Nonlinear renewal theory under growth conditions
Let \(X_ 1,X_ 2,..\). be i.i.d. square integrable non-lattice r.v. with positive mean; let \(Z_ 1,Z_ 2,..\). be r.v. such that the \(\sigma\)- algebras \(\sigma (X_ 1,Z_ 1,...,X_ n,Z_ n)\) and \(\sigma (X_{n+1},X_{n+2},...)\) are independent. The sequence \(Y_ n=S_ n+Z_ n\) with \(S_ n=X_ 1+...+X_ n\) is a perturbed or nonlinear renewal process. At first the author takes \(Z_ n-g(n+z)\) with appropriate g, a deterministic perturbation. A Blackwell type theorem is established as well as an asymptotic law for the excess over the boundary. Secondly stopping times \(T(a)=\inf \{n\); \(Y_ n>a\}\) are treated for \(Z_ n=\xi_ n+g(n)\); now excess over the boundary and E T(a) are discussed in detail. The paper generalizes results for the case \(g(n)=n^{\alpha}\), obtained by \textit{C. Hagwood} and \textit{M. Woodroofe} [Ann. Probab. 10, 844-848 (1982; Zbl 0493.60089)], and by \textit{C. Zhang} [ibid. 16, No.2, 793-824 (1988; Zbl 0643.60067)].
scientific article; zbMATH DE number 4107885
scientific article