The WLLN is said to hold for a zero-mean sequence {Sn; n ≥ 1} if Sn >E lim Pr n→∞ n = 0 for every E > 0. The CLT is said to hold for {Sn; n ≥ 1} if for some σ > 0 and all z ∈...

The WLLN is said to hold for a zero-mean sequence {S
n;
n
≥ 1} if

S
n

n→∞

n

= 0 for every
E

>
0.

The CLT is said to hold for {S
n;
n
≥ 1} if for some
σ >
0 and all
z
∈ R,

S
n

n→∞

o
√ ≤
z

=
δ(z),

where
δ(z) is the normal CDF. Show that if the CLT holds, then the WLLN also holds. Note 1: If you hate
E,
δ
arguments, you will hate this. Note 2: It will probably ease the pain if you convert the WLLN statement to: For every
E

>
0,
δ

>
0 there exists an
n(E,
δ) such that for every
n
≥
n(E,
δ),

Pr
S
n

n

−E

≤
δ
and Pr

S
n

n

>
1 −
E

≤
δ. (i)