Let \(\{(X_ i,Y_ i),i\geq 1\}\) be a sequence of independent identically distributed nonnegative bivariate random variables, where \(Y_ i\) represents the time-interval between the (i-1) th “shock” and the i th “shock”, and \(X_ i\) is the magnitude of the i th “shock”.
Let M(t) be the maximum magnitude of “shocks” occurred in (0,T] and let \(T_ a\) be the first time the magnitude of “shock” exceeds \(a>0\). The author proves limit theorems for \(T_ a\) as \(a\to \sup \{x;P(X_ 1<x)<1\}\) and M(t) as \(t\to \infty\) assuming \(EY_ i=\infty\) and some additional conditions.