6.10 Let Y be a random p-vector, and let Y(t) denote its characteristic function. Suppose log ,...

6.10 Let Y be a random p-vector, and let φY(t) denote its characteristic function. Suppose log , for some positive integer m, where Ωj : p × p, Ωj ≥ 0 for every j, and 0 < α ≤ 2. Show that log φY(t) is concave if and only if 1 ≤ α < 2, and that for 0 < α < 1, log Y(t) is neither concave nor convex. [Hint: Calculate the Hessian matrix and show that its latent roots are positive if and only if 1 < α ≤ 2. Then apply Property (8) of 2.11.2, and the results on concavity in 2.14.3. This result will be useful in portfolio analysis (see Chapter 12).] Why must the case of α = 1 be studied separately?