STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
ID. The Dependent Case
A set of random variables r = (ru ...,r„) is said to be multivariate stable
if every linear combination of r is univariate stable. The characteristic function
of r is
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0, and the central tendency measure S is linear
homogeneous. The characteristic exponent a is assumed to satisfy 0 < a ^ 2.
With specific choices of 1. Press sets 5 (t) == f't
and y(t) = iZA (f'Jty) a/2 where n^m^l and each Q,- is a positivesemidefinite
matrix of order n x n. The log characteristic function is then
m
(9) \m/,r(t) = if't-$Z(t'njty' 2 .
J=I
The m in (9) indicates the number of independent partitions of the random
variables (/•,, ...,rn). Hence if m = 1, all the random variables are dependent
(as long as Ql is positive definite). Then if a = 2, the case of normal
distributions,
which is precisely half the variance of the linear sum t'r. If m = n, the
random variables are independent. In this case with a = 2 one may interpret
y(t) as half the variance by setting the7jth coefficient of ilj equal to coj > 0
(and all other coefficients equal to zero). Then
m n
J=I j=i
Thus the class of distributions defined by (9) allows for the decomposition
of the (r J,..., r„) into independent parts in a way consistent with and motivated
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 257