Utilities of the finite family of random variables
Yutaka Matsushita · 2003
In this paper, we first represent addition of independent random variables by the symmetric product on the symmetric algebra S(V), and then develop two utilities of random vectors: 1) an additive utility of the sum of independent random variables (Proposition 1) deduced by a utility over vector space V (Theorem 2); and 2) a utility over a tensor space T/sup k/ (Theorem 3). The distinction between two utilities over V and over T/sup k/(V) is dependent on two (weaker and stronger) solvability conditions. The weaker one deduces a utility of random variables which are connected by a non-commutative operation. Finally, we give a "holistic" order to such random vectors that they are decomposed into two types of components represented by the convolution and the non-commutative operation.