Stability of maxima over randomly deleted sets (Q1382236)

Let \(0\leq t_1< t_2< \dots\) be a fixed sequence. At time \(t_k\) a unit with random magnitude \(X_k\) enters a system, remains there for a random period \(L_k\) and then departs. Assume that \(X_1,X_2, \dots\) are i.i.d. random variables and independent of \(L_1, L_2, \dots\), an i.i.d. sequence of nonnegative random variables with \(P(L_1 >0)>0\). The author compares weak and strong behaviours of the maximum magnitude \(M_n= \max\{X_k: I(t_k\leq t_n< t_k+ L_k)= 1\}\) among the units present in the system at time \(t_n\) with that of other maxima.