any ip G $>. However, 5 need not be continuous in order for being an element of some £^j0(7r)- As pointed out in Wu and Woodroofe (2000), if g = I s is the indicator function of some B € B(X), then
ôlB>x) = dj^y (30)
where dB denotes the topological boundary of B and d^{dB,x) :=
infy€dBd(x,y). They further show that, if B(x,R) = {y : |a; — y\ < R}
is the closed J?-ball with center x £X, ip(t) — i1/4 and A denotes Lebesgue measure, then, for each x eX, 1M(X,R) — 7r(B(x, i?)) G £^>0(7r) for A-almost all R > 0, see their Theorem 3.
Theorem 3.1. Let r £ (l,oo] with conjugate number s > 1, given by
\ + j = 1. Let aZso ip e \I> 6e satisfying
r ^ d t < 00. (31)
0 *
7/ E l o g+ l a < 0 and fityj holds for some p > s, then each g e £r(7r) n
£^0(7r) admits a solution h € £0(71") to the Poisson equation h = g + P/i.
With the help of the Poisson equation, one may write
Sn(g) = Wn + Rn, 71 > 1 (32) where
n
Wn := 5 3 ( / i ( Mf c) - P / i ( Mf c_ 0 ) , n > 0 (33)
fc=i
l
forms a zero mean martingale under Pn with stationary increments from
£r(7r) and
Rn := Ph(M0) - Ph(Mn), n > 1 (34) is stochastically £r-bounded under P„- in the sense that
s u p P ^ d i ^ l > t) < 2P7 r(Z>£) (35)
n > l
for all t > 0 and some Z G Cr(ir); take any random variable Z > 0 with distribution function P^(|P/i(M0)| < t/2) for t > 0.
In the stationary regime, that is under P„., the following central limit theorem now follows exactly as in Benda (1998) from Theorem 3.1 and a martingale central limit theorem. However, an additional argument is needed to show that the same result holds true under Fx for 7r-almost all x G X. While this extension is not considered in Wu and Woodroofe (2000), its proof in Benda (1998) fails to work here because it draws on the continuity of g and a moment condition like (10).
Corollary 3.1. Given the assumptions of Theorem 3.1 with r > 2 and p > s, Sn(g)/y/n is asymptotically normal with mean 0 and variance
s2(flO : = J{h2 — {Ph)2)dit under F„ as well as under Px for n-almost all i € l
So if g G £2(7r)n£j, 0(7r) we need moment condition (11) for some p > 2, to conclude asymptotic normality of Sn(g)/y/n. By using the result of the existence of a solution for the Poisson equation (21), the following corollary is taken from Theorem 2 in Fuh and Zhang (2000).
Corollary 3.2. Given the assumptions of Theorem 3.1 with p > s > 1, Sn(g)/n converges (3-quickly to 0 for (3 = r — 1, i.e.
^ ] nr-2P7 r( s u p r1| ^ (5) | > £N) < co (36)
n > l V^n J
for all e > 0.
References
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4. M. Benda, J. Appl. Prob. 35, 200 (1998).
5. P. Diaconis and D. Freedman, SIAM Review 4 1 , 45 (1999).
6. M. Duflo, Random Iterative Models, Springer-Verlag, New York. (1997).
7. J. H. Elton, Stoch. Proc. Appl. 34, 39 (1990).
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Springer-Verlag, New York. (1993).
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STRONG A P P R O X I M A T I O N S *
LI-XIN ZHANG
Department of Mathematics, Zhejiang University, Xixi Campus,
Hangzhou 310028, China, E-mail: Ixzhang@mail. hz. zj. en
Various adaptive designs have been proposed and applied t o clinical trials, bioas- say, psychophysics, etc. More and more people have been paying attention to these design methods. Via strong approximations, this paper presents asymptotic properties of several broad families of designs, such as t h e play-the-winner rule, randomized play-the-winner rule and its generalization to t h e multi-arm case, dou- bly biased coin adaptive design, Markov chain model.
AMS 1999 subject classification. 60F15, 60F05, 62G10, 60G10.
Key words: clinical trial, adaptive designs, (Randomized) play-the-winner rule, biased coin design, Markov chain, asymptotic properties, urn model.
1. Introduction
Traditional designs for clinical trials use the balanced (or 50%-50%) allo- cation of patients to treatments. For example, in a trial to compare ex- perimental therapy (drug) to placebo (control), a standard feature of most designs is to distribute half of patients to each arm. It is reasonable that one may want to reduce the total number of failure outcomes in a trial, and keep the capability of making a comparison between experimental therapy and placebo as well. Hence the idea of adaptive designs has been proposed to serve the purpose.
Adaptive design, an important subdivision of experimental designs, are designs in which the probability a treatment assigned to the coming patient depends upon the results of the previous patients in the study. The goal is to show assignment probabilities to favor better treatment performance. This
* Research supported by National Natural Science Foundation of China (No. 10071072).
112
kind of design has also been applied to bioassay, psychophysics, etc. This paper gives some asymptotic properties of several broad families of designs via strong approximations, such as the play-the-winner rule, randomized play-the-winner rule and its generalization to the multi-arm case, the drop- the-loss rule, doubly biased coin adaptive design, Markov chain model.
We consider the K-treatment clinical trial (K > 2). Suppose the pa- tients are recruited into the clinical trial sequential. Let Xnk = 1 if the n-th patient is assigned to the treatment k and = 0 if otherwise, and let Nnk = S m = i Xmk be the number of patients assigned to the treatment k after n assignments, k = 1,...,K. Write Xn = ( Xni , . . . ,Xnx) and Nn = ( i Vn l, . . . , Nn K) . Denote {£n = ( £ „ i , . . . , £n K) , n = 1 , 2 , . . . , } to be the sequence of responses of patients, where £nk is the response of the n- th patient on the treatment k, k = 1,...,K. {£„} is usually assumed to be a sequence of independent random vectors. In the clinical trial, only those £„fcs for which Xnk = 1 appear. But we assume that all £ni , . . . , £„#, n = 1,2,..., are there and only those non-zero ones of Xnk£nkS are selected in the trial.