An iterative approximation procedure for the distribution of the maximum of a random walk
Let \(I(F)\) be the distribution function of the maximum of a random walk whose i.i.d. increments have the common d.f. \(F\) and a negative mean. A recursive sequence of embedded random walks is derived whose underlying d.f.'s \(F_k\) converge to the d.f. of the first ladder variable and satisfy \(F\geq F_1\geq F_2\geq \cdots\) on \([0,\infty)\) and \(I(F)= I(F_1)= I(F_2)= \cdots\). By using these random walks, improved upper bounds for the difference of \(I(F)\) and the d.f. of the maximum of the random walk after finitely many steps are obtained.
scientific article; zbMATH DE number 1556858