We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

A perturbation in the carbon-isotope record at the time of the Pliensbachian–Toarcian boundary (~ 184 Ma) in the Early Jurassic is reported, based on new data from Yorkshire, England. Two sharp δ13Corg negative excursions, each with a magnitude of ~ −2.5 ‰ and reaching minimum values of −28.5 ‰, are recorded in the bulk organic-matter record in sediments of latest Pliensbachian to earliest Toarcian age. A similar pattern of negative carbon-isotope excursions has been observed at the stage boundary in the SW European section at Peniche, Portugal in δ13Ccarbonate, δ13Cwood and δ13Cbrachiopod records. The isotopic excursion is of interest when considering the genesis and development of the later Toarcian Oceanic Anoxic Event (OAE), as well as the second-order global extinction event that spans the stage boundary. Furthermore, the isotope excursion potentially provides a chemostratigraphic marker for recognition of the stage boundary, which is currently achieved on the basis of different ammonite faunas in the NW European and Tethyan realms.

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

The Laplace transform is obtained for the joint length and height of an M/M/∞ queue excursion above occupancy level C.

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class of subexponential distributions are asymptotically of two types as x →∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞, or it starts from a level of the form x – O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 – ρ.

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.