This paper discusses a general framework common to some varied known and new results involving measures of threshold exceedance by high values of stationary stochastic sequences. In particular these concern the following. (a) Probabilistic modeling of infrequent but potentially damaging physical events such as storms, high stresses, high pollution episodes, describing both repeated occurrences and associated ‘damage’ magnitudes. (b) Statistical estimation of ‘tail parameters’ of a stationary stochastic sequence {Xj}. This includes a variety of estimation problems and in particular cases such as estimation of expected lengths of clusters of high values (e.g. storm durations), of interest in (a). ‘Very high’ values (leading to Poisson-based limits for exceedance statistics) and ‘high’ values (giving normal limits) are considered and exhibited as special cases within the general framework of central limit results for ‘random additive interval functions’. The case of array sums of dependent random variables is revisited within this framework, clarifying the role of dependence conditions and providing minimal conditions for characterization of possible limit types. The methods are illustrated by the construction of confidence limits for the mean of an ‘exceedance statistic’ measuring high ozone levels, based on Philadelphia monitoring data.

%B Journal of Statistical Planning and Inference %V 45 %P 247-280 %G eng %R 10.1016/0378-3758(94)00075-1