Some applications of the classical formula on ruin probabilities
Let \(\{X_ i,i=1,2,...\}\) be a sequence of i.i.d. r.v.'s with \(P(X_ 1=+1)=P(X_ 1=-1)=1/2\). Put \(S_ 0=0\), \(S_ i=X_ 1+...+X_ i\) \((i=1,2,...)\). Starting from the well-known formula for \(P(-a<S_ j<c-a\), \(j=1,...N\), \(S_ N=k)\) (where \(a>0\), \(c>1\), \(N>0\), \(-a<k<c-a\) are integers) the author gives a rich collection of formulas for various joint distributions for unconditional and conditional random walk concerning maxima, minima and their indices. This collection and the long list of references shows, that the classical results still provide interesting new results. As an illustration, in the last section the author presents the recent results on the so-called narrowest tube problem.
joint distributions for unconditional and conditional random walk
scientific article; zbMATH DE number 3971864
scientific article