13-Fat Tails-Scaling-Stabl Page 359 Wednesday, February 4, 2004 1:00 PM
--------------- ---------------
----------------
--------------
Fat Tails, Scaling, and Stable Laws 359
n
∑ Xi
X i =^1 A.S. [] = μ
n = ---------------- n → ∞ → EX
n
where convergence is in the sense of almost sure convergence.
- If the variables Xi are only independently distributed (ID) but have
finite means and variances (μ i,σ i) and
n
(^12)
lim ------∑σ i ∞ <
n ∞ → 2
ni = 1
then the following relationship holds:
n n n
∑ Xi ∑ Xi ∑ μ i
Xn i =^1
n
= -
n ∞ →
A.S.
→ i =^1
n
- = i =^1
n
Suppose the variables are IID. If the scaling factor n is replaced with
n , then the limit relation no longer holds as the normalized sum
n
∑ Xi
i = 1
n
diverges. However, if the variables have finite second-order moments,
the classical version of the Central Limit Theorem (CLT) can be demon-
strated. In fact, under the assumption that both first- and second-order
moments are finite, it can be shown that
Sn – nμ D
---------------------- Φ →
σ n
n
Sn = ∑ Xi
i = 1