Let X-1, X-2,... be random claim sizes with continuous distribution functions F and N((.)) an independent point process on [0, infinity). Denote by X-N([0,X-t):N([0,X-t) the maximal claim size occurring during the time interval [0,t], t>0 and K-t (a) the number of claims that excess the random barrier XN([0,t])-a(t), with a(t)>0. For iid claim sizes, both distributional and asymptotic properties of K-t (a) are investigated in Li and Pakes (2001) [Li, Y, Pakes, A.G., Insur.: Math. Econ. 28 (3), 309-323]. In this paper, by dropping the iid assumption we extend and simplify limit results of Li and Pakes (2001). Further, we show that K-t (a)/t is a strongly consistent estimator of certain tail probability.