On central limit theory for random additive functions under weak dependence restrictions

Abstract

Random additive functions defined on intervals provide a general framework for varied applications including (dependent) array sums, and level-exceedance measures for stochastic sequences and processes. Central limit theory is developed in Leadbetter and Rootz?n (1993) for families {?t(?) : ? > 0} of such functions under (array forms of) standard strong mixing conditions. One objective of the present paper is to introduce a potentially much weaker and more readily verifiable form of strong mixing under which the limiting distributional results are shown to apply. These lead to characterization of possible limits for such ?t(?) as those for independent array sums, i.e. the classical infinitely divisible types. The conditions and results obtained for one interval are then extended to apply to joint distributions of {?t(Ij) '? 1 < j < p} of (disjoint) intervals 7i,/2,... /P, asymptotic independence of the components being shown under the extended conditions. Similar results are shown under even slightly weaker conditions for positive, additive families. Under countable additivity this leads in particular to distributional convergence of random measures under these mixing conditions, to infinitely divisible random measure limits having independent increments.