Theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science, probability and statistics. In this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable Banach spaces. Both independent and dependent cases are covered to provide a wide range of applications.

Let {X,Xn; n≥1} be a sequence of i.i.d. random variables with ΕX=0 and ΕX2=σ2＜∞. Set Sn=Σnk=1 Xk, Mn=maxk≤n|Sk|, n≥1. Letr＞1, then we obtain limε↘√r−1 1/−log(ε2 − (r −1)) ∞Σn=1 nr−2−1/2Ε{Mn −σε√2n log n}+ =2σ/(r−1)√2π holds, if and only if ΕX=0, ΕX2=σ2＜∞and Ε(|X|2r/(log|X|)r)＜∞.

With large volatility observed in stock markets around the world over the last few years, many actuaries are now being urged to employ stochastic models to measure the solvency risk generated from insurance products with equity-linked guarantees. There are a large number of potential stochastic models for equity returns. Insurance regulators, both in Europe and North America, normally do not restrict the use of any stochastic model that reasonably fits the historical baseline data. However, in the U.S. and Canada, the final model must be calibrated to some specified distribution percentiles. The emphasis of the calibration process remains on the tails of the equity return distribution over different holding periods. In this paper, we examine the effect of temporal aggregation on classes of stochastic equity return models that are commonly used in actuarial practice. The advantages of choosing a closed (under temporal aggregation) class of processes for modelling asset returns and equity-linked guarantees are discussed. Actuarial applications of temporal aggregation using S&P500 data are given.

In clinical studies, patients are usually accrued sequentially. Response-adaptive designs are then useful tools for assigning treatments to incoming patients as a function of the treatment responses observed thus far. In this regard, doubly adaptive biased coin designs have advantageous properties under the assumption that their responses can be obtained immediately after testing. However, it is a common occurrence that responses are observed only after a certain period of time. The authors examine the effect of delayed responses on doubly adaptive biased coin designs and derive some of their asymptotic properties. It turns out that these designs are relatively insensitive to delayed responses under widely satisfied conditions. This is illustrated with a simulation study.

The strong approximations of a class of Rd-valued martingales are considered. The conditions used in this paper are easier to check than those used in [3] and [9]. As an application, the strong approximation of a class of non-homogenous Markov chains is established, and the asymptotic properties are established for the multi-treatment Markov chain adaptive designs in clinical trials.