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Unformatted text preview: RS AND CENTRAL LIMIT THEOREM 4.9.1 157 Law of large numbers There are many forms of the law of large numbers (LLN). The law of large numbers is about the
sample average of n random variables: Sn , where Sn = X1 + . . . + Xn . The random variables have
n
the same mean, µ. The random variables are assumed to be independent, or weakly dependent, and
some condition is placed on the sizes of the individual random variables. The conclusion is that as
n → ∞, Sn converges in some sense to the mean, µ. The following version of the LLN has a simple
n
proof.
Proposition 4.9.1 (Law of large numbers) Suppose X1 , X2 , . . . is a sequence of uncorrelated random variables such that each Xk has ﬁnite mean µ and variance less than or equal to C. Then for
any δ > 0,
Sn
C n→∞
P
− µ ≥ δ ≤ 2 → 0.
n
nδ
Proof. The mean of Sn
n is given by E
The variance of Sn
n Sn
=E
n n
k=1 Xk n = n
k=1 E [Xk ] n = nµ
= µ.
n is bounded above by: Var Sn
n = Var n
k=1 Xk n = n
k=1 Var(Xk )
n2 ≤ nC
C
=.
2
n
n Therefore, the propos...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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