In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

Consider an autoregressive-moving-average process of given order where it is known that a number of moving-average roots are of unit modulus. Such a situation might arise, for example, when a time series has been differenced to induce stationarity by removing a non-stationary polynomial or seasonal trend. A band-limited spectral estimation procedure is proposed for estimating the coefficients of such a process and the asymptotic properties of the estimators investigated. The asymptotic theory is illustrated with reference to simulated and real data. A preliminary investigation of the use of Akaike's AIC criterion and this procedure to determine the number of roots of unit modulus (in the case where this is unknown) is also carried out by means of simulation.

The proposed band-limited spectral estimation procedure can also be used to take account of other possible effects met in practice. These include, for example, the band-limited response of a recording device or trend-contaminated low-frequency components.

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.