Continuous Probability Distributions Commonly Used in Financial Econometrics 357
Xn be identically distributed and independent of each other. Then, assume
that for large n ∈ N, there exists a positive constant an and a real constant
bn such that the normalized sum Y(n)
Yn()=+aXnn() 12 XX++... +αbSn~(,,βσ,)μ (B.9)
converges in distribution to a random variable X, then this random variable
X must be stable with some parameters α, β, σ, and μ. The convergence in dis-
tribution means that the distribution function of Y(n) in equation (B.9) con-
verges to the distribution function on the right-hand side of equation (B.9).
In the context of financial returns, this means that α-stable monthly
returns can be treated as the sum of weekly independent returns and, again,
α-stable weekly returns themselves can be understood as the sum of daily
independent returns. According to equation (B.9), they are equally distributed
up to rescaling by the parameters an and bn.
From the presentation of the normal distribution, we know that it
serves as a limit distribution of a sum of identically distributed random
variables that are independent and have finite variance. In particular, the
sum converges in distribution to the standard normal distribution once the
random variables have been summed and transformed appropriately. The
prerequisite, however, was that the variance exists. Now, we can drop the
requirement for finite variance and only ask for independent and identical
distributions to arrive at the generalized central limit theorem expressed by
equation (B.9). The data transformed in a similar fashion as on the left-hand
side of equation (B.2) will have a distribution that follows a stable distri-
bution law as the number n becomes very large. Thus, the class of α-stable
distributions provides a greater set of limit distributions than the normal
distribution containing the latter as a special case. Theoretically, this justi-
fies the use of α-stable distributions as the choice for modeling asset returns
when we consider the returns to be the resulting sum of many independent
shocks with identical distributions.