Stochastic bounds for the Sparre Andersen process (Q812976)

Let \(X(j)\), \(j>0\), be independent and identically distributed random variables, and let \(N(t)\) be a counting process, independent of the \(X(j)\). The random sum \(S(t)\) of the first \(N(t)\) members of the sequence \(X(j)\) is frequently appearing in the actuarial and insurance literature. \(S(t)\) is called a Sparre Andersen process if \(N\) is a renewal process. Computation with \(S(t)\) is usually difficult, due to the lack of knowledge on \(X(j)\) and \(N(t)\). The authors set stochastic bounds on \(S(t)\) which bounds are computable. Several examples are given, and the sharpness of the bounds is discussed.