In this respect, one is interested in the subclass C-�� of C- of all distributions satisfying besides ��^=X- the mean orthogonal property �̦�. This so-called mean orthogonal class is characterized as follows.Theorem 2 (Characterization of the mean orthogonal class) ��Let X be a random variable with cgf C(t; ��, ��) satisfying the above assumptions. Then, one has selleckbio X��C-�� if and only if the following quasi-linear partial differential equation is satisfied:��2??C?��?(?C?t?��)=0.(2)Proof ��This is shown in H��rlimann [16]. Hudson [17, Theorem 1] has shown that the class C- is closed under convolution. In fact, convolution invariance holds under the more stringent mean orthogonal property.Theorem 3 (Convolution invariance of the mean orthogonal class) ��If X1,X2��C-�� are independent, then X=X1+X2��C-��.
More precisely, if Xi��C-�� has cgf Ci(t; ��i, ��(i)), with ��i��(i), i = 1,2, then the cgf C(t; ��, ��, ��) of X = X1 + X2, with �� = ��1 + ��2, �� = |��1��22 ? ��2��12|, and �� = (��(1), ��(2)), satisfies (2) and one has ��^=X-, ��(��, ��).Proof ��Without loss of generality we assume that �� = ��1��22 ? ��2��12 > 0. Since �� = ��1 + ��2 and ��2 = ��12 + ��22, one can express (��1, ��2) as a function of (��, ��) through the parameter transformation ��1 = (�̦�12 + ��)/��2, ��2 = (�̦�22 ? ��)/��2. Since X1,X2��C-�� and C(t; ��, ��, ��) = C1(t; ��1, ��(1)) + C2(t; ��2, ��(2)), one =?C(t;��,��,��)?t?��,(3)which?=?C1(t;��1,��(1))?t?��1+?C2(t;��2,��(2))?t?��2?=��12??C1(t;��1,��(1))?��1+��22??C2(t;��2,��(2))?��2?=��2??��1?��??C1(t;��1,��(1))?��1+��2??��2?��??C2(t;��2,��(2))?��2?obtains��2??C(t;��,��,��)?�� implies the result by (2) of Theorem 2.
Example 4 ��Binomial random variables and their convolutions belong to the class C-��. For two binomials this is shown in H��rlimann [16, Example 2] (see also [21]). For an arbitrary number of binomials this is derived in the Appendix of Hudson [17].3. Mean Orthogonal Characterization of the Compound Gamma DistributionConsider random sums of the typeX=��i=1NYi,(4)where the Yi’s are independent and identically distributed nonnegative random variables, and N is a counting random variable defined on the nonnegative integers, which is independent of the Yi’s. The mean and variance of X, N, and Y ~ Yi are denoted, respectively, by ��, ��2, ��N, ��N2, and ��Y, ��Y2. The coefficient of variation of X is denoted by �� = ��/��. In some applications, it is convenient to scale the severity Y by the mean �� such that the mean Brefeldin_A scaled severity Z = Y/�� ~ Zi = Yi/�� has mean 1/��N. The resulting sumX=��?��i=1NZi(5)is called mean scaled compound random sum. The mean scaled compound model has important insurance risk applications.