On the maximal distance between two renewal epochs (Q1095506)

Let \(X_ 1,X_ 2,..\). be a sequence of positive i.i.d. r.v.'s with d.f. F and mean \(\mu\) generating the renewal points \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\). Let N(t) be the number of renewals up to time t. The maximal renewal is then \(M_ t=\max (X_ 1,X_ 2,...,X_{N(t)}\), \(t- S_{N(t)})\). The authors assume \(1-F(x)=x^{-\alpha}/L(x)\) for \(\alpha\geq 0\) and some slowly varying L. We outline their results: (i) if \(\mu <\infty\) then upper and lower class results are obtained for \(M_ t;\) (ii) if \(\mu =\infty\) and \(\alpha =1\) there is an a.s. lower limit for \(M_ t;\) (iii) if \(0<\alpha <1\) there are a.s. upper and lower limits; (iv) the case \(\alpha =0\) is quite different and intriguing: it is illustrated by a number of examples.