The rate of convergence for subordinated densities (Q1600621)

Let \(F\) be a subexponential d.f., and suppose \(F\) has a bounded subexponential density \(f\). Let also \(N\) be a nonnegative integer-valued r.v. with \(p(n)= P(N=n)\), \(n\geq 1\), and put \(q(m)= \sum_{n\geq m+2}(n- 1-m)p(n)\), \(m\geq 1\). The author studies rates of convergence, as \(x\to\infty\), of the following quantities: \[ r_n(x)= f^{*n}(x)- nf(x),\quad R_n(x)= 1- F^{*n}(x)- n(1- F(x)), \] \[ r_N(x)= \sum_{n\geq 1} p(n) r_n(x),\quad R_N(x)= \sum_{n\geq 1} p(n) R_n(x), \] and \(-h'(x)\), where \(h(x)= \sum_{m\geq 1}q(m) f^{*m}(x)\). The results he obtains can be applied in transient probability theory.