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If An rv cn P as n -+ 00 and p(1- a) > 1, (see (3.13)), then it is elementary to
show that (2.1) and (2.2) are satisfied for n sufficiently large. However the asymptotic
behavior of An, An rv cn P as n -+ 00, is not sufficient to actually determine the values of
n for which conditions (2.1) and (2.2) are satisfied. Some indications can be obtained
if more informations on the An are available, for instance
Lemma 3.1. If An = cnP for all n and if
tben, for any rl, r2
for some n sucb tbat
p(l-a) > 1,
An+l - An ~ rl (A~+l + A~),
A~-n ~ r2,
wbere C3 is an appropriate constant.
Proof. Choose no such that
-lip IIp(l-n) < < 1 + -lip IIp(l-n)
C
r 2
_ no
c
r 2
,
so that (3.15) is satisfied for all n ~ no. The lemma is proved if
If not we observe that (3.14) and (3.15) are satisfied by n ~ N + 1,
if N < 00. To show that N < 00 and estimate N we add the inequalities
for j = no, ... , N; this yields
N
AN+l - Ano ::; rl L (Aj+l + Aj)
j=no
N+l
::; 2rl L Aj;
j=no
(3.13)
(3.14)
(3.15)
(3.16)