Note on the Background to the Subject: Theory of Risk

By a general theorem the necessary and sufficient condition for a function (formula) being completely monotonic for , lying in the right semi-plane, i.e. that the n (th) derivative with respect to (variable) has the sign of (variable) is that the function may be represented by the Laplace-Stieltjes integral (formula), where U (v) is a non o decreasing function of v, independent of (variable) and bounded in every finite interval, (variable) a real or complex variable represented in the right semi-plane, is a real constant the real part of ,. By the notation q~n (~) we designate (formula), which for U (v) being independent of a, as assumed above, is equal to (formula).