The rate of convergence for subexponential distributions and densities (Q1873233)

This paper builds on the previous literature and extends the results presented there, especially those of the authors [Lith. Math. J. 38, 1-14 (1998) and Liet. Mat. Rink 38, No.~1, 1-18 (1998; Zbl 0935.60005)]. A distribution function \(F\) on \([0,+\infty[\) is called subexponential if \(\lim_{x\to+\infty} (1-F^{*n}(x))/(1-F(x))=n\) for all \(n\geq 2\); here \(F^{*n}\) denotes the \(n\)-fold Stieltjes convolution of \(F\) with itself. \(F\) is here assumed to have a density \(f\). The authors study the asymptotical behaviour of the remainder term \(R_n(x):=1-F^{*n}(x)-n (1-F(x))\) and of its density analogue \(r_n(x)=-R_n'(x)\).