Abstracts for the Workshop on Extremes in Space and Time, May 31 On the extremal behavior of random variables ob- served at renewal times

s for the Workshop on Extremes in Space and Time, May 31 On the extremal behavior of random variables observed at renewal times Bojan Basrak (University of Zagreb) This is joint work with Drago Špoljarić (University of Zagreb). We consider the asymptotic extremal behavior of iid observationsX1, X2, . . ., until a random time τ(t) which is determined by a renewal process, possibly dependent on Xi’s. The maximum of these observations M(t) = max i≤τ(t) Xi , has been studied for decades. The first advances have already been made in the 1960s in the relatively straightforward case of a renewal process with finite mean interarrival times. Anderson [1] was the first to study the limiting behavior of M(t) in the case of a renewal process with infinite mean interarrival times. More recently, his result has been extended to describe the limiting behavior of (M(t)) at the level of processes (see [2], for instance). Using point processes techniques, we show how one can recover these known results and characterize the asymptotic behavior of the upper order statistics in the sequence Xi until time τ(t). We also allow for certain types of dependence between the observations and interarrival times, relaxing the conditions previously used in the literature. Finally, we show how our approach yields some well known and apparently new results concerning e.g. the longest run of heads and the maximal sojourn time of a continuous time random walk.