The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 358 Wednesday, February 4, 2004 1:00 PM

358 The Mathematics of Financial Modeling and Investment Management

Many properties of power-law distributions are distinctly different in

the three following ranges of α: 0 < α≤1, 1 < α≤2, α> 2. The threshold

α= 2 for the tail index is important as it marks the separation between

the applicability of the standard Central Limit Theorem; the threshold α

= 1 is important as it separates variables with a finite mean from those

with infinite mean. Let’s take a closer look at the Law of Large Numbers

and the Central Limit Theorem.

The Law of Large Numbers and the Central Limit Theorem

There are four basic versions of the Law of the Large Numbers (LLN),

two Weak Laws of Large Numbers (WLLN), and two Strong Laws of

Large Numbers (SLLN).

The two versions of the WLLN are formulated as follows.

1. Suppose that the variables Xiare IID with finite mean E[Xi] = E[X] = μ.
   Under this condition it can be demonstrated that the empirical average
   tends to the mean in probability:

n

∑Xi

X i=^1 P [] = μ

n = ---------------- → EX

n n ∞ →

2. If the variables are only independently distributed (ID) but have finite
   means and variances (μi,σi), then the following relationship holds:

n n n

∑Xi ∑Xi ∑μi

X i=^1 P i=^1 i=^1

n = ---------------- → ---------------- = --------------

n n ∞ → n n

In other words, the empirical average of a sequence of finite-mean finite-

variance variables tends to the average of the means.

The two versions of the SLLN are formulated as follows.

1. The empirical average of a sequence of IID variables Xitends almost
   surely to a constant aif and only if the expected value of the variables
   is finite. In addition, the constant ais equal to μ. Therefore, if and only
   if EX[] i = EX[] = μ < ∞ the following relationship holds: