The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 359 Wednesday, February 4, 2004 1:00 PM

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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.

2. 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