As mentioned in class, the problem needs to be
restated.
Let 
, 
be independent
Bernoulli random variables with success probability
, and put 
= 
,
= 
. Assume
Actually, since 
we have 
and of course
1, so in fact condition (i)
implies condition (ii), and the latter is unnecessary.
We will apply the Lindeberg CLT given in class. That is stated as follows:
We will use our triangular array of Bernoulli random
variables, where 
. Conditions (I), (II), and
(IV) hold immediately. Of course, (III) and (V)
are achieved by centering and normalizing: define
which is the variance of the row sum 
.
Then put
Then (III) and (V) hold. Thus, we only need
to do any work to verify the Lindeberg
condition, condition (VI). Now the event
in the Lindeberg
condition is the same as
But 
only takes on the values
0 and 1, and 
is between
0 and 1, so
However, 
(condition (i) in the problem),
so for each 
there
is an 
such that for all 
,
, i.e. for all
, 
and hence
by (38)
the event in (37) is empty.
Thus, the indicator inside the expectation
in (34) is 0 and so for all
,
We see then that all the conditions in
the Lindeberg theorem hold, so 
.
Observe that 
is the l.h.s. of
(33), and the problem is done.
