A followup to this construction for the individual model of risk theory is H��rlimann [19]. We clarify and simplify the original proof to obtain a characterization of (4), which will be used in Section 4. In particular, (3.20) in H��rlimann [18] is not a consequence but an assumption. Since this equation is satisfied in the provided examples, this error does not harm the obtained FTY720 result but must be rectified from a mathematical logical point of view. Also, the proof of Lemma 7 there will be simplified (proof of Lemma 8 below).Theorem 5 (Compound gamma characterization) ��Let N��C-�� be a counting random variable with cgf CN(t; p, ��N). Suppose there exists a one-to-one coordinate transformation mapping (p, ��N) to (��N, ��N) such that ��N = ��N(p, ��N)��N, and set ��N = ?lnPr(N = 0).

Suppose the cgf CY(t) of the severity Y exists, and let C(t) = CN(CY(t)) be the cgf of the random sum X = ��i=1NYi. Assume the cgf of the mean scaled severity CZ(t) = CY(t/��) is functionally independent of ��, and set �� = (��N��2)?1, and �� = ��/��. If X��C-�� and ��2 ? (?��N/?��) = ��, then Y is gamma distributed with cgf CY(t) = �� ? ln ��/(�� ? ��Yt).To show this, some preliminaries are required. First, we review conditions under which N��C-��. Given the probability generating function (pgf) P(s) = E[sN] = ��n=0��p(n)sn of N, it is very useful to consider the associated so-called cumulant pgf defined ��N=?ln?P(0)=G(1).(6)The given name stems???byG(s)=ln?P(s)+��N=��k=1��g(k)?sk, from the following series representation of the cgf:CN(t)=ln?P(et)=G(et)?��N=��k=1��(ekt?1)?g(k).

(7)Remark 6 ��The sequence g(k), k = 1,2,��, is the unique solution of the system of equations (e.g., [22, Corollary 2], [23], and [24, Theorem p(0)=e?��N.(8)If g(k) �� 0, k = 1,2,��,?1]):n?p(n)=��k=1nk?g(k)?p(n?k),n=1,2,��, the distribution of N is compound Poisson with parameter ��N and severity distribution h(k) = g(k)/��N, k = 1,2,��. Otherwise, one speaks of the so-called pseudo-compound Poisson representation of the distribution.Lemma 7 �� Let N be a counting random variable with cgf CN(t; p, ��N) of the form (7). Suppose there exists a one-to-one coordinate transformation mapping (p, ��N) to (��N, ��N), and set ��N = ?lnPr(N = 0). Then N��C-�� with ��N = ��N(p, ��N)��N is equivalent to the following k=1,2,��.

(11)Proof?conditions:��N2??CN(t)?��N=?CN(t)?t?��N,(9)��N??G(s)?p=?��N?p?s?G��(s),(10)��N??g(k)?p=?��N?p?kg(k), ��The condition (9) is a restatement of Theorem 2. Applying the chain rule of differential calculus, this condition transforms to��N2??CN(t)?p=?��N?p?(?CN(t)?t?��N).(12)Now, by Lemma 8 below and the chain rule, one has��N2??��N?p=?��N?p?��N.(13)Inserting Carfilzomib into (12) shows that��N??CN(t)?p=?��N?p?(?CN(t)?t?��N).(14)The statements (10) and (11) follow by using the representation (7).Lemma 8 ��If N��C-��, then the partial differential parameter equation ��N2 ? (?��N/?��N) = ��N holds.