Use of optical tweezers for colloid science

A space-borne optical tweezer apparatus for use with colloidal crystallization experiments has been characterized. The trapping force has been measured as a function of index mismatch between colloidal microspheres and the surrounding fluid and as a function of particle size. This work also presents a method to determine the refractive index of a colloidal microsphere, which is then used to calculate the applied trapping force for the case of an arbitrary background fluid. This is useful for work with dense colloidal suspensions when the usual (e.g., Stokes flow) trap force measurement methods do not apply, as well as microrheological studies of complex soft matter.     

Transition kernels and the conditional extreme value model

Classical multivariate extreme value modelling assumes that the joint distribution belongs to a multivariate domain of attraction and this assumption requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The Heffernan and Tawn (J R Stat Soc Ser B (Stat Methodol) 66(3):497–546, 2004) alternative extremal model for multivariate data does not require all the components belong to an extremal domain of attraction but assumes instead the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components is extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation of the probability of certain risk regions. The original focus on conditional distributions has technical drawbacks but is a natural assumption in many contexts. The technical drawbacks are overcome by relying on convergence of measures and the theory of extended regular variation Heffernan and Resnick (Ann Appl Probab 17(2):537–71, 2007); Das and Resnick (Extremes 14(1):29–61, 2000a); Das et al. (Adv Appl Probab 45(1):139–163, 2013). We compare the two approaches and describe in what way relying on variational limit properties of conditional distributions restricts the class of limit approximations.     

Characterizations and Examples of Hidden Regular Variation

Random vectors on the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn (1996, 1997). We provide characterizations and examples of such distribution in terms of mixture models and product models.Sidney Resnicks research was partially supported by NSF grant DMS-0303493 and NSA grant MSPF-02G-183 at Cornell University.This revised version was published online in March 2005 with corrections to the cover date.     

Modeling multiple risks: hidden domain of attraction

A sub-model of multivariate regular variation called hidden regular variation facilitates more accurate estimation of joint tail probabilities in the presence of asymptotic independence. A related concept called hidden domain of attraction can sometimes offer similar estimation assistance in circumstances where hidden regular variation is absent. Examples and discussion illustrate strengths and limitations of this concept. We outline estimation techniques where applicable.