n - Kybernetika

262 P.K. SEN
The varying probability structure introduces additional complications in the study of
asymptotics for such estimators. Rosen [32], [33] considered an alternative approach
(via the coupon collector's problem) and presented deeper results. Let {Jfcj k > 1}
be a sequence of i.i.d.r.v.'s where
Let then
P{J k = r}=p r , l 1;
v k = inf{m(> k) : number of distinct J\,... ,J m = k}, k > 1. (3.14)
Then Rosen showed that for every n > 1,
n
HT n =Y,Yn» k = Bnv n , (3.15)
k=i
where for each m > 1, H nm is the bonus sum at the mth stage in a coupon collector's
problem with the set {a nr , p r ; 1 < r < n}, with a nr = a r /A(r,n), for
r = 1,..., N. Thus, the asymptotic behavior of randomly stopped bonus sums provides
the access to the general asymptotics for HT n and other related estimators.
Rosen's formulation rests on sophisticated nonstandard mathematical analysis, and
some simplifications and generalizations based on martingale approximations are
due to Sen [38]. By reference to a coupon collector's model, we consider a sequence
{QN} where for each N, QN = {(^N(l),Pi(l)), • • • ,(a N (N),p N (N))} and the nonnegative
p N (s) add upto 1. Define the JNk as in (3.12) and the YNk as in (3.13)
(with the ajfc/A(Jfc,n) being replaced by a N (J Nk ))- Let then
ZNn = ^Ynfc, n>l, Z NO = 0. (3.16)
k