Joint distribution of the first hitting time and first hitting place for a random walk
\textit{W. Feller} [``An introduction to probability theory and its applications. Vol. II'' (1971; Zbl 0219.60003)] gave a general formula to calculate the Laplace-Fourier transform of a joint distribution of the first hitting time and the first hitting place of \((-\infty,0)\) for a general random walk on the real line starting from 0. The present paper obtains a representation of Laplace-Fourier transform of the joint distribution of the first hitting time and the first hitting place of the set \((-\infty,-a)\) \((a>0)\), which gives a relation to the above mentioned Feller's result, i.e. with the joint distribution of those of the set \((-\infty,0)\). The leading idea is the Wiener-Hopf's factorization theorem. The result is applied to a random walk example frequently occurring in queueing theory.
scientific article; zbMATH DE number 1269768