Homework Set No. 7 – Probability Theory (235A), Fall 2009
Posted: 11/10/09 — Due: 11/17/09
1. (a)
Read, in Durrett’s book (p. 63 in the 3rd edition) or on Wikipedia, the statement
and proof of
Kronecker’s lemma
.
(b)
Deduce from this lemma, using results we learned in class, the following rate of con
vergence result for the Strong Law of Large Numbers in the case of a ﬁnite variance: If
X
1
,X
2
,...
is an i.i.d. sequence such that
E
X
1
= 0,
V
(
X
1
)
<
∞
, and
S
n
=
∑
n
k
=1
X
k
, then
for any
± >
0,
S
n
n
1
/
2+
±
a.s.
→
n
→∞
0
.
Notes.
When
X
1
is a “random sign”, i.e., a random variable that takes the values

1
,
+1
with respective probabilities 1
/
2
,
1
/
2, the sequence of cumulative sums (
S
n
)
∞
n
=1
is often
called a
(symmetric) random walk on
Z
, since it represents the trajectory of a walker
starting from 0 and taking a sequence of independent jumps in a random (positive or
negative) direction. An interesting question concerns the rate at which the random walk
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 Fall '11
 DanRomik
 Probability, Probability theory, Sn, xk, SLLN

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