Metric entropy of the class of probability distribution functions on [0, 1] with a k-monotone density is studied through its connection with the small ball probability of k-times integrated Brownian motions.

We consider a one-parameter family of expanding interval maps {Tα}α∈[0,1] (japanese continued fractions) which include the Gauss map (α = 1) and the nearest integer and by-excess continued fraction maps (α = 1 2 , α = 0). We prove that the Kolmogorov-Sinai entropy h(α) of these maps depends continuously on the parameter and that h(α) → 0 as α → 0. Numerical results suggest that this convergence...

We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both a...

In decision under uncertainty, the Choquet integral yields the expectation of a random variable with respect to a fuzzy measure (or non-additive probability or capacity). In general, for the discrete setting, this technique allows to integrate functions taking values on a finite n-set with respect to a (fuzzy) measure taking values on subsets of such a set. Yet, the integrand may well be treate...

We consider generalisations of tree width to directed graphs, that attracted much attention in the last fifteen years. About their relative strength with respect to “bounded width in one measure implies bounded width in the other” many problems remain unsolved. Only some results separating directed width measures are known. We give an almost complete picture of this relation. For this, we consi...

|This paper studies minimax aspects of nonparametric classiication. We rst study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f; is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L 2 loss is determined by the massiveness of the class as measured by metric entropy. The seco...

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (“cash”). As a main result, for any function f , we find the greatest closed convex monotone and cash-invariant function major...

A strong entanglement monotone, which never increases under local operations and classical communications (LOCC), restricts quantum entanglement manipulation more strongly than the usual monotone since the usual one does not increase on average under LOCC. We propose new strong monotones in mixed-state entanglement manipulation under LOCC. These are related to the decomposability and 1-positivi...