An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s Fk converge to the d.f. of the first ladder variable and satisfy F[greater-or-equal, slanted]F1[greater-or-equal, slanted]F2[greater-or-equal, slanted]... on [0,[infinity]) and I(F)=I(F1)=I(F2)=.... Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.Random walk Maximum Approximation Embedded random walk